AP Calculus AB 2024 by Edvantage Science

Edvantage Math

AP® Calculus AB

2025 Volume 1

AP Calculus AB

Authors

Bruce McAskill, BSc, BEd, MEd, PhD Deanna Catto, BSc, BEd Mathew Geddes, BSc, BEd, MMT Steve Bates, BEd, MEd

Introduction i

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ii Introduction

Contents

AP Calculus AB

Unit 1: Differential Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1: Analyzing Functions Using Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Rates of Change and Limits – Determining Limits Graphically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Exploring Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Asymptotic Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 End Behavior of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.5 Function Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6 Intermediate Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.7 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Chapter 2: The Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1 Slope of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 Definition of the Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.3 Alternate Definition of the Derivative at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.4 Differentiability of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.5 Symmetric Difference Quotient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.6 Graphing the Derivative of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.7 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Chapter 3: Differentiation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.1 Derivatives of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.2 Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.3 Velocity & Other Rates of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.4 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.5 Introducing the Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.6 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Chapter 4: Advanced Differentiation Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4.1 The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.2 Composite Functions and Function Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.3 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.4 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.5 Derivatives of Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.6 Logarithmic Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.7 Derivatives of Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 4.8 L’Hôpital’s Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 4.9 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Unit 1 Project - Design a Roller Coaster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Unit 2: Applications of Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Chapter 5: Analyzing Functions Using Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 5.1 The First Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 5.2 Modelling and Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.3 The Second Derivative Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.4 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 5.5 The Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 5.6 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Chapter 6: Solving Problems Using Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6.1 Related Rates Involving Shape and Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 6.2 Related Rates Involving Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 6.3 Related Rates Involving Periodic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 6.4 Linearization and Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 6.5 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Unit 2 Project: Shrinking Lollipop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Introduction iii

Unit 3: Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Chapter 7: Antidifferentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 7.1 The Antiderivative – Working Backwards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 7.2 Antiderivative of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 7.3 Antidifferentiation Using Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 7.4 Advanced Antidifferentiation Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 7.5 Antidifferentiation Involving Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 7.6 Antidifferentiation Involving Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 7.7 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Chapter 8: Solving Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 8.1 Solving Differential Equations Analytically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 8.2 Solving Differential Equations Graphically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 8.3 Approximating Area Under a Curve Using Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 8.4 Calculating Area Under Functions Graphically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 8.5 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Chapter 9: Integrals and the Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 9.1 Integrating Using the Fundamental Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 9.2 Derivative of a Definite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 9.3 The Average Value of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 9.4 The Integral as Net Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 9.5 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Chapter 10: Area and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 10.1 Area Between Two Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 10.2 Area Between Two Curves Using Horizontal Slices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.3 Volumes of Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 10.4 Finding Volume Using the Washer Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 10.5 Finding Volume Using Cross-Sectional Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 10.6 Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

iv Introduction

Unit 1: Differential Calculus This unit focuses on the following AP Big Idea from the College Board: Big Idea 1: The idea of limits is essential for discovering and developing important ideas, definitions, formulas, and theorems in calculus By the end of this unit, you should be able to: • • • • • • •

Express limits symbolically using correct notation Interpret limits expresses symbolically Estimate limits of functions Determine limits of functions Deduce and interpret behavior of functions using limits Analyze functions for intervals of continuity or points of discontinuity Determine the applicability of important calculus theorems using continuity

By the end of this unit, you should know the meaning of these key terms: • • • • • • • • • • • • • • • • • • • •

Continuity at a point Continuous function Difference rule End behavior models (left-end and right-end) Extended function Horizontal asymptote Infinite discontinuity Intermediate Value Theorem (for continuous functions) Jump discontinuity Left-hand limit Limit One-sided limit Oscillating discontinuity Product rule Quotient rule Right-hand limit Sandwich Theorem Sum rule Two-sided limit Vertical asymptote

When the Mars Rover lands, it’s important to know its velocity at all times.

Chapter 1: Analyzing Functions and Using Limits 1

Chapter 1: Analyzing Functions Using Limits

Limits tell you where you’re going, even if you never actually get there Students often wonder “What is Calculus?” Calculus uses the logic of mathematics to analyze situations involving continuous change by breaking them down into the smallest components (differential calculus) and then rebuilding them (integral calculus). As such, Calculus is considered the science and the art of change. It provides a mechanism to understand the world around us, make predictions, and interpret behavior. One type of behavior involves a limit. What is a limit and how is it useful? The mathematical answer is that a limit is the y-value that a function, f(x), approaches as the value of x approaches some number. A few “real life” examples can be found in: Physics - pressure at a point is calculated as the average pressure (force per unit area) applied to an area that is shrinking to zero (i.e., shrinking to a point).

Ecology - sustainable population (or carrying capacity) is often extrapolated by determining, over an extended period of time (i.e., infinite), the upper bound on some population that can be sustained by a given ecosystem.

Chemistry - if you drop an ice cube into a glass of warm water and measure the temperature against time, the temperature will eventually approach the room temperature where the glass is stored. Determining the final temperature is a limit as time approaches infinity. A limit describes the behavior of a function at a particular point based on the points around it.

EXPLORING THE BIG IDEA In this chapter you should be able to: • Determine the limits of functions using limit theorems. • Interpret limits expressed in analytic notation. • Represent limits analytically using correct notation.

Chapter 1: Analyzing Functions and Using Limits

1.1 Rates of Change and Limits – Determining Limits Graphically Warm Up The concept of a limit is CENTRAL to calculus. Limits can be used to describe how a function behaves as the independent variable moves towards a certain value. Consider the functions f ( x) = sin x and g ( x) = x Graph f ( x) = sin x on [ −2π , 2π ] , using an appropriate range. On the same grid graph g ( x) = x . (Be careful…when does x = 1 ?)

When does g ( x) = f ( x) ?

What are the values of g ( x) and f ( x) at the point of intersection?

How do g ( x) and f ( x) behave near x = 0 ?

What do you think the value of the quotient function y =

sin x is at the point of intersection? x

Chapter 1: Analyzing Functions and Using Limits 3

Look graphically and numerically: Using your calculator, graph both f(x) and g(x), then complete the tables of values below.

g(x)

f(x)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

g(x)

f(x)

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Consider the y-value as the x-value gets closer and closer to 0. How do the functions behave as x approaches 0? (use the ZOOM function on your calculator to explore)

What value will the quotient approach as the values of the functions become more and more alike?

Definition: Indeterminate form If functions g ( x) and f ( x) are both 0 at x = a then lim x→a

f ( x) cannot be found by substituting g ( x)

x = a as the result would be 0 . Substitution produces a meaningless expression known as an indeterminate form.

Other indeterminate forms include:

∞ , (∞)(0), ∞ − ∞,1∞ , 00 , ∞ 0 . ∞

To determine the value of an indeterminate form, other methods, such as graphing, analyzing numerically or using algebraic manipulations, must be used.

Chapter 1: Analyzing Functions and Using Limits

Example 1: Determining a Limit Graphically and Numerically sin x approaches as x approaches 0 ( x → 0 ) . Use a graph and a table of x values to justify your answer.

Find the value that f ( x) =

How to Do it...

What to Think About How does the function behave as x gets closer to zero from the left? From the right?

What is the value of the function at x = 0 ?

What value does the function approach as x gets closer to zero from the left? From the right?

Definition: Limit

lim f ( x) = L is read, “the limit of f of x as x approaches a equals L”. It means that as x gets closer and x→a

closer to a (from the left-side and the right-side of a), the function’s y – value gets closer and closer to the number L.

Chapter 1: Analyzing Functions and Using Limits 5

Your Turn ...

sin 2 x Compare y = sin 2 x and y = 2 x graphically then graph f ( x) = and find the value 2 x that y approaches as x → 0 . Confirm using a table of values.

y -0.03 -0.02 -0.01 0 -0.01 -0.02 -0.03

Example 2: The Existence of a Limit Versus the Existence of the Function at a Point. 3, x = 2 determine lim f ( x) . Does lim f ( x) equal to f (2) ? x→2 x→2 2 x + 1, x ≠ 2 

Given f ( x) = 

How to do it...

What to Think About How does the function behave as x gets closer to 2 from the left? From the right?

What is the value of the function at x = 2?

Is the function continuous at x = 2?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... For each of the following functions, determine if lim f ( x) = f (a ) x→a

 x 2 − 1, x ≤ 1 at a = 1 f ( x) =   x − 1, x < 1

 x 2 − 1, x < 1  f= ( x) = 1, x 1 at a = 1   x − 1, x > 1

Did You Know? The modern notation of placing the arrow below the limit symbol is attributed to the English mathematician G.H. Hardy, who used it in his 1908 book A Course of PureMathematics.

SUMMARY The value of a limit is the y – value that the function approaches as x → a . The existence of a limit as x → a never depends on whether the function is defined or undefined at x = a . The limit exists when the function approaches the same y – value on both sides of x = a .

Chapter 1: Analyzing Functions and Using Limits 7

1.1 Practice For questions 1 to 8, determine the limit graphically.

x +1 lim 2 x →1 x − 1

1 1 − lim 3 − x 3 x →0 x

lim

x2 − 5x + 6 x →3 x2 − 9

(6 − x) − 9 lim

lim

7 x3 − 3x 2 x →0 5 x 4 − 9 x 2

lim

Chapter 1: Analyzing Functions and Using Limits

x →3

x →0

x−3

sin 2 x x

x + sin x x →0 x

lim

sin 2 x lim x →0 x

For questions 9 to 12, use a graph to show that the limit does or does not exist 9.

lim

x2 − 4 x →1 x − 1

10.

lim

11.

lim

1 x →3 ( x − 3) 2

12.

lim

x2 + 1 x→2 x 2 − 4

x 2 + 16 x + 64 x →8 x +8

Chapter 1: Analyzing Functions and Using Limits 9

1.2 Exploring Limits Warm Up Consider the following functions:

h( x)= x + 1

f ( x) =

x −1 x −1

 x2 −1 ,x ≠1  g ( x) =  x − 1 1, x = 1 

Without a graphing calculator sketch the graph of each function. Make sure to consider the points where the function does not exist. a)

Using the graphs, determine each of the following limits and confirm algebraically. a)

lim h( x) = x →1

lim f ( x) = x →1

lim g ( x) = x →1

Why does factoring help to determine the limit algebraically?

Chapter 1: Analyzing Functions and Using Limits

Did You Know? A removable discontinuity (or point discontinuity) occurs when a function has a hole at one point c on an open interval such that lim f ( x) = L , but f (c) ≠ L or f (c) does not exist. You will explore x →c the concept of continuity in Section 1.5.

Example 1: Determining the Limit of Function Find the limit of each of the following functions, if it exists. a)

lim x3 + 4 x 2 − 3 x→2

How to Do it...

x2 + x − 2 x →1 x −1

lim

lim x →1

1 x −1

x −5 x → 25 x − 25 lim

What to Think About How does the function behave as x gets closer to zero from the left? From the right?

What is the value of the function at x = a ?

What value does the function approach as x gets closer to a from the left? From the right?

Chapter 1: Analyzing Functions and Using Limits

Your Turn... Find the limit of each of the following functions, if it exists.

3 = x →−2 x + 2

a) a)

x4 + x2 −1 = lim x →1 x2 + 5

bb) )

c) c)

lim

x −3 x →3 x 2 − 5 x + 6

d) lim

1 −1 h + 1 lim = h →0 h

lim

x →16

x −4 x − 16

Look at b) and c) graphically. What do you think makes a limit exist?

Chapter 1: Analyzing Functions and Using Limits

Definition: One-sided limit The existence of a limit depends on the one-sided limits about x = a. So when the lim = f (fx()x) lim = f (fx()x) L,Lthen lim then lim = lim = , then limf (fx()x) exists and lim f ( x) = L where L ∈  . When the right− −

+ + x →xa→ a

x →xa→ a

x→a

hand limit equals the left-hand limit at a point, the limit exists.

Example 2: One-sided and Two-sided Limits Given the graph of f determine each limit

How to Do It...

What to Think About What does the notation + “ x → 1 ”mean?

What does the notation − “ x → 1 ”mean?

Is the left-side limit the same as the right-side limit?

What does the abbreviation “DNE” represent?

Chapter 1: Analyzing Functions and Using Limits

Your Turn... 1) Given the graph of g determine each limit. a) a)

c) c)

e) e)

lim g ( x) =

b) b)

lim g ( x) =

d) d)

lim g ( x) =

f f) )

x →1+

x →1

x →−1+

lim g ( x) =

x →1−

lim g ( x) = x →0

lim g ( x) =

x →−1−

2) Given the graph of f determine each limit. a) a)

b) b)

c) c)

lim f ( x) =

x → 0+

lim f ( x) =

x → 0−

lim f ( x) = x →0

Chapter 1: Analyzing Functions and Using Limits

3) Given the following function, sketch the graph of the function and then use one-sided limits to show that the lim f ( x) does not exist. x→2

3 − x, x < 2  f= ( x) = 5, x 2  x, x ≥ 2 

Chapter 1: Analyzing Functions and Using Limits

Example 3: The Greatest Integer Function y = [x] or y = int(x), is defined to be the largest integer to the “left” of x on the number line (or the greatest integral part of x). What is the result when the greatest integer function is applied to each of the following? a) int(2.3)

b) int(2.9)

c) int(3)

d) int(–1.2) e) int(–3.5) f ) int(0.2)

What to Think About

How to Do it...

What is the greatest integer less than the number 2.3? What is the greatest integer less than –1.2? Try plotting this on a number line.

Did You Know? The greatest integer function is commonly used when calculating utility bills (gas, electricity, water, etc.) as well as postage rates. 2015 Canada Postage for Lettermail, Letter Post, Light Packer (LP), Small Packet Air (SP) Maximum dimensions: 380 mm x 270 mm x 20 mm for lettermail, letter post, light packet

Weight ≤30g ≤50g ≤100g ≤150g ≤200g

Canada 0.85+tax 1.20+tax 1.80+tax 2.95+tax

≤250g ≤300g ≤400g ≤500g

4.10+tax 4.70+tax 5.05+tax

≤1000g ≤1500g ≤2000g

Chapter 1: Analyzing Functions and Using Limits

USA 1.20+tax 1.80+tax 2.95+tax 5.00(LP) 5.15 8.13(SP) 7.0(LP) 10.30 10.79(SP) 11.75(LP) 16.23(SP) -

International 2.50+tax 3.60+tax 5.95 7.50(LP) 10.30 9.49-10.67(SP) 11.50(LP) 18.98-21.68(SP) 20.60 21.00(LP) 37.22-40.44(SP) 46.57-52.78(SP) 55.93-61.40(SP)

Your Turn... 1. Graph the greatest integer function (also called a step function):

2. Use the graph to help determine the following limits: a)

lim int( x) =

x → 0.6

x →1.5

x →1−

lim int( x) =

x →−1.4

lim int( x) =

x →1+

lim int( x) = x →1

SUMMARY: For the greatest integer function, ( n ∈ Z , n is an integer) lim[ x] = n

x →n+

lim[ x]= n − 1

x →n−

lim[ x] = DNE x→n

Chapter 1: Analyzing Functions and Using Limits

1.2 Practice 1. What is lim(2 x 3 − 3 x 2 + x − 1) ? x →c

Determine each limit by substitution and support the result graphically. 2.

lim 4 x 2 (2 x − 1)

x →−

1 2

lim( x − 6) 2020 x →5

Chapter 1: Analyzing Functions and Using Limits

lim( x 4 − 3 x 3 + 2 x − 5)

y2 + 5 y + 6 y →−1 y2 +1

lim int( x)

x →1

lim

x→

1 3

Chapter 1: Analyzing Functions and Using Limits

7. Explain why you cannot use substitution to determine each limit. Use a graph to support your explanation a.

lim x − 3

Did You Know? The signum function is a mathematical function that extracts the sign of a real number. It is often represented as

x →−2

x −1 if x < 0 = sgn( = x)  x>0 x 1 if

x x →0 x

lim

Determine each limit. 8.

10.

lim int( x)

x →1−

lim+

x →0

x x

lim int( x)

x →−0.001

Chapter 1: Analyzing Functions and Using Limits

Which of the statements are true about the function y = f(x) graphed below and which are false?

lim f ( x) = 0

12.

1 2

14.

lim f ( x) = 1

16.

17.

lim f ( x) = 1

18.

19.

There is a removable discontinuity at x = 0 .

20.

11.

13.

15.

x →−1+

lim+ f ( x) =

x →0

x →1−

x →1

lim f ( x) = 1

x → 0−

lim f ( x) = 1

x →1+

lim f ( x) = 1 x →0

lim f ( x) = 1

x → 2−

There is a jump discontinuity at x=1.

Chapter 1: Analyzing Functions and Using Limits

Given that lim f ( x) = 0 and lim g ( x) = 4 determine the following limits. x →−2

21.

23.

x →−2

lim ( g ( x) + 2)

22.

lim ( x ⋅ f ( x))

24.

x →−2

lim g 2 ( x)

x →−2

lim

x →−2

(Note: g 2 ( x) = [ g ( x) ] ) 2

f ( x) g ( x)

Complete the following parts for each of questions 25 and 26. a)

Draw the graph of f(x).

Determine lim+ f ( x) and lim− f ( x) .

Does lim f ( x) exist? If so, what is it? If it does not exist explain why.

25.

c = 1 , f ( x) = 

x →c

 2 + x, x < 1  4 − x, x > 1

Chapter 1: Analyzing Functions and Using Limits

26.

 3 − x, x < 2  c = 2 , f ( x) = 5, x =1 x  − 1, x > 2 2

Complete the following parts for question 27. a)

Draw the graph of f(x).

At what points c in the domain of f does lim f ( x) exist?

At what point “c” does the limit exist and why?

27.

cos x, 0 < x ≤ 2π f ( x) =  sin x, −2π ≤ x ≤ 0

x →c

Chapter 1: Analyzing Functions and Using Limits

1.3 Asymptotic Behavior Warm Up Sketch the graph of y =

1 and determine each limit: x

1 = x →0 x

lim

1 = x

lim

1 = x →−∞ x

lim+

x →0

lim−

x →∞

1 = x

lim

Next sketch the graph of y =

1 and determine each limit: x2

lim+

1 = 2 x →0 x

1 = x →0 x 2

lim

1 = x →−∞ x 2

lim−

1 = x →∞ x 2

lim

What can you conclude are the conditions necessary for a vertical or a horizontal asymptote to exist?

Chapter 1: Analyzing Functions and Using Limits

Definition: Horizontal asymptote The line y = b , where b is a real constant, is a horizontal asymptote of the graph of a function y = f(x) if either lim f ( x) = b or lim f ( x) = b . x →∞

x →−∞

Example 1: Finding a Horizontal Asymptote Find the horizontal asymptote(s) for each function. Use a graph and limits to help. a)

f ( x) =

1 x −1

How to Do It...

f ( x) =

x 2

x +1

What to Think About Numerically, what happens to the value of the rational expression as x becomes very large?

What value does the function approach as the denominator gets positively or negatively very large?

Does the +1 have an effect on the value of the denominator x →∞ as x → ∞ ?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... Find the horizontal asymptote(s) for each function using a graph of the function and limits. a)

1 = x →0 x lim+

1 = x →0 x lim−

Note: Important limits to remember sin u =1 u →0 u

lim

sin u = 0 , where u = f ( x) u →±∞ u lim

Chapter 1: Analyzing Functions and Using Limits

Definition: Vertical Asymptote or Infinite Discontinuity The line x = a is a vertical asymptote of the graph of a function y = f(x) if either

lim f ( x) = ±∞ or lim− f ( x) = ±∞

x→a+

x→a

The function has an infinite discontinuity at a vertical asymptote.

Example 2: Finding a Vertical Asymptote Find the vertical asymptote(s) for each function. Use a graph and limits to help a)

f ( x) =

1 x −1

How to Do It...

f ( x) =

1 ( x + 1) 2

What to Think About Are there any values of x that make the function undefined?

As x approaches this value, what happens to the value of the denominator?

What is the limit of the function at the value(s) where the denominator equals zero? What is the shortcut for determining if there is a vertical asymptote?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... Find the vertical asymptote(s) for each function using a graph of the function and limits. a)

f ( x) =

1 x +1 2

Chapter 1: Analyzing Functions and Using Limits

f ( x) = tan x

Example 3: Finding the Asymptotes of a Function. Determine the asymptote(s) of f ( x) =

How to Do It...

x2 − 4 . Justify your answer. x2 + 2x

What to Think About Are there values that make the denominator equal to zero?

Does the numerator also equal zero at any of these points?

What happens at x = 2?

Is it necessary to show both sides of an infinite limit to justify a vertical asymptote?

Do either –4 or +2x affect the numerator or denominator as x →∞ ?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... Determine the asymptote(s) of each function. Justify your answer. a)

x2 − 9 x 2 − 3x

sin x 2x2 + x

Chapter 1: Analyzing Functions and Using Limits

1.3 Practice For questions 1 to 4 determine each limit graphically and numerically. 1.

1 x→2 x + 2

[ x] x →0 x

lim−

Chapter 1: Analyzing Functions and Using Limits

lim csc x

x → 0+

2 x →3 x − 3 lim+

Chapter 1: Analyzing Functions and Using Limits

For questions 5 to 8: Determine the asymptote(s) of f(x). Justify your answer using limits. Support your solution graphically and/or numerically 5. f ( x) =

1 x −9

6. f ( x) =

3x − x 2 x+2

Chapter 1: Analyzing Functions and Using Limits

7. f ( x) = sec x

8. f ( x) =

tan x sin x

Chapter 1: Analyzing Functions and Using Limits

1.4 End Behavior of Functions Warm Up Imagine standing 2 meters away from any wall in your classroom. Now, take a step that is exactly 1/2 of the way towards the wall. Next, repeat that process taking another step that is exactly 1/2 of the way towards the wall. Then, keep repeating this 1/2 step process until you touch the wall with your foot. Graph what this looks like using the number of steps, n, as the independent variable and the distance from the wall, d, as the dependent variable.

Express this process as a limit. What number of steps would you use in the limit?

In theory, if the pattern is followed forever, will you ever reach the wall?

Did You Know? The end behavior of a function describes the shape of the curve as x → ∞ . It can be used to predict behavior in the distant future or distant past.

Chapter 1: Analyzing Functions and Using Limits

Example 1: End Behavior Function Models: Let f ( x) = 3 x 4 − 2 x3 + 3 x 2 − 5 x + 6 and g ( x) = 3 x 4 . Show that while f and g are quite different for small values of x, they are essentially identical for |x| large.

How to Do It...

What to Think About Compare the graphs of the functions for small and large values of x.

What happens to the shape of the graphs as x increases?

Does g(x) represent a good model for the behavior of f(x) at either end or both.

Chapter 1: Analyzing Functions and Using Limits

Definition: End Behavior Model The function g is a good approximation for the behavior of f and is called a right end behavior f ( x) model for f, if and only if lim =1 . x →∞ g ( x )

The function g is a good approximation for the behavior of f and is called a left end behavior f ( x) model for f, if and only if lim =1 . x →−∞ g ( x )

If g approximates f well at either end, then they behave the same way as x → ∞ . By dividing them and taking the limit, the quotient must equal 1 because they behave the same.

Your Turn... Let f ( x)= x + e − x Show graphically and analytically that g ( x) = x is a right-end behavior model for f and that h( x) = e − x is a left-end behavior model for f.

Chapter 1: Analyzing Functions and Using Limits

Example 2: Finding End Behavior Models Find a function that is an end behavior model for each function.

f ( x) =

3x 4 + x3 − 2 x 2 − 5 4x2 + 7 x − 3

How to Do It...

f ( x) =

4 x4 − 5x2 + 6 5 x 4 + 3x3 + 7

What to Think About Which are the dominant terms in the numerator and denominator?

When x is very large, what happens to the rest of the terms

Does the function have a limit as x → ∞ ?

What does this limit tell you about the function?

Chapter 1: Analyzing Functions and Using Limits

Example 2: Finding End Behavior Models - continued

Your Turn... For each function determine the end behavior model to find the limits as x → ∞ . Identify any horizontal asymptotes. 2 x5 + x 4 − x 2 + 1 3x 2 − 5 x + 7

f ( x) =

2 x3 − x 2 + x − 1 f ( x) = 3 5x + x2 + x − 5

f ( x) =

4 x 2 − 3x + 5 2 x3 + x − 1

Chapter 1: Analyzing Functions and Using Limits

Example 3: Determining End Behavior Using the Squeeze Theorem What is the right end behavior of f ( x) =

sin x ? x

How to Do It...

What to Think About Can you sketch a graph of the function?

What limit are you using to determine the end behavior?

What functions can be used that squeeze (or sandwich) f(x) as x → ∞ ?

What inequality expresses how the two functions squeeze f(x)?

Can you evaluate the limits that squeeze f(x)?

What conclusion can you now make?

Chapter 1: Analyzing Functions and Using Limits

Theorem: Sandwich (Squeeze) Theorem = f ( x) lim = g ( x) L If f ( x) ≤ h( x) ≤ g ( x) for all x ≠ c in some interval about c, and lim x →c

x →c

then lim h( x) = L x →c

Your Turn ... 1 − cos x =0 x →∞ x

Use the Squeeze Theorem to show that lim

Did You Know? The squeeze theorem was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, but it was Carl Friedrich Gauss who formulated the modern version.

Chapter 1: Analyzing Functions and Using Limits

1.4 Practice For questions 1 to 4 match the function with the graph of its end behavior model. 1.

lim

3x5 − 2 x 2 + 3x − 1 x →∞ 2 x 2 − 3x + 4

lim

2 x 4 − x3 − 6 x 2 + x + 2 x →∞ 5− x

lim

− x 4 + 2 x3 + 5 x →∞ 1 + x2

1 3 x + x2 − 4x − 7 lim 2 x →∞ x−2

Chapter 1: Analyzing Functions and Using Limits

For questions 5 to 10: a) Determine a power function, g ( x) = kx n , end behavior model for f ; and, b) Identify any horizontal asymptotes, if they exist. 5.

f ( x) = 4 x 2 − 7 x + 2

f ( x) =

f ( x)

2x − 6 2 2 x − 5x + 9

−4 x3 + 2 x − 6 x −5

f ( x) = −2 x3 + 5 x 2 − 8 x − 3

4x2 − 2 x + 3 f ( x) = 3 x 2 − 12

10.

x 4 − 2 x3 − x 2 + 6 x − 7 f ( x) = − x2 + 5

Chapter 1: Analyzing Functions and Using Limits

For questions 11 and 12 sketch a graph of a function y = f(x) that satisfies the stated conditions. Include any asymptotes. 11. lim f ( x) = 0, x →1

12. lim f ( x) = 0, x →0

lim f ( x) = ∞,

x → 4−

lim f ( x) = ∞,

x →3−

lim f ( x) = ∞,

x → 4+

x →3−

lim f ( x) = ∞,

lim f ( x) = 6,

x →3+

Chapter 1: Analyzing Functions and Using Limits

x →∞

lim f ( x) = ∞,

x →3+

lim f ( x) = 0,

x →±4

lim f ( x) = 6,

x →−∞

For questions 13 to 18 use the Squeeze Theorem to determine each limit. 13.

lim

2 − cos x x →∞ x+3

14.

lim

15.

5 x 2 − sin(3 x) x →−∞ x 2 + 10

16.

1 lim x 2 sin   x →0 x

17.

 1  lim x 2 sin  2  x →0 x 

18.

lim

cos 2 (2 x) x →∞ 3 − 2 x

lim x e

1 sin    x

x →0

Chapter 1: Analyzing Functions and Using Limits

1.5 Function Continuity Warm Up For what intervals is the function f continuous? Use interval notation to express your answer.

For which points is f discontinuous?

For each point of discontinuity determine the value of the function.

Find the limit at each point of discontinuity or state that it does not exist.

Compare the value of the function at each discontinuity to the limit of the function. What do you notice?

Determine lim f ( x) and compare this to the value of the function at x = 3. x →3

What is the relationship between the limit and the value of the function at each point where the graph is continuous?

Chapter 1: Analyzing Functions and Using Limits

Definition: Continuity at a Point Interior Point: A function y = f(x) is continuous at an interior point c of its domain if

lim f ( x) = f (c) x →c

EndPoint: A function y = f(x) is continuous at its left endpoint a of its domain if

lim f ( x) = f (a )

x→a+

Similarily, a function y = f(x) is continuous at its right endpoint b of its domain if

lim f ( x) = f (b)

x →b −

Note: If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Sometimes a point c may be classified as a point of discontinuity even though it is not in the domain of f. Can you provide an example?

Example 1: Finding Points of Continuity and Discontinuity Find the intervals of continuity and the points of discontinuity of the greatest integer function.

How to Do It

What to Think About Where is the function discontinuous?

Can you generalize the statement?

What does this mean in terms of the intervals of continuity and points of discontinuity?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... Find the points of discontinuity and intervals of continuity for each function. a)

f ( x) =

1 x −1 2

f ( x) =

x x

Example 2: Types of Discontinuities Write the equation of each function. Use a piece-wise function if necessary. Then, state the type of discontinuity or state that it is continuous.

How to Do It

What to Think About What function will generate each graph?

What algebraic expression results in a hole in a graph?

Chapter 1: Analyzing Functions and Using Limits

Example 2: Types of Discontinuities - continued

Which graphs can be represented using a piecewise function?

What happens to these functions as x → 0 ?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... For each function identify the point(s) of discontinuity and the type of the corresponding discontinuity. Determine the interval(s) of continuity.

1 x +1

g ( x) =

x +1 4x − 2

f ( x) =

h( x ) =

x −3 x − x−6

k= ( x) ln x − 4

 x 2 + 3, x < 1  f ( x)= 10 − x,1 ≤ x ≤ 2 6 x − x 2 , x > 2 

1 m( x) = cos    x

Chapter 1: Analyzing Functions and Using Limits

Example 3: Extending a Function to Create a Continuous Function Extend f ( x) =

x3 − 7 x − 6 to create a new function, g(x) that is continuous at x = 3. x2 − 9

How to Do It

What to Think About Can you factor the denominator?

What is the domain of f?

What is happening around x= 3?

Can you graph it and use your calculator zoom to determine numerically if there is a removable discontinuity at x = 3?

Is (x − 3) a factor of the numerator of f?

What is the limit of the simplified function asx?→ 3

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... a) For the given function find the value of a that makes the function continuous.

 x 2 − a, x < 5 g ( x) =   4 x + 2a, x ≥ 5

b) Determine the extended function for f ( x) =

Chapter 1: Analyzing Functions and Using Limits

x−2 x2 − 4

Definition: Continuous function A function is continuous on an interval if and only if it is continuous at every point of the interval. A function is called a continuous function if it is continuous at every point of its domain. Note: A continuous function need not be continuous on every interval, since there may be a point of discontinuity at a point outside of the domain of the function.

For example, y = is not continuous on [−1, 1] since it has a point of discontinuity at x = 0; x but the function is still considered to be a continuous function, because x = 0 is not part of the domain of the function!

Example 4: Continuous Functions

Is each of the following functions considered a continuous function? Justify your answer.

1 y= x+2

How to Do It...

 x + 1, x < 2  2 = f ( x) = x ,x 2 2 x − 1, x > 2 

What to Think About Are there any values of x where the function does not exist?

Is the x-value part of the domain of the function?

What do you know about each part of the piecewise function

Which value(s) of the domain need to be checked?

Chapter 1: Analyzing Functions and Using Limits

Example 4: Continuous Functions - continued What are the limits of each part of the function at x = 2 and what is the value of the function at x = 2

What do the results tell you?

Your Turn ... Determine if each function is a continuous function. Justify your reasoning. a)

 3 2 x , −2 ≤ x ≤ 4 f ( x) =  6 − x, 4 < x ≤ 6

f ( x) =

x x −1

Chapter 1: Analyzing Functions and Using Limits

SUMMARY: Properties of Continuous Functions If functions f and g are continuous at x = a, then the following combinations are continuous at x = a. 1. Sum rule:

f +g

2. Difference rule:

f −g

f ⋅g 3. Product rule:

4. Constant multiples: 5. Quotient rule:

c ⋅ g , for any number c f , g (a) ≠ 0 g

1.5 Practice For questions 1 to 6 identify the interval(s) of continuity, the point(s) of discontinuity, and each type of discontinuity.

1 ( x − 3) 2

y=e

3x − 7

= y

= y 2x − 8

 x2 − 9 ,x ≠3  y =  x −3  x=3 9,

1 x2

3x − 6 x2 − 4

Chapter 1: Analyzing Functions and Using Limits

For questions 7 to 8 use the following function f and the corresponding graph of f.

f ( x)

 − x 2 + 1, −1 ≤ x < 0  x, 0 < x <1 2  x 1 1, =  3 − x + , 1< x < 2 2   1 2< x<3 − ,  2

7)a.

Does f (−1) exist?

Does lim+ f ( x) exist?

Does lim f ( x) exist?

Is f continuous at x = −1 ?

8)a.

Does f (1) exist?

Does lim+ f ( x) exist?

Does lim f ( x) exist?

Is f continuous at x = 1 ?

x →−1−

x →1−

Chapter 1: Analyzing Functions and Using Limits

x →−1

x →1

For questions 9 to 11: a) determine each point of discontinuity. b) identify which discontinuities are removable or not removable and state your reasons.

 2 + x, f ( x) =  −2 x + 6,

10.

 1 ,  f ( x) =  x + 1  x3 − 8, 

11.

 −2 + x,  f ( x) = −1,  x 2 − + ,  3 3

x<3 x≥3

x < −1 x ≥ −1

x<2 x= 2 x>2

Chapter 1: Analyzing Functions and Using Limits

For questions 12 and 13 identify the extended function, g(x), that makes the given function continuous at the indicated point. x2 − 4 12.= f ( x) = , x 2 x−2

= f ( x) 13.

x −9 = , x 9 x −3

Chapter 1: Analyzing Functions and Using Limits

For questions 14 and 15 sketch a possible graph for a function f that has the stated properties. 14.

f(5) exists but lim f ( x) does not. x →5

15. f(x) is continuous for all x except x = −2 , where f has a non-removable discontinuity.

Chapter 1: Analyzing Functions and Using Limits

1.6 Intermediate Value Theorem Warm Up a) On the grid provided below draw a graph of f ( x)= x + 1 using a scale and domain [a, b] of your choice. Label the left endpoint of your graph as (a, f(a)) and the right end point as (b, f(b))

b) Place a ruler on your graph so that it is parallel to the x-axis and passes through (a, f(a)).

c) Keeping your ruler parallel to the x-axis slide it upwards until you reach the point (b, f(b)).

d) Is your graph continuous on the interval [a, b]?

e) Was there any y-value (i.e., any value of f(x)) between f(a) and f(b) that your ruler did not pass through?

f ) Do you think the same obervations would be made for any function? Can you think of a function where parts d) and e) do not hold true? Explain

Chapter 1: Analyzing Functions and Using Limits

Definition: Intermediate Value Theorem for Continuous Functions (IVT) If y = f(x) is continous on closed interval [a,b] and f (a ) ≠ f (b) , then for every value y0 between f(a) and f(b), there exists at least one value c ∈ [a, b] such that f (c) = y0 .

Example 1: Applying the IVT to Determine if a Function Has a Root on a Given Interval Does y = x5 + x3 − 5 x + 2 have at least one root in [0, 1]?

How to Do It...

What to Think About What conditions do you know exist?

What can you conclude from the conditions?

What do you know about the y-values of the end points and what does this tell you?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... Is any real number on the closed interval [1, 2] exactly 1 less than its cube? (Hint: Is there a solution to = x x3 − 1 ?)

Definition: Extreme Value Theorem for Continuous Functions (EVT) If f(x) is continous on closed interval [a,b] then f has both a maximum value and minimum value on that interval.

Example 2: Applying the EVT to Determine if a Function Has a Maximum or Minimum 1 Let f ( x) = . Although f is continuous on [1, ∞) , it has no minimum value on the interval. x Does this contradict the EVT? Justify your answer.

How to Do It...

What to Think About What conditions are stated?

What can you conclude from the conditions?

How does this affect the given statement?

Chapter 1: Analyzing Functions and Using Limits

Your Turn ... For each graph identify the x-value at which a maximum or minimum occurs. Explain how your answer is consistent with the Extreme Value Theorem.

Chapter 1: Analyzing Functions and Using Limits

1.6 Practice 1. The function k(x) is continuous on the interval [8, 11]. If k(8) = 3 and k(11) = −2, can you conclude that k(x) is ever equal to 0? Justify your answer.

2. Is there a solution to x 5 − 2 x 4 − x − 3 = 0 in [2,3] ? Justify your answer.

3. Let f ( x) =x 4 − 3x 2 + 6 x . Show that there is a number c such that f(c) = 1.

π  4. If f ( x) = sin( x) , is there a number in the interval  , π  where f(x) = 0.4? Justify your reasoning. 2 

Chapter 1: Analyzing Functions and Using Limits

5. Suppose that function f has every value between y = 0 and y = 1 on the interval [0, 1]. Must f be continuous on that interval? Explain your reasoning.

6. Between which of the following two values does the equation 5 x 4 + 7 x 2 + 1 = 0 have a solution? a) Between −2 and −1

b) Between −3 and −2

c) Between 0 and 1

d) Between −1 and 0

7. Prove that the function f ( x) = x 2 − 4 x + 2 intersects the x-axis on the interval [0, 2]. Can the same be 2x − 3 said for the function g ( x) = ? Explain your reasoning. x −1

Chapter 1: Analyzing Functions and Using Limits

1.7 Chapter Review For questions 1 to 3, use your calculator to graph each function and estimate the limit numerically to two decimal places or state that the limit does not exist.

1 − cos3 ( x) lim x →0 x2

xx − 4 lim 2 x→2 x − 4

lim x

1 x −1

x →1

For questions 4 to 11, evaluate the limit if it exists. If not, determine whether the onesided limit exists. 4 + x2 x →1 2 x − 3

lim

x −4 x →16 x − 16

lim

1 x →−1 x + 1

lim

lim 5 − x

1 2

x →9

lim

Chapter 1: Analyzing Functions and Using Limits

x3 − x x →1 x − 1

3x3 + 5 x − 2 2 1 x→ 6 x − 5 x + 1 3

10.

lim x sec x x→

11.

cos x − 1 x →0 sin x

lim

12. Determine the left- and right-hand limits of the function f(x) shown below at x = 0, 2, and 5. 8 6 4 2 −4

−2

−2 −4 −6

13. Determine: 6

lim g ( x)

x → 6−

b) c)

g (6) whether g(x) is continuous at x = 3.

d) the points of discontinuity of g(x). −4 e) whether any points of discontinuity are removable.

y=g(x)

−2

6,2.206

−2

−4

Chapter 1: Analyzing Functions and Using Limits

14. Find all points of discontinuity for the given functions. a)

f ( x) =

x−2 x2 − 9

g= ( x)

16.

f= ( x) ln

2x −1

For questions 15 and 16 find: i) a right-end behavior model; and, ii) a left-end behavior model for the function. 15.

f ( x= ) 2x − ex

Chapter 1: Analyzing Functions and Using Limits

1 − cos x x

17. What value of k makes f a continuous function?

 x2 + k , x < 1 f ( x) =  2( x − 2), x ≥ 1

18. Show that f ( x) = 2 x3 − 5 x 2 − 10 x + 5 has a root somewhere in the interval [−1, 2].

19. Find all asymptotes for each of the following: a)

2x −1 4x2 −1

1 xe x − x

Chapter 1: Analyzing Functions and Using Limits

Chapter 2: The Derivative How things change over time

The average velocity of a dragster can be calculated given its position-distance time graph or a table of values.

Time

Distance 0 1 2 3 4 5 6 7

0 30 65 160 290 480 720 1000

What is the car’s average speed between 0 and 5 seconds as it begins to reach top speed? How could you calculate it? What is the car’s speed exactly at 4.3 seconds? How would you calculate it?

EXPLORING THE BIG IDEA In this chapter you should be able to: • Calculate the instantaneous speed of a moving object. • Interpret a limit as a definition of a derivative. • Calculate derivatives of familiar functions.

Chapter 2: The Derivative

2.1 Slope of a Function Warm Up On a road trip through Tennessee, a group of college students were 80 miles from South Padre Island at 3pm. By 4:15pm they were only 10 miles away. What was their average speed? Do you believe this speed accurately reflects their trip? Why? How might you estimate the instantaneous velocity of the dragster at exactly 3.5 seconds given the finite data in the graph?

Note: Refer to applet that follows the curve of a function and solves for the instantaneous rate of change. https://qrs.ly/91gge2r

Chapter 2: The Derivative

In general, the average rate of change of a function over an interval, is the amount of change (the net change) divided by the length of the interval, written as a difference quotient. Average rate of change =

y − y1 ∆y ∆d or or 2 x2 − x1 ∆t ∆x

Example 1: Finding Average Rate of Change Find the average rate of f ( x= ) x3 − x change of over the interval [1, 3].

How to Do It f (3) = 24 f (1) = 0

What to Think About What are values of the function at the endpoints?

average rate of change

24 − 0 = 12 3 −1

How are these used to calculate the difference quotient?

Graphically, what does the value represent?

Chapter 2: The Derivative

Your Turn ...

Find the average rate of change of f (= x) 2 x 2 − 3 x over the interval [2, 5].

Graph any function f(x) and illustrate the slope between any two points evaluated at x = a and x = b .

Definition: Secant Line A line through two points on a curve is called a secant line. The slope of the secant line of a continuous function y = f(x) on the closed interval [a, b] is

f (b) − f (a ) b−a

The slope of the secant line represents the average rate of change of f on [a, b].

Chapter 2: The Derivative

Example 2: Growth in Technology According to Moore’s Law, the computing power, related to the number of transistors in a dense integrated circuit, doubles every 2 years indicating that the speed of computer processing has been doubling during that time frame. This trend has been used to set goals within the industry.

Given the data listed above, determine the average rate of change in transistor growth between 1993 and 2000 once the Pentium processor was introduced.

How to Do It...

What to Think About

42, 000, 000 − 3,100, 000 Average rate of change = 2000 − 1993

38,900, 000 7

= 5,555,143 Transistors/year

What are the applicable units for this rate of change?

When is the processor computing speed growing at its slowest speed? Fastest speed? Why?

Chapter 2: The Derivative

Your Turn ... The rise of bitcoin as an alternate currency has been a highly volatile and intriguing investment. Over the long term, we can assess its average rate of change. Here is a graphical representation of its value. Estimate the average rate of change from June 2017 to January 2018. Estimate the average rate of change from January 2018 to March 2018. How does this compare to the Standard & Poor’s 500 which is an American stock market index based on the market capitalizations of 500 large companies having common stock listed on the NYSE or NASDAQ.

Chapter 2: The Derivative

Slope of a Function at a Point To calculate the exact value of the instantaneous change of a function we need to use the concept of infinitesimals and limits. In the development of calculus, Newton and Leibniz used the concept of infinitesimals but did not define it explicitly. It is now defined as an indefinitely small quantity approaching zero. The idea is to break down a changing function into tiny pieces, considering each piece to occur over an instantaneous or infinitesimal change in the independent variable. Over an infinitely small interval we can consider the change to have a constant slope.

Leibniz notation

dy ∆y = lim dx ∆x →0 ∆x

Note: Refer to applet with slider that draws a second point on the curve back to the point of tangency https://qrs.ly/begge2s Definition: Slope of a Curve at a Point f ( a + h) − f ( a ) The slope of a curve y = f ( x) , at a point is the value m = lim provided the h →0 h limit at point P(a, f (a )) exists. The tangent to the curve at P is the line through P with this slope. The slope of the tangent indicates the instantaneous rate of change at that point.

Did You Know? Leibniz Notation Gottfried Leibniz (1646-1716) discovered calculus through his use of inifinitesimals, which is expressed in differentials. Thus, the derivative in Leibniz notation dy is expressed as and read as “dy by dx.” dx

Chapter 2: The Derivative

Example 3: Exploring Slope and Tangent a) Find the slope of the secant line to the curve y = x 2 between x = a and x = a + h. b) Given the function y = x 2 , find the slope of the tangent at any point x = a. Use a limit as h → 0 to calculate the slope at x = a. c) Find the slope for x = 2

How to Do It... ( a + h) 2 − a 2 h →0 h

= lim

a 2 + 2ha + h 2 − a 2 h →0 h

= lim

What to Think About Therefore, at x = 2 the slope is equal to = 2(2) =4

What is the significance of 2x as it relates to the original function?

2ha + h 2 h →0 h

= lim

= lim 2a + h h →0

= 2a

What is the significance of the slope of the tangent to the function at x = 2?

Draw a rough sketch to illustrate your calculations.

Chapter 2: The Derivative

Your Turn ... y x2 + 3 . Given the function = a) Determine the slope of the tangent at any point x = a. b) Evaluate the function and the slope at x = −3.

2.1 Practice Find the average rate of change of the function over each interval.

2 x + 5 over [2,10]

y = e x over [1, 4]

π π y= 2 + sin x over  − ,   2 2

= y x 2 + 3 x over [2,5]

1 over [−3,1] x−2

y =− x − 1 over [3,5]

= y

Chapter 2: The Derivative

7. You’ve developed a new algorithm for machine learning that is quickly receiving interest in the AI marketplace. Your customer base for your new company has risen at a record pace over the last few months as illustrated in the table below. What is the average growth rate per month for your consumer interest over the months of September and December? Month # of customers

Aug 2022

Sept 2022

Oct 2022

Nov 2022

Dec 2022

Jan 2024

Feb 2024

Mar 2024

256

512

1,024

2,048

4,096

8,192

16,384

32,768

1 8. Describe what happens to the line tangent to the function of y = over the interval [1, 5] between x consecutive points with a whole value of x.

9. Describe what happens to the tangent as a changes for the function y= 4 − x 2 .

Chapter 2: The Derivative

10. The height, in feet, of a roller coaster entering a full corkscrew is given by the graph below. Estimate its instantaneous rate of change at 6 seconds.

11. The Dow Jones Industrial Average rises and falls during the day. Use the data below to estimate its instantaneous rate of change at 2:05pm.

Minutes after 1PM 1.7 65.0 90.0 98.3 103.3 105.0 105.8 106.3 106.7 107.0

DJIA (Points) 10800 10700 10600 10500 10400 10300 10200 10100 10000 9869.62

Chapter 2: The Derivative

12. The curve y = f ( x) has been analyzed with certain calculations yielding the following results f ( a + h) − f ( a ) lim = ∞ . What do you expect to happen in the graph of the function at x = a ? h →0 h

y 2 x + 5 at x = 4 . 13. Determine the slope of the tangent to the graph =

14. Determine the slope of the tangent to the graph y = −2 x at x = 3 . 2

Chapter 2: The Derivative

y 5 x 2 − 3 x at x = 2 . 15. Determine the slope of the tangent to the graph=

16. Determine the slope of the tangent to the graph = y

x + 3 at x = 2 .

Chapter 2: The Derivative

2.2 Definition of the Derivative Warm Up y x 2 − x and calculate the slope of the tangent x = 2. Sketch a graph of the function =

Calculate the slope of the curve at x = a using a limit. Use this information and the value of the function at x = 2 to write the equation of a line that is tangent to the curve.

Definition: Tangent line equation Tangent line equation at a point (a, f (a )) with a slope of f ′(a ) yy−−f f((aa)=)= fff'(′('(aaa)( ))(xx−−aa)) Note: Newton notation for the slope function uses an apostrophe.

Chapter 2: The Derivative

Definition: Derivative The derivative of a function, f ′( x) represents the slope of the curve of f ( x) at any point of the domain. rise f ( x + h) − f ( x ) ′'(( x) lim Slope== f= h →0 h run Note: The process of determining the derivative of a function is called differentiation.

Example 1: Exploring Slope and Tangent 3 a) Given the function f ( x) = x , find the derivative function. b) What is the equation of the tangent at x = 1?

How to Do It...

What to Think About Did you apply any shortcuts to expand the brackets?

What is the significance of the expression = 3x 2 ?

What is the point-slope form of a linear equation?

Draw a rough sketch to illustrate your calculations.

Chapter 2: The Derivative

Your Turn ... y x 3 + 2 , find the slope of the tangent any point x = a. a) Given the function =

b) What is the equation of the tangent x = 2?

Did You Know? Newton Notation Sir Isaac Newton (1642-1727) invented calculus through his method of fluxions. His notation is useful when the independent variable is time. Newton’s notation for derivatives is expressed as y’ and read as “y prime.” y’ represents the slope of the equation of the tangent for the graph of y respectively.

Chapter 2: The Derivative

Example 2: Differentiating Functions a) Use the definition of the derivative to differentiate (find the slope of ) y =

1 . x

b) What is the equation of the tangent when the slope of the tangent equals −

How to Do It...

1 ? 4

What to Think About What does differentiate mean?

How can we deal with the complex fraction?

Which method do you prefer?

Chapter 2: The Derivative

Example 2: Differentiating Functions - continued

What does it mean if there are two x - values where the slope 1 is − ? 4

What does this look like graphically?

Chapter 2: The Derivative

Your Turn ... 1 at x = −3 to calculate the slope of the tangent at that point. Use the result to x+2 determine the equation of the tangent at that point. Differentiate y =

Example 3: Differentiating Trigonometric Functions at a Point What is the slope of the function y = sinx at x = 0 ? Use the definition of the derivative to justify your result.

How to Do It...

What to Think About Differentiate the function using the concept of limits.

Do you recognize this limit?

What does this look like graphically?

Chapter 2: The Derivative

Your Turn ... What is the slope of the function y = 2sinx at x = 0?

Example 4: Differentiating Piece-wise Functions y 2 x + 1 . Evaluate y′(−2) using two different methods. Consider the function =

How to Do It...

What to Think About How do you write the function using piece-wise notation?

What are two ways to identify the slope of the tangent at x = 2?

Why must the derivative of an absolute value function be written in a piece-wise fashion? Consider this graphically and algebraically.

Chapter 2: The Derivative

Your Turn ... Consider the function y =− x + 2 . Evaluate y′(−2) using two different methods.

2.2 Practice In questions 1 to 4, at the indicated point, identify the slope of the curve, the equation of the tangent, and draw a graph representing the relationship between the tangent and function.

y = x 2 at (2, 4)

= y 3 x − 4 at (2, 2)

Chapter 2: The Derivative

1 1 at  6,  x−2  4

= y x 2 − 3 x at (2, −2)

In questions 5 and 6, describe what happens to the derivative for the entire domain of the following functions 5.

y = x2

1 x−2

7. At Wrigley Field in Chicago, IL it is customary for a fan to throw an opposing team’s home run ball back onto the field. Suppose an eager fan in the left field stands throws a ball that follows a flight pattern where its height is measured as a ) 80 − 16t 2, how fast is the ball function of time where h(t= falling 2.5 seconds after it leaves their hand? NB: The rate of change of height (i.e., how height changes over time) is the slope of the tangent to the curve at each point in dh time and is defined as = h′(t ) . This represents the velocity dt of the ball.

Chapter 2: The Derivative

8. Cliff jumping is a very popular pastime in Mexico. Suppose that a diver jumps from a 100ft cliff, what will his speed be when he enters the water if his height is measured as a function of time and dh h(= t ) 100 − 16t 2 ? The rate of change of height = h′(t ) represents the dt velocity of the diver.

9. At what point is the tangent to the graph of y = x + 6 x − 1 horizontal?

10. Find an equation for each tangent to the curve y =

1 that has a slope of −1. x −1

Chapter 2: The Derivative

2.3 Alternate Definition of the Derivative at a Point Warm Up If a derivative does exist at a certain point, we can use the slope formula in another way to find the derivative or slope at a point. Consider the slope formula as change of y (∆y ) over the change of x (∆x) . What is the slope of the secant line in this case?

What would happen if the value of x approached the value a? How can you write these infinite operations in one mathematical expression?

Which notation is Leibniz and which is Newton?

Chapter 2: The Derivative

Example 1: Determine the Derivative of a Function Using the Alternate Definition Differentiate f ( x) = x using the alternate definition.

How to Do It...

What to Think About What does the alternate definition solve for?

How do you simplify an expression that involves a radical binomial?

Chapter 2: The Derivative

Definition: Alternative Definition of a Derivative at a Point dy dy ff((xx))−− ff((aa)) ′((aa)) lim = = f= f= lim x→ x →aa dxx =x =aa dx xx−−aa

Your Turn ... x) Differentiate f ( =

x + 2 using the alternate definition.

Chapter 2: The Derivative

Example 2: Use the Alternate Definition to Determine a Derivative Differentiate f ( x) =

1 using the alternate definition. x

How to Do It...

What to Think About What are the two different algebraic manipulations used?

Chapter 2: The Derivative

Your Turn ... Differentiate f ( x) =

1 using the alternate definition. x −3

Did You Know? (“The Controversy” 1709-1716) Two 17th century mathematicians both claimed that they invented calculus. Isaac Newton began his work on the development of calculus in 1666 with manuscripts that highlighted his method of fluxions and fluents, which he called the derivative of a continuous function. Newton published his work in 1687 in his book the Philosophiae Naturalis Principia Mathematica. Leibniz first produced his findings in 1684 and 1686 with his writings on differential calculus and integral calculus in his book Nova Methodus pro Maximis et Minimus. However, there was circumstantial evidence that Leibniz had access to Newton’s manuscripts. In the end, both men have been noted as independent inventors of calculus early in the 18th century after a decade old controversy about the intellectual property related to the invention of modern calculus.

Chapter 2: The Derivative

Example 3: The Function and its Derivative Expressed as a Limit 2

( x + h) − x 1 a) Given f ′( x) represented as a limit, determine f ( x) =given f ′'(( xx)) = lim . h → 0 h x −3 b) Express the derivative of y = cosx as a limit.

How to Do It...

What to Think About What is the relationship between the original function and the derivative as a limit?

Your Turn ... Given the derivative in the form of a limit, determine the original function:

x+h − 3 x h

ff ′('(xx)) = lim

x+h x − 2 2 ( x + h) + 1 x + 1 ff ′('(xx)) = lim h →0 h

h →0

f ′'(( xx)) = lim

f ′'(( x)) = lim

h →0

tan( x + h) − tan x h

e x+h − e x h →0 h

1 1 1 e) = ff ′('(xx)) lim  −  h →0 h x + h x 

Chapter 2: The Derivative

2.3 Practice For questions 1 to 4, find the derivative of the given function at the indicated point using the definition

f ( x + h) − f ( x ) h →0 h

f ′'(( xx)) = lim

1 at x = 3 x

f ( x= ) x 2 − 3 at x = 2

f ( x) =

f ( x)= 4 − x 2 at x = 2

f ( x= ) x 2 + 5 x at x = 1

100

Chapter 2: The Derivative

For questions 5 to 8, find the derivative of the given function at the indicated point using the alternative definition f ( x) − f (a) ff ′('(aa)) = lim x→a x−a

f ( x= ) x 2 − 3 at a = 2

f ( x) =

1 at a = 3 x

f ( x)= 4 − x 2 at a = 2

f (= x)

x + 4 at a = 1

Chapter 2: The Derivative

101

For questions 9 and 10 determine the derivative of each function.

d (3 x 2 ) dx

11.

What is f ′( x) if f ( x) =

10.

d (5 x) dx

1 ? x+2

12. Estimate f ′(4) for the graph of the piece-wise function (Hint: Pythagorean Theorem)

102

Chapter 2: The Derivative

2.4 Differentiability of a Function Warm Up Zooming in to “See” Differentiability Graph the following functions and assess the value of their derivative at x = 0.

f ( x= ) x +1

g ( x) =x 2 + 0.0001 + .99

What do you notice happens to your graphs as you zoom in at x = 0 ?

Is it possible to calculate the slope at x = 0 in f(x)? g(x)?

What would the slope be at a point that is a corner?

Definition: Local linearity All functions that are differentiable at a point x = c are well modeled by a unique tangent line in a neighborhood of c and are thus considered locally linear.

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Definition: Right-hand limit and left-hand limit The Right-hand derivative of a function f(x) at x = a f ( x + h) − f ( x ) f ( x) − f (a) lim+ f ′( x) = lim+ or lim+ x→a h →0 x→a x−a h The Left-hand derivative of a function f(x) at x = a f ( x + h) − f ( x ) f ( x) − f (a) lim− f ′( x) = lim− or lim− x→a h →0 x→a x−a h

Example 1: One-Sided Derivatives Can Differ at a Point Show that the following function has left-hand and right-hand derivatives at x = 0.

 x2 , f ( x) =   2 x,

x≤0 x>0

How to Do It...

What to Think About What does the graph of the function look like?

What can you conclude about the derivative x = 0?

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Chapter 2: The Derivative

Your Turn ... Show that the following function has left-hand and right-hand derivatives at x = 0.  x 2 + 2 x, f ( x)  0.5 x,

−4 ≤ x ≤ 0 0< x≤4

Graphically what does this look like?

What happens when the left-hand and right-hand derivatives are not the same?

What is the left-hand derivative at x = 0?

What is the right-hand derivative at x = 0?

What is the derivative at x = 0?

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105

Four ways in which a function fails to be differentiable:

A: Corners Graph y = x and analyze its derivative.

How to Do It...

What to Think About What is the relationship between the left and righthand derivatives at the corner?

Your Turn ... Graph = y

106

3 x − 1 and analyze its derivative.

Chapter 2: The Derivative

B: Cusp 2 3

Graph y = x and analyze its derivative graphically

How to Do It...

What to Think About What is the relationship between the left and righthand derivatives at the cusp?

How does the graph of the derivative relate to the graph of the function?

Did You Know? Your graphing calculator can graph the derivative function. In the graphing screen, choose 8: nderiv( from the MATH menu.

Use x as the independent variable, then enter Y1 or the function of your choice.

Finally, choose “for all x”. ON A TI-83 CALCULATOR ENTER Y2=nderiv(Y1,X,X).

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107

Your Turn ... Graph y =

x − 2 − 3 x + 2 and analyze its derivative graphically (specifically at x = −2).

C: Vertical Tangent 1 3

Graph y = x and analyze its derivative graphically.

How to Do It...

What to Think About What is the relationship between the left and righthand derivatives at the vertical tangent?

Does the function have a tangent line at x = 0?

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Chapter 2: The Derivative

Your Turn ... y Graph =

2 − x and analyze its derivative graphically.

D: Discontinuity a) Graph y =

x x

and analyze its derivative graphically. 1  

b) Graph y = sin   and analyze its derivative graphically. x

How to Do It...

What to Think About What is the relationship between the left and righthand derivatives at the discontinuity?

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109

Your Turn ... Graph y =

[ x ] and analyze its derivative graphically. x

Note: Continuity at a point does not guarantee differentiability as there may be a corner, cusp or vertical tangent. Conversely, however, if a function is differentiable at a point, then it must be continuous at the point.

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Chapter 2: The Derivative

2.4 Practice For the following graphs, identify the points at which the graph is (a) differentiable (b) continuous but not differentiable (c) neither continuous nor differentiable. Write your solution using set notation or interval notation.

Chapter 2: The Derivative

111

For what values of x do the following functions not have a derivative? 2

y= x

y = x3

y= x

y= 4 − 3 x

For questions 10 to 13 compare the left-hand derivative and right-hand derivatives to justify that the function is differentiable at point P.

 x2 , x<0 10. = f ( x) = ,P 0 4 x , x ≥ 0 

1 x <1  , = f ( x) = ,P 1 11. x 4 x − 3, x ≥ 1

x<0 − x, = f ( x) = ,P 0 12. x≥0  x ,

− x 2 , x < 0 = f ( x ) ,P 0 13. = 2 x≥0  x ,

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Chapter 2: The Derivative

2.5 Symmetric Difference Quotient Warm Up

For each interval and point determine whether the derivative is positive, negative, zero, or does not exist? (−∞, a − h)

x= a − h

( a − h, a + h )

x= a + h

( a + h, b )

x=b

(b, ∞)

Draw a secant joining the points on the curve with x-values a – h and a + h. What is the slope of this secant?

What would happen when the value of h approaches 0?

Express this using proper mathematical notation.

What does this expression approximate? Note: Refer to applet illustrating symmetric difference quotient https://qrs.ly/4bgge2t

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Definition: Symmetric difference quotient The Symmetric Difference Quotient can be used to find the slope at x = a.

f ( a + h) − f ( a − h) h →0 2h

f ′(a ) = lim

Example 1: Using the Symmetric Difference Quotient to Estimate the Derivative at a Point Given the following table, estimate each of the derivatives. x f(x)

0 5

1 2

2 1

3 3

f’(1) =

f’(2) =

f’(3) =

f’(4) =

f’(0) =

f’(5) =

How to Do It...

4 7

5 4

What to Think About Which two points are special cases in determining the derivative of the function?

Can you still apply the symmetric difference quotient at the endpoints?

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Chapter 2: The Derivative

Your Turn ... Given the following table, estimate each of the derivatives. x f(x)

0 3

1 -4

2 -6

3 0

f’(1) =

f’(2) =

f’(3) =

f’(4) =

f’(0) =

f’(5) =

4 2

5 3

Example 2: Compute a Numerical Derivative Using Your Calculator Compute the derivative of x 3 at x = 2 .

How to Do It...

What to Think About

nvDeriv( x3 , x, 2) = 12 This notation indicates that you are taking the derivative of the cubic function, with respect to x when x = 2.

Did you know that your calculator can approximate the derivative at a point?

The calculator may only be accurate to 5 decimal places in which case the true answer is x 3 ′ = 12 .

( )

x=2

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115

Your Turn ... 3

a) Compute the derivative of y =x − 3 x − 4 at x = −3 using your calculator.

b) Compute the derivative of x at x = 0 . What value do you expect the calculator to confirm?

Note: The calculator uses the symmetrical difference quotient to calculate the derivative, therefore the calculator IS NOT ALWAYS CORRECT!! Be smarter than your calculator!

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Chapter 2: The Derivative

2.5 Practice 1. a) Given the numerical values shown, find approximate values for the derivative of f(x) at each of the x-values given. x f(x)

0 18

1 13

2 10

3 9

4 9

5 11

6 15

7 21

8 30

b) Over what interval does the rate of change of f(x) appear to be positive?

c) Where is it negative?

d) Where does the rate of change of f(x) seem to be greatest?

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117

2. Given f ( x) =

1 3 x fill in the table below then use it to estimate f’(2), f’(3), f’(4). 3

(Round your answers to 3 decimal places) x 2 3 4

f(x)

What do you notice? Can you guess a formula for f ′( x) ?

3. Sketch a possible graph of y = f(x) given the following information. ff ′('(xx)) > 0 on 1 < x < 3 ff ′('(xx)) < 0 for x < 1, x > 3 ff ′('(xx)) = 0 at= x 1,= x 3

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Chapter 2: The Derivative

Use your calculator to find the derivative of the function at the indicated point. Is the function differentiable at the indicated point?

y= 6x − x2 , x = 3

6x − x2 , x = 0

y = x − 2 + 3, x = 3

y = x − 2 + 3, x = 2

3 = y x= ,x 1

3 = y x= ,x 0

Find all values of x for which the given function is differentiable. − x,  10.= y  2 x,  x2 , 

12.

x<0 0 ≤ x ≤1

11.

y = 2sin x

13.

x >1

y = x+3 −4

x3 − 8 (Hint: Synthetic division) x2 + x − 2

Chapter 2: The Derivative

119

 4 − x,

x <1

x ≥1

14.Let f be the function defined as y = 

ax + bx,

Justify that the function is continuous at x = 1 and that it is differentiable at x = 1.

15.True or False. If f has a derivative at x = a, then the function is continuous. If f is continuous at x = a then the function is differentiable at x = a. Justify your answer.

16.Which of the following is true about the graphs of y = x 3 and y = 3 x at x = 0? a.

y has a corner.

y has a vertical tangent

y has a cusp

y has a discontinuity

y(0)=DNE

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Chapter 2: The Derivative

2.6 Graphing the Derivative of a Function Warm Up −2 x + 1, 3  Graph the piece-wise function f ( x= )  x + 1, 2 4,

x≤0 0< x<2 x≥2

Graph the derivative of f on the second grid.

What would the vertical axis be labelled?

How could you draw a function if you were given the graph of its derivative? Note: Refer to applet that identifies the value of the derivative https://qrs.ly/r1gge2u

Chapter 2: The Derivative

121

Example 1: Sketch f’ Given the Function f Given the graph of a function sketch a graph of the derivative.

How to Do It...

What to Think About When f is increasing what do we know about f’?

When f is decreasing what do we know about f’?

What happens to the slope of f over x ∈ (−∞, 2) ?

What happens to the slope at the vertex?

What happens to the slope of f over x ∈ (2, ∞) ?

122

Chapter 2: The Derivative

Your Turn ... Sketch a graph of f‘ for each function shown below

Chapter 2: The Derivative

123

Example 2: Sketch f Given the Derivative Function. Given the graph of f ′( x) , graph f. Assume f(0) = 6

How to Do It...

What to Think About If f’ > 0, then what can you assume about the graph of f?

If f’ < 0 then what can we assume is true of f ?

If f’ = 0 then what can we assume of the original graph?

What is the connection between the degree of the polynomial function and the degree of its derivative?

124

Chapter 2: The Derivative

Your Turn Sketch f given the graph f’.

Chapter 2: The Derivative

125

2.6 Practice Match the graph of the function with its derivative. 1.

126

Chapter 2: The Derivative

5. Given the function shown below sketch a possible graph of f(x). f ′( x) > 0 for x < 1 and 1 < x < 3 f ′( x) < 0 for x > 3 f ′( x) = 0 for x = 1, x = 3

6. Sketch a possible graph of f(x). f ′( x) > 0 for 2 < x < 4, x > 4 f ′( x) < 0 for x < 2 f ′( x) = 0 for x = 2 f ′( x) = DNE for x = 4

Chapter 2: The Derivative

127

7. The declination of a celestial body is its angular distance north or south of the equator. The declination of the Sun changes from 23.5o N to 23.5o S and back again during the course of the year. Answer the following questions by estimating the slopes of the graph given below. (Graph is for Northern Hemisphere)

a) On what day is the angle of the sun increasing at the greatest rate?

b) On what day does the angle of the sun have no rate of change?

c) On what day is the angle of the sun decreasing at the greatest rate?

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Chapter 2: The Derivative

8. Over the course of a 12-hour day, the value of a stock on the TSX was charted on the graph below. a) What is the value of the stock when the value of the derivative is at its largest? Most negative?

Stock Value

Hours

b) What is the value of the stock when the derivative is zero?

Chapter 2: The Derivative

129

9. A father and son have a foot race at the family picnic. The father orchestrates a photo finish where the race is a tie and he meets his son at the finish line. The position of each runner in 10s of metres is given in relation to their time in seconds in the graph below. The father’s progress is graphed under the straight line. a) Describe the speed of each runner throughout the race.

b) At what point are they the farthest apart?

c) When are they running at the same speed?

d) When does the father start running faster than his son and closing the gap?

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Chapter 2: The Derivative

10. Given the graph of each function f below, sketch the graph of f’ below it

Chapter 2: The Derivative

131

11. Given the graph of the derivative, f’ below, sketch the graph of the function f. a.

132

Chapter 2: The Derivative

12. Identify in order of least to greatest the following values. g ′(−3), g ′(2), g ′(0), g ′(8)

Chapter 2: The Derivative

133

2.7 Chapter Review Find the average rate of change of each function over the given interval.

= y x 2 − 3 x [ 2,5]

 π y = sin x 0,   2

Estimate the slope of the curve at the indicated point using three separate difference quotients

134

f ( x= ) x 2 − 3 x at [ 2, f (2)]

Chapter 2: The Derivative

4. The following graph shows the number of active Uber drivers in the USA. Find the average rate of change related to this increase between July 2013 and July 2014.

5. An object in free fall begins to pick up speed according to the data below. Use the concept of infinitesimals to estimate the speed of the object at 3 seconds.

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135

6. The intensity of a sound as it moves away from its source will resemble an inverse square law. This is useful in measuring the strength of a radio signal. Estimate how fast the signal is losing its strength at 3 metres away from its source

7. Use the definition of the derivative to determine the slope of the tangent y x − 1 at x = 3. to the graph =

136

Chapter 2: The Derivative

8. Use the definition of the derivative to determine the slope of the tangent 1 to the graph y = at x = 5. x −3

9. Use the definition of the derivative to determine the slope of the tangent y 4 x − 8 at x = 1. to the graph =

y 3 x 2 − 12 x horizontal? 10. At what point is the tangent to the graph of=

Chapter 2: The Derivative

137

11. At what point does the tangent to the curve y =

1 1 have a slope of − ? x −1 9

12. Determine whether the function has a tangent at the indicated points. If it does, indicate the slope. If not, indicate why.

( x + 1) 2 ,  y= 2 x + 1, (6 − x) 2 , 

x≤0 0 < x < 3 at x = 0, 3 x≥3

) x 2 + 2 at x = 5 using the alternate definition. 13. Find the derivative of f ( x=

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Chapter 2: The Derivative

14. Find the derivative of f ( x) =

1 at x = 1 at the indicated point using the alternate definition. x+3

15. Identify points on the graph where the function is not differentiable

Chapter 2: The Derivative

139

16. Sketch the graph of the function that satisfies the following conditions f (0) = 1 f ′(−2) = f ′(2) = f ′(5) = f ′(8) f ′( x) < 0 on (−∞, −2] ∪ [2,5] ∪ (5,8) f ′( x) > 0 on (−2, 2) ∪ [8, ∞)

17. Determine the derivative at the point x = 3.5 using the symmetric difference quotient and data below. x f(x)

140

Chapter 2: The Derivative

0 6

1 1

2 2

3 8

4 5

5 4

For questions 18 and 19, use your calculator to find the derivative of the function at the indicated point.

18.

= y x 3 − 4 x at x = −1

19.

4 5

y = x at x = 3

20. Given the graph of the function f, sketch the graph of the derivative f’.

Chapter 2: The Derivative

141

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Chapter 2: The Derivative

Chapter 3: Differentiation Techniques

Learn shortcuts to find derivatives quickly and easily

Although the concept of a derivative as the slope of the tangent to a function can be attributed back to mathematicians such as Euclid, Archimedes and Apollonius in the 3rd century BC, their differentiation techniques were limited to the use of infinitesimals. Modern techniques of differentiation are generally credited to Leibniz and Newton, though these are largely built on earlier works submitted by other great minds in the field of mathematics. Newton was the first to apply his techniques to theoretical physics and Leibniz’s notation is still widely used today. Today derivatives are used by: governments in population censuses; economists to ascertain the marginal cost; and, pharmaceutical scientists as they test the body’s reaction to certain medications. Any time you are discussing a rate of change, you are discussing derivatives.

EXPLORING THE BIG IDEA

In this chapter you should be able to: • Express limits symbolically using correct notation. • Interpret limits expressed symbolically. • Analyze functions for intervals of continuity or points of discontinuity.

Chapter 3: Differentiation Techniques 143

3.1 Derivatives of Polynomial Functions Warm Up Use your knowledge of the definition of the derivative to find the derivative of each function: f ( x) = x 2

f ( x) = x3

f ( x) = x 4

f ( x) = 5 x

f ( x) = mx

f ( x) = 8

f ( x) = a

Can you find a pattern to avoid using the definition?

Definition of the Derivative (the derivative function): f ( x + h) − f ( x ) h →0 h

f ′'(( x) = lim The Derivative at a Point (2 formulas) f ′'((a )) = lim h →0

f ( a + h) − f ( a ) h

Right-Hand Derivative at x = a f ′'( (aa)) = lim+ h →0

f ( a + h) − f ( a ) h

The Difference Quotient

f ( a + h) − f ( a ) h

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Chapter 3: Differentiation Techniques

f ′(a ) = lim x→a

f ( x) − f (a) x−a

Left-Hand Derivative at x = a f ′'( (aa)) = lim− h →0

f ( a + h) − f ( a ) h

The Symmetric Difference Quotient

f ( a + h) − f ( a − h) 2h

The Power Rule

As with all other derivative formulas, has its origins using limits. = ( x n )′ nx n −1 , n ∈  Proof #1 of the Power Rule Proof Using Limits d n ( x + h) n − x n x ) = lim ( h →0 dx h apply binomial theorem 1  n  x + nx n −1h + n(n − 1) x n − 2 h 2 + ... + nxh n −1 + h n  − x n  d n 2  x ) = lim  ( → 0 h dx h

Subtract the x n

d n ( x ) = lim h →0 dx

1 nx n −1h + n(n − 1) x n − 2 h 2 + ... + nxh n −1 + h n 2 h

Factor out the h d n 1 = x ) lim nx n −1 + n(n − 1) x n − 2 h + ... + nxh n − 2 + h n −1 ( h → 0 dx 2 d n ( x ) = nx n−1 dx

Proof #2 of the Power Rule Using Limits

f ′(a ) = lim x→a

f ( x) − f (a) x−a

xn − an x→a x − a

f ′'((a )) = lim

Given xn − an = ( x − a )( x n −1 + ax n − 2 + a 2 x n −3 + ... + a n −3 x 2 + a n − 2 x + a n −1 )

∴ ( x − a )( x n −1 + ax n − 2 + a 2 x n −3 + ... + a n −3 x 2 + a n − 2 x + a n −1 ) x→a x−a

f ′'((a )) = lim

f ′'((a) = lim x n −1 + ax n − 2 + a 2 x n −3 + ... + a n −3 x 2 + a n − 2 x + a n −1 x→a

f ′'((a ) = a n −1 + aa n − 2 + a 2 a n −3 + ... + a n −3 a 2 + a n − 2 a + a n −1 f ′'((a ) = a n −1 + a n −1 + a n −1 + ... + a n −1 + a n −1 + a n −1 f ′'((a ) = na n −1 ∴ f ′( x) = nx n −1 © Edvantage Interactive 2025

Chapter 3: Differentiation Techniques 145

Example 1: Application of the Power Rule for Differentiation 2

Find f’(x) for the function f ( x) = 3 x .

How to Do It...

What to Think About

f ( x)= (3 × 2) x 2−1

Application of the power rule = ( x n )′ nx n −1 , n ∈  , How does this connect the graph of the function and its derivative?

f ( x) = 6 x

Your Turn ... a.

f ( x) = x3

f ( x) = 2 x5

f ( x) = 3x

f ( x) = mx3

f ( x) = 6

f ( x) = a

f ( x) =

f ( x) = 3 x

1 x3

Sum rule

d d d f ( x) + g ( x) [ f ( x) + g ( x)]= dx dx dx

Difference rule

d d d f ( x) − g ( x) [ f ( x) − g ( x)]= dx dx dx

Constant multiple

d d k⋅ f ( x) [ k ⋅ f ( x)] = dx dx

Constant rule

d k =0 dx

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Chapter 3: Differentiation Techniques

Example 2: Application of the Power Rule Within the Sum, Difference & Constant Rules of Differentiation Find

dy 5 if y =x 3 + 6 x 2 − x + 14 dx 3

How to Do It...

What to Think About Application of the power rule = ( x n )′ nx n −1 , n ∈  , How does this follow the graph of the function and its derivative?

Your Turn ... Find

dy if: dx

x4 x2 + + 5x 4 8

x + 3x − 9

Chapter 3: Differentiation Techniques 147

Example 3: Determining Tangent Lines x2 + 3 Find the equation of the tangent to the curve f ( x) = at the point (1, 2). Support your answer 2 x graphically.

How to Do It...

What to Think About What does

dy represent? dx

dy Are there any points where dx can’t exist? Why?

To illustrate, use 2nd Draw 5: Tangent

To get the Y1 function on the calculator use VARS, YVARS, 1:Function, 1: Y1

148

Chapter 3: Differentiation Techniques

Example 3: Determining Tangent Lines - Continued dy To illustrate, use 2nd Calc 6: dx and enter value for x.

Your Turn ... Find the equation of the tangent to the curve y = graphically.

x4 + 2 at the point (−1, 3). Support your answer x2

Chapter 3: Differentiation Techniques 149

Example 4: Finding Horizontal Tangents 4

Does the curve y =x − 2 x + 2 have any horizontal tangents? If so, at which x-values? What are the equations of the horizontal tangents?

How to Do It...

What to Think About What condition must exist for a graph to have a horizontal tangent?

How can you use the graph of the derivative to find where a horizontal tangent line exists on?

Remember Your graphing calculator can graph the derivative function. In the graphing screen, choose 8: nderiv( from the MATH menu.

Use x as the independent variable, then enter Y1 or the function of your choice.

Finally, choose “for all x”. On a TI-83 calculator enter Y2 = nderiv(Y1, X , X )

150

Chapter 3: Differentiation Techniques

Your Turn ... 3

Does the curve y = x − x − 8 x + 12 have any horizontal tangents? If so, at which x-values? What are the equations of the horizontal tangents?

Example 5: Finding Horizontal Tangents Using Technology y 0.2 x 4 − 0.7 x3 − 2 x 2 + 5 x + 4 has horizontal tangents. Determine the values of x where the curve =

What to Think About

How to Do It...

What is the slope of a horizontal tangent?

Your Turn ... 6

Determine the function values where the curve f ( x) = .1x + .1x − 2.5 x − 2.5 x + 18 x + 10.8 x − 43.2 has horizontal tangents.

Chapter 3: Differentiation Techniques 151

3.1 Practice Find

dy for each equation: dx

= y 3x 2 − 5 x

x6 = y +x 3

y = −x

y = x2 + x + 7

y= − x2 −

y = x + 3 x2 − 5

1 x

Determine the values of x for which each curve has a horizontal tangent.

152

y = x3 − 4 x 2 − 3x + 6

Chapter 3: Differentiation Techniques

2 3 x2 x − − 3x + 5 3 2

11.

y =x 4 − 6 x 2 + 7

10.

y = 6 x 2 − 4 x3 + 1

1 5 x −x 5

12.

y =

1 5 2 3 x + x +x 5 3

y 3 x 2 − 4 x at x = 2? 13. What is the value of the slope of the tangent to the equation=

14. Find the equation of the line tangent to the equation 3 x − 4 y = 12 at x = 2?

15. What is the value of the slope of the tangent to the equation y = 3 at x = 2?

Chapter 3: Differentiation Techniques 153

16. Find the equation of the line perpendicular to the tangent to the curve y = x 2 − 5 x + 4 at the point (−2, 18)

x3 − 4 x when the slope of the tangent is 5. What feature does this 3 point correspond to on the original function? 17. Find the points on the curve = y

18. Find the intercepts of the tangent to the curve = y x 3 + 6 x when the point of tangency is (2, 16).

3 y x 2 + 8 x parallel? 19. Where are the tangent lines of y = x and =

20. What condition must occur for a function to have a vertical tangent line? Consider both the derivative and the original function.

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Chapter 3: Differentiation Techniques

3.2 Higher Order Derivatives Warm Up The heights in feet of a water skier’s jumps are modelled by the equation h(t ) = −16t 4 + 120t 3 − 150t 2 + 74t − 12 . How would you determine an equation that represents how the height is changing over time in seconds? What are the units of this equation?

How would you determine an equation that represents how fast the change in height is changing? What are the units of this equation?

What real-world measurement do each of these new equations represent?

Note: This model would not be true unless there was a variable gravity environment. © Edvantage Interactive 2025

Chapter 3: Differentiation Techniques 155

Example 1: Second and Higher Order Derivatives 3

Find the first four derivatives of y =x − 5 x + 2

What to Think About

How to Do It...

How do you denote the different derivatives in both Leibniz and Newton notation?

Did You Know? The 4th derivative is known as the “jounce” and is useful in determining the cosmological equation of state, which is characterized by a dimensionless number w. It is defined as the ratio of its pressure to its energy density. The 5th and 6th derivatives, while useful in applications of theoretical physics, have no universally accepted name. Recently, the use of the fifth derivative and curve fitting were used to perform DNA analysis and population matching. 156

Chapter 3: Differentiation Techniques

Your Turn ... Find the first four derivatives of the following functions.

y = 6x2 − 4x + 3

y= −2 x + 5 x − 7 x + 2

= y x −2 − x 5

x2 −1 y= 4 x

Example 2: Differentiating a Product 2 Does the derivative= of y ( x )( x + 1) equal 2x?

What to Think About

How to Do It...

Is it possible to rewrite the function in order to have only power terms?

Your Turn ... 2

Find the derivative of y = ( x − 1)(2 x − 5 x + 3)

Chapter 3: Differentiation Techniques 157

Definition: Product Rule The product of two differentiable functions u and v is differentiable

d du dv (= uv) v+ u dx dx dx ∴ (uv)′ =u ′v + v′u

Example 3: Differentiating a Product of Functions ( x + 2)( x − 3) Find f ′( x) if f ( x) = 2

How to Do It...

What to Think About Does it matter which function is u and which v?

For the product of polynomials, how can we verify the derivative?

Can you simplify the function first?

Your Turn ...

Differentiate y = x 3 x

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Chapter 3: Differentiation Techniques

Example 4: Working with Numerical Values Let y = fg be the product of the functions f and g. Find y’(2) if f(2) = 4, f’(2) = − 5, g(2) = 3, and g’(2) = 2.

How to Do It...

What to Think About What differentiation rule must you use for this function?

Your Turn ... Let y = gf be the product of the functions f and g. Find y’ (3) if f (3) = 4 , f ′(3) = −8 and g (3) = 6 , g ′(3) = 4 .

Definition: Quotient Rule At a point where v ≠ 0 , the quotient y = du dv v− u d  u  dx dx  = dx  v  v2

u of two differentiable functions is differentiable v  u ′ u ′v − v′u ∴   =2 v v

Chapter 3: Differentiation Techniques 159

Example 5: Differentiating a Quotient Find the derivative of a) g ( x) =

1 x2 −1 and b) f ( x) = 2 . x x +1

How to Do It...

What to Think About What differentiation rule must you use for this function?

Does it matter which term goes first in the numerator?

Is there an advantage to keeping your denominator in a factored form?

Your Turn ... t 2 − 3t + 2 Find the derivative of the following function g (t ) = t −4

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Chapter 3: Differentiation Techniques

Example 6: The Quotient Rule With an Application of Numerical Values Let y =

f be the quotient of the functions f and g. Find y′(2) if f (2) = 4 , f '(2) = −5 , g (2) = 3 and g

g ′(2) = 2

How to Do It...

What to Think About How would you determine the derivative in terms of x for this function?

Your Turn ... g be the quotient of the functions f and g. f Determine y’(2) if f(2) = 4, f’(2) = –5, g(2) = 3 and g’(2) = 2

Let y =

Chapter 3: Differentiation Techniques 161

3.2 Practice For questions 1 to 4 determine the first four derivatives of each function.

y = 3x 2 − 5 x + 7

= y x −1 − x 3

y = 3x 4 + 2 x3 − 4 x − 7

x2 + 1 x3

2x + 5 3x − 2

For questions 5 to 12 determine

162

dy . dx

y =(2 x3 + 3)( x 4 − 2 x)

Chapter 3: Differentiation Techniques

y= ( x − x + 7)(2 x − 3 x − 5)

x2 y= 3 − 4 x3

y= ( x −2 + x −3 )( x 4 − 3 x 2 )

10.

y =( x − 1)( x + 5) −1

12.

y 11. =

x ( x + 2)

x3 + 3x 2 − 5 x2

Chapter 3: Differentiation Techniques 163

Given that u and v are functions that are continuous and differentiable, and that u(3) = 2, u’(3) = 3, v(3) = 4, and v’(3) = 5, determine the values of the derivative at x = 3.

13.

d (uv) dx

15. Determine the derivative of y =

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Chapter 3: Differentiation Techniques

14.

d u   dx  v 

x +2 using the quotient rule. x −3

For questions 16 and 17, determine the equation of the tangent line to the curve at the given point.

16.

y= (1 + 3 x) 2 at (1, 16)

17.

3x + 1 at (1, 2) x2 + 1

18. Explain why (π 4 )′ ≠ 4π 3 ?

19. Justify why f ( x) =

1 has a horizontal tangent at y = 0? x

20. How many horizontal tangents does y =( x − 1)( x + 1) have?

Chapter 3: Differentiation Techniques 165

3.3 Velocity & Other Rates of Change Warm Up Analyzing a Car’s Performance Suppose you drive your car for 3 hours and travel exactly 180 mi. What is your average velocity? ) 30 + 5t 2 during the first 3 hours, is there a time that the car’s veIf the car’s velocity function is v(t= locity is exactly 60 mph? Justify your answer. Instantaneous Velocity = first derivative of the position function = s’(t)

How does Liebniz notation help to determine the units of a derivative?

Definition: Velocity function (rectilinear motion)

ds Let s(t) be a position function for an object. Then the velocity function is = s′(t ) represents dt the change in position over time. When s’(t) = 0, the object is not moving. The object is stopped or at rest. When s’(t) > 0, the object is moving in the positive direction (forward/to the right/up). When s’(t) < 0, the object is moving in the negative direction (backward/to the left/down). The net change in position over [a,b] is called the displacement and equals s(b) – s(a). The average velocity over [a,b] equals s (b) − s (a ) b−a Some objects move one-dimensionally. This motion is called rectilinear motion. 166

Chapter 3: Differentiation Techniques

Example 1: Relating Velocity & Position 3

Given the position function s (t ) =t − 6t + 9t (t is measured in sec, s is measured in ft) a) find the velocity function.

b) find the displacement over the first 2 secs.

c) find the average velocity after 2 secs.

d) find the velocity at 2 secs and at 4 secs.

e) determine when the object is at rest.

f ) determine when the object is moving in a positive direction and in a negative direction.

How to Do It...

What to Think About What is the relationship between the position function and its velocity?

What is the relationship between the position function and displacement?

What is the relationship between average velocity and displacement?

What is the relationship between the velocity at a point and the position function?

What do zero, positive and negative values of the derivative imply about the function? for t over © Edvantage Interactive 2025

Chapter 3: Differentiation Techniques 167

Your Turn ... Baby Kaya’s first steps were filmed last week in a 5 second video. The position of her right foot, relative to her left, was recorded as the following 3 2 function s (t ) = t − 5t + 8t − 4 (t is measured in sec, s is measured in cm). a) What is the velocity at time t?

b) When is the right foot at rest?

c) What is the velocity after 1 seconds? 1.5 seconds? 3 seconds?

d) When is the right foot moving forward (in a positive direction)?

e) When is the right foot moving backward (in a negative direction)?

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Chapter 3: Differentiation Techniques

Example 2: Relating Velocity to a Position-time graph Given the position function s (t ) =t 3 − 6t 2 + 9t from example 1, use the previous solution to determine the following. a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero.

b) Draw a diagram to represent the rectilinear motion of the object for t ≥ 0 .

c) graph the position-time graph.

d) find the displacement after the first 5 secs.

e) find the total distance traveled over the first 5 secs.

How to Do It...

What to Think About What direction is the object moving during the first second?

When does the object change directions?

In between stopping points, is it possible for an object to change direction?

Does the position-time graph indicate which direction the object is moving?

Does the position-time graph indicate when an object is stopped?

Chapter 3: Differentiation Techniques 169

Example 2: Relating Velocity to a Position-time graph - Continued How do displacement and total distance differ?

Your Turn ... 3

Consider baby Kaya’s first steps and the position of her right foot moddeled by s (t ) = t − 5t + 8t − 4 from the last example. a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero. Does this table of values describe all important intervals? Please add any additional points that would be useful. t

0 1 2 3 4 5

b) Draw a diagram to represent the rectilinear motion of the t ≥ 0 .

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c) Graph the position-time graph.

d) Find the displacement after the first 5 secs.

e) Find the total distance traveled over the first 5 secs.

Chapter 3: Differentiation Techniques 171

Example 3: Free-Falling Object The vertical height y (in feet) of a coin thrown upwards from the obsertion deck of the Empire State Building, in time t seconds, is described by the −16t 2 + 96t + 640 position function: y (t ) = a) What is the initial velocity at t = 0? b) What is the height of the coin when the velocity is zero? c) What is the speed of the coin when it returns to earth?

How to Do It...

What to Think About What does the word initial imply?

What is important about the height when the velocity is zero in this situation?

When does the coin return to earth (hit the ground)?

How does speed relate to velocity?

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Chapter 3: Differentiation Techniques

Your Turn ... On a construction site, a dynamic blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of = s 160t − 16t 2 ft after t seconds.

a) How high does the rock go?

b) What is the velocity of the rock when it is 256 ft above the ground on the way up? on the way down?

Did You Know? Free falling to maximum velocity skydivers are familiar with the term “terminal velocity” which states that terminal velocity is the highest velocity attainable by an object as itfalls through a fluid (air being the most common example).

c) What is the speed of the rock when it is 256 ft above the ground on the way up? on the way down?

As a skydiver falls towards the earth, his velocity continually gets faster. This change in velocity is known as acceleration. Acceleration is the rate of change of the velocity. Acceleration = a= (t ) v′= (t ) s′′(t )

d) When does the rock hit the ground?

dv d 2 s = dt dt 2

Chapter 3: Differentiation Techniques 173

Example 4: Relating Position to Velocity and Acceleration 3 2 Consider the position function s (t )= t − 6t for a particle at time, t.

a) Determine v(t) and a(t) b) Graph s(t), v(t) and a(t) over the interval 0 < t < 8. Use a vertical scale between –40 and 60. c) Describe the motion of the particle.

How to Do It...

What to Think About What is the relationship between velocity, acceleration and position?

What points on the velocity function correspond to minimum and maximum points on the position function?

What feature of the velocity function do the zeros of the acceleration function correspond to?

When the acceleration function is zero and the velocity function is at an extreme value, what is happening on the position function?

174

Chapter 3: Differentiation Techniques

Example 4: Relating Position to Velocity and Acceleration - continued

How do displacement and total distance differ?

Your Turn ... Consider the horizontal position function of a hummingbird as it 4 2 approaches a feeder s (t ) =t − 2t + t a) Determine v(t) and a(t).

Continued on next page © Edvantage Interactive 2025

Chapter 3: Differentiation Techniques 175

b) Graph s(t), v(t) and a(t) and describe the horizontal motion of the hummingbird over the interval 0<t<6.

SUMMARY A particle is speeding up when its velocity and acceleration move in the same direction (both v and a have the same sign). A particle is slowing down when its velocity and acceleration move in the opposite directions (v and a have different signs).

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Chapter 3: Differentiation Techniques

3.3 Practice 1. You are training your new puppy Hugo to walk with a leash. His velocity in the first few seconds is shown in the graph below.

a) When is he moving forwards?

b) When is he moving backwards?

c) When did he stop?

d) When is his acceleration negative?

e) When is his acceleration positive?

f ) When is he moving at his greatest speed?

g) When is he speeding up?

h) When is he slowing down?

Chapter 3: Differentiation Techniques 177

2. Slacklining is the act of walking along a suspended length of taut, flat ribbon that is tensioned between two anchors. It resembles tightrope walking and is growing in popularity due to its workout benefits, simplicity and versatility. A slackliner in his backyard is walking across the rope on a summer evening. The velocity, v(t), of his movements during time t, 0 ≤ t ≤ 11, is illustrated in the graph below.

a) At what time(s) t does the walker change direction?

b) At what time(s) t is the speed of the walker greatest?

c) When is the walker moving forward? Moving backward?

d) When is the walker speeding up? Slowing down?

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Chapter 3: Differentiation Techniques

3. The height of a football of an NFL punt is given by the function h(t ) = −4.9t 2 + 24t .

a) What is the ball’s velocity as a function of time?

b) What is the ball’s acceleration as a function of time?

c) What was the maximum height of the ball?

d) When did the ball hit the turf?

e) When was the ball at a speed of 4.4 mph?

Chapter 3: Differentiation Techniques 179

4. A golfer in a long distance drive contest hits a ball whose height is given by the equation h = −1.5t 2 + 60t . What is the velocity of the ball at 2 seconds?

5. Chris hits a ball out of the sand trap with a height of h = −d 2 + 6d where h is the height and d represents the horizontal distance the ball travels. What is the maximum height the ball reaches?

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Chapter 3: Differentiation Techniques

6. A bungee jumper leaps off a bridge with an initial velocity of 3.6 mph and reaches heights of h =3.6t − 0.6t 2 + 9.6 .

a) What is his velocity at time t?

b) When does he reach the peak of his jump?

c) When does he hit the water below?

d) What is his velocity when he hits the water?

Chapter 3: Differentiation Techniques 181

w(t ) 200(25 − t ) 2 liters at a 7. In fighting a fire with a hose, a firefighter has access to a tank that holds= time, t seconds after turning it on full blast. a) What is the average velocity of the water in the first 10 seconds of use?

b) What is the instantaneous velocity of the water at t = 10 seconds?

8. On the show American Ninja Warrior, during the first hooks of the Sky Hang obstacle, the velocity of a contestant’s body is represented by the function v = t 3 − 6t 2 + 9t + 4 . Find the contestant’s velocity every time their acceleration is zero.

182

Chapter 3: Differentiation Techniques

9. An NFL football offensive lineman, while doing a wave drill moving forwards and backwards to work on his foot speed, had his position from 2 the goal line marked with the function s (t ) = 2t − 8t + 30 . a) What is the displacement during the first 5 seconds?

b) What is the average velocity in the first 5 seconds?

c) What is the instantaneous velocity at 3 seconds?

d) What is his acceleration at 3 seconds?

e) When did he change direction?

Chapter 3: Differentiation Techniques 183

10. A teenager flyboarding in San Diego has his height in feet above the water represented by the function h = (t − 4)(t − 10)(t − 12) for 0 ≤ t ≤ 15 . In order to propel the rider upwards, the instructor must increase the throttle.

a) When does the instructor increase the throttle?

b) When does the instructor let off the throttle?

c) What is the total vertical distance traveled by the flyboarder over the first 15 seconds?

d) What is the average velocity of the flyboarder between 0 & 7 seconds?

e) What is the instantaneous velocity of the flyboarder at time 4 seconds?

f ) How far below the surface does the fly boarder end up after he lost his balance the first time?

g) What is the maximum height the flyboarder reaches above the surface of the water?

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Chapter 3: Differentiation Techniques

11. Given the graph of functions a, b, and c, indicate which represents the function and its first two derivatives.

12. Given the functions a, b, c and d, indicate which represents the original function and its first three derivatives. b a

Chapter 3: Differentiation Techniques 185

x2 − 9 13. What is the instantaneous rate of change of the function f ( x) = at x = 3? x+2

14. What is the instantaneous rate of change of the area of a circle when its radius is 2 in?

15. Identify the derivatives of f ( x) = x 2 and g ( x= ) x 2 + 3 . Identify the family of functions, h, such that have the derivative h′( x) = 2 x .

16. What is the derivative of the product of the three differentiable functions y = abc?

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Chapter 3: Differentiation Techniques

3.4 Derivatives of Trigonometric Functions Warm Up Sketch the graph of y = sinx

d (sin x) by considering the slope for several points Create a sketch of the derivative graph for dx along the graph above.

Repeat the process again, this time by considering the slope for several points along the graph that you just created.

What can you conclude about the derivative of the sine function?

Chapter 3: Differentiation Techniques 187

Note: See applet to help identify the value of the derivative of the sine function https://qrs.ly/3zgge2v Derivatives of basic trigonometric functions

d (sin x) = cos x dx

d (cos x) = − sin x dx

Example 1: Derivatives Involving Trigonometric Functions 3 + 5sin x − cos x Differentiate y =

How to Do It...

What to Think About What differentiation rule must be used?

Your Turn ... Find the derivative of:

y = x 3 sin x

188

x 1 + cos x

cos x 1 − sin x

Chapter 3: Differentiation Techniques

Example 2: The Motion of a Weight on a Spring (Simple Harmonic Motion) A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is s (t ) = 5cos t . What is its velocity and acceleration at time t?

How to Do It...

What to Think About What are other examples of periodic motion?

Your Turn ... a) A dog in a swing with an appetite for extreme sports sees his height above the ground follow the parameters of the simple harmonic function h = 2sin t . What is his velocity and acceleration at time t?

Did You Know? b) Given his thrill-seeking nature and desire for the rush of adrenaline, your dog emphatically requests that you push him through each swing from the back. Mathematically, how do we account for this external factor?

Simple Harmonic Motion is periodic motion that can be modeled with a sinusoidal position function.

Chapter 3: Differentiation Techniques 189

Definition: Jerk A sudden change in acceleration is called “jerk”. Therefore, a Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is:

da d 2 v d 3 s j (= t) = = = a′(= t ) v′′(= t ) s′′′(t ) dt dt 2 dt 3

Example 3: Deriving the Jerk a) Determine the jerk caused by the constant acceleration of gravity. b) Determine the jerk of the simple harmonic motion in Example 2.

How to Do It...

What to Think About What differentiation rule must be used?

Your Turn ... Determine the jerk felt by the dog in the swing

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Chapter 3: Differentiation Techniques

Did You Know? Jerk is felt as a change in force such as the boost that a space shuttle gets during its launch as it exits the atmosphere. For those of us that will never visit the moon, we can still experience the jerk. Roller coasters are designed to simulate the system by elevating the acceleration through controlling its rate of change, magnitude, and duration. These changes will prey upon your body’s ability to handle g forces as it demands the designers to have an impeccable understanding of the jerk!

Example 4: Determine How to Do It...

d (tan x) dx

What to Think About How can you rewrite the tangent function in order to derive it?

Which differentiation rule must we apply to find the derivative?

Your Turn ... Find

d (cot x) . dx

Chapter 3: Differentiation Techniques 191

Derivatives of Other Basic Trigonometric Functions

d (tan x) = sec 2 x dx

d (cot x) = − csc 2 x dx

d (sec x) = sec x tan x dx

d (csc x) = − csc x cot x dx

Example 5: Finding Tangent and Normal Lines Find the equations for the lines that are tangent and normal to the graph of f ( x) =

tan x at x = 2. x

Use a calculator to round all significant values to the nearest hundredth and support your conclusion graphically. Use the window [−5, 5] by [−4, 4]

How to Do It...

What to Think About 2

How do you enter sec x into your calculator?

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Chapter 3: Differentiation Techniques

Your Turn ... 2

Find the equations for the lines that are tangent and normal to the graph f ( x) = x sin x of at x =

π 2

3.4 Practice For questions 1 to 10 determine

dy . dx

y =2 − x 2 − sin x

= y cos x + tan x

y = x csc x

y= 3 − x 2 sin x

3x cos x

tan x 1 + tan x

Chapter 3: Differentiation Techniques 193

7. A bungee jumper is bouncing in simple harmonic motion with its position s= (t ) cos t − sin t . What is the jerk at time t?

8. What is the equation of the line that is tangent to the graph of = y cos x + 2 at x = π ?

9. Over the domain − ≤ x ≤ , find the point(s) on the curve y = tanx that have tangents with a slope 2 2 of 2.

1 + 2 csc x + cot x ? 10. What is the equation of the horizontal tangent to the curve y =

194

Chapter 3: Differentiation Techniques

 x + a, x < 0 11. What is the value of a in which the function f ( x) =  will be differentiable at x = 0? cos x, x ≥ 0

12. Evaluate the derivative of y = 2sin x at x = . (Hint: use your trigonometric identities for 2 sin 2 x & cos 2 x ).

13. A child playing with a yoyo has its position at s (t ) = −2 cos t . Is it traveling upwards or downwards at 3π ? t= 2

14. Is it speeding up or slowing down at t =

5π ? 4

Chapter 3: Differentiation Techniques 195

3.5 Introducing the Chain Rule Warm Up A composite function is a nested function of the form y = f ( g ( x)) . Identify the inside and the outside functions in each composition

Outside Function

Inside Function

y = cos 2 x

b) = y cos( x 2 − 5)

= y ( x 2 + 3) 4

y = sin(tan x)

y = tan x sin x

Find

d (2 x 2 )3  dx

Find

d d (u )3  ⋅ (2 x 2 ) where u = 2x2 du dx Why can’t you simply use the power rule on the original function?

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Chapter 3: Differentiation Techniques

Definition: The Chain Rule (for the Power Rule) d n du (= u ) nu n −1 ⋅ dx dx [= u n ]′ nu n −1 ⋅ u ′

Example 1: Taking the Derivative of a Composite function Given the function= y (3 x 2 + 1) 2 what is y′ ?

What to Think About

How to Do It...

What differentiation rule can you use to find the derivative of this function?

Your Turn ... Determine the derivative of the following functions

y = cos3 x

 cos x  y=   1 + sin x 

f= ( x)

4 x3 − 5 x

Chapter 3: Differentiation Techniques 197

Example 2: Application of the Chain Rule in the Context of a Trigonometric Derivative a) Graph the function y = sin x and its derivative using your graphing calculator b) Graph y = sin 2 x to see the graph of the derivative of this function

How to Do It...

What to Think About What should you input into your calculator? From the graphs what do you think the derivatives are? d (sin x) = dx

What should you input into your calculator? From the graphs what do you think the derivatives are?

d (sin 2 x) = dx Why does the amplitude of the derivative graph change and what does it mean?

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Chapter 3: Differentiation Techniques

Your Turn ... a) Plot the function y1 = cos x and its derivative using your graphing calculator and show the graph on the axes below.

b) Now graph y2 = cos 3 x and its derivative.

c) Why does the amplitude increase by a factor of 3? Explain in your own words.

Chapter 3: Differentiation Techniques 199

New Rules for Composite Trigonometric Functions

d (sin u ) = dx

d (cos u ) = dx

d (tan u ) = dx

d (cot u ) = dx

d (sec u ) = dx

d (csc u ) = dx

3.5 Practice For questions 1 to 8 determine

dy . dx

π  1. = y cos  − 3 x  2 

2. = y x sin(π − 2 x)

200

− cos 3 x + y=

2π sin 2 x 3

Chapter 3: Differentiation Techniques

y= ( x − x ) −3

= y (sec x − cot x) −1

= y cos3 x − sin −2 x

7. = y 3 x 2 (2 x − 5)3

y = tan 3 x sin 2 x

Chapter 3: Differentiation Techniques 201

9. An Olympic swimmer doing the butterfly has his center of gravity found  3π  π  by the function = h(t ) 2sin  t  − cos  t  .  2  4  What is the velocity of the swimmer as a function of time?

For questions 10 and 11 determine y’’.

10.

y = tan x

11. = y cot(5 x − 3)

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Chapter 3: Differentiation Techniques

Determine the derivative of each function.

12.

y = sin x 2

13. = y cos( x 2 + 3)

14. = y tan(2 x − 1)

15.The pedals on a spin bike during a recreational spin class rise and fall such that their position is satisfied by the equation h = r cos(2π st ) where r is the length of the pedal arm, s is the number of revolutions per second and t is the time in seconds. If you double the number of revolutions per second on the bike, how does this affect the rotational velocity and acceleration of the pedals?

Chapter 3: Differentiation Techniques 203

3.6 Chapter Review Differentiate each equation:

y = x3 − 5 x + 7

y= −3 x −4 + x −3 +

4 V = π r3 3

204

−

1 4x2

x + x 4 − 3x 3 + 3 x 2

Chapter 3: Differentiation Techniques

3 12 x − x −π 4

= y

y = cx 5 −

8 1 − 4 x x

x + x2 x3

For questions 8 to 10 determine the values of x for which the function has horizontal tangents.

y = x3 + 4 x 2 + 4 x + 7

10.

y = x 4 − 4 x3 + 4 x 2 + x − 4

2 3 x2 y= x − − 6x + 4 3 2

For questions 11 and 12 determine the slope of the tangent at the indicated point?

11.

y = 2 x3 − 4 x + 1 at x = 2

12.

3x 2 − 4 y = 8 at x = 2

13. Show that y = 5 x3 + 7 x + 3 has no tangents with a slope of −2.

Chapter 3: Differentiation Techniques 205

14. Determine the equation of the tangent line and the normal line for y = x 2 − 5 x + 2 at (1, −2).

15. For what value(s) of x are the tangent lines of y = x 3 and y = − x 2 + 8 x parallel?

For questions 16 and 17 determine the first four derivatives of the given equation.

16.

y = 3x3 − 9 x 2 + 7

17.

= y x −2 − x 3

For questions 18 to 21 differentiate the given equation.

18.

206

y =(2 x3 − 4)( x 4 − 3 x)

Chapter 3: Differentiation Techniques

19.

 1 2 2 y=  2 − 3  ( x + 3x ) x  x

20.

4 + 3x 2 − 5x

21.

y =( x + 2)(1 + x 2 ) −1

For questions 22 to 25, given that u and v are continuous and differentiable functions, and that u (5) = 1 , u ′(5) = 1 , v(5) = 2 , and v′(5) = 6 , determine the values of the derivatives at x = 5.

22.

d (uv) dx

23.

d u   dx  v 

24.

d (3u − 4v) dx

25.

d (4uv) dx

Chapter 3: Differentiation Techniques 207

Use the following information to answer questions 26 to 33: A particle is moving along the x-axis with its velocity function, v(t), shown in the figure below.

26. When is the particle moving to the right?

27. When is the particle moving to the left?

28. When did it stop?

29. When is the particle’s acceleration negative?

30. When is the particle’s acceleration positive?

31. When is the particle moving at its greatest speed?

32. When is the particle speeding up?

33. When is the particle slowing down?

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Chapter 3: Differentiation Techniques

Use the following information to answer questions 34 to 39: A particle in rectilinear motion follows the position function x(t ) = t 3 − 5t 2 + 8t + 5 in the first 7 seconds of its journey.

34. What is the velocity at time t?

35. What is the velocity after 2 seconds? 5 seconds?

36. When is the particle at rest?

37. When is the particle moving to the right (in a positive direction)?

38. What is the displacement during the 7 seconds?

39. What is the total distance traveled from time t = 0 to t = 7 sec?

Chapter 3: Differentiation Techniques 209

Use the following information to answer questions 40 to 42: The outside sphere on a pendulum on a desk continues in a simple harmonic motion in which its height is measured as h(t ) = 1 + 6sin t .

40. Determine its velocity and acceleration at time t.

41. Find the velocity, acceleration at time t =

4π 3

42. Identify the direction of the pendulum and whether it is speeding up or slowing down.

210

Chapter 3: Differentiation Techniques

Differentiate each function:

43.

y = 3x5

44.

= y (2 x − 1) 4

45.

y = sin 2 x

46.

y = cot π x

47.

y = tan x

Chapter 3: Differentiation Techniques 211

48.

 2 y = cot  2  x 

49.= y sin 2 (3 x − 5)

50.

y = sin 3 x cos x

51.

y = sin 2 (cos sin 3 x ))

212

Chapter 3: Differentiation Techniques

Chapter 4: Advanced Differentiation Techniques

Learn the advanced techniques that let you find rates of change even when things get really complicated Noise cancelling headphones take the noise accumulated, measured in decibels, outside of the ear phones and constructs a function to represent it. It order to cancel it, that function is embedded within a function that builds its opposite. It would require an application of the chain rule to determine the instantaneous change in decibels required to cancel out random sounds.

Entering a corkscrew turn in a hanging roller coaster, the center of gravity of one of the riders follows the pathway defined by x3 + y 3 = 9 xy … it requires an application of implicit differentiation to determine how fast the height of the person is changing (and the G forces applied on the body!)

EXPLORING THE BIG IDEA In this chapter you should be able to: • Express limits symbolically using correct notation. • Interpret limits expressed symbolically. • Analyze functions for intervals of continuity or points of discontinuity.

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4.1 The Chain Rule Warm Up Determine the derivative of the following functions. b) a) y = sin 2 x

y = sin x 2

Which rule must be applied to differentiate each of the above functions? Write each function as a composition of functions. Write each with an outer function of y = f(u) and an inner function of u = g(x). a)

Example 1: Using the New Rules and Beyond Determine

dy for y = sin x3 and y = sin 3 x dx

How to Do It...

What to Think About Which differentiation rule is needed for this type of function?

What type of u-substitution can be made to simplify the composite function?

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Your Turn ... Differentiate the following functions.

y = sin(cos x)

b) = y tan(5 x − x3 )

1 csc x + cot x

Example 2: Using Multiple Laws of Differentiation Differentiate = y x 2 cos( x 2 + 3)

How to Do It...

What to Think About Which laws of differentiation are applicable?

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Your Turn ... Differentiate each function.

sin x sec 3 x

b) = y sin 3 (3 x − 2)

x2 + 6 sin 3 x

d) = y x 4 (5 x − 3)5

Example 3: Rectilinear Motion An object moves along the x-axis so that its position at any time t ≥ 0 is given by= x(t ) cos(t 2 + 1) . What are the velocity and acceleration of the object as a function of t?

How to Do It...

What to Think About How are position, velocity and acceleration functions related?

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Your Turn ... An object moves along the x-axis so that its position at any time t ≥ 0 is given by = x(t ) sin 2 (2t − 3) . Find the velocity and acceleration of the object as a function of t.

Example 4: Equation of the Tangent Line What is the equation of the line tangent to the curve y = sin 3 x at the point where x =

How to Do It...

π 3

What to Think About What is the tangent line equation in point slope form?

Which special triangle will apply in the evaluation of slope of the function at the point of tangency?

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Your Turn ... a) What is the equation of the line tangent to the curve y = cos3 x at the point where x =

b) Show that the slope of every line tangent to the curve y =

π 6

1 1 is positive, for x ≠ . 3 2 (1 − 2 x)

Did You Know? Biologists employ the Chain Rule when they determine the speed of growth of a bacterial population

A = 100e5t

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Example 5: Composition of Multiple Functions What is the derivative of = y tan(7 − sin 3 x) ?

How to Do It...

What to Think About How many times will the chain rule be applied?

Could we use a substitution method to make the problem simpler?

Your Turn ... Differentiate the following functions

y= (1 + cos3 4 x) 2

b) = y

sin 3 (2 − 3 x 2 )

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4.1 Practice For questions 1 to 5 determine dy . dx

y = sin 2 x

2. = y tan( x3 + 2 x)

 sin x  y=   1 − cos x 

y = − cos3 (tan 3 x)

4. = y sin 2 ( x3 − 3 x 2 )

x 6. What is the largest value of the slope of the curve y = sin   , [−4π , 4π ] ? 4

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4.2 Composite Functions and Function Notation Warm Up Given f ( x) = sin x and g ( x) = x 2 , determine a)

( f  g )( x) =

( g  f )( x) =

( g  f )(0) =

Given f ( x) =2 + x + x and g ( x)= 5 − x , determine

( f  g )(0) =

Does the order of function matter when using this notation?

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Example 1: Derivative of a Linear Composite Function Using two different methods, determine h′( x) if f ( x= ) 2 x + 3 , g ( x= ) 3 x − 1 and h( x) = f ( g ( x))

How to Do It...

What to Think About Is it advantageous to determine the expression for the composite function before differentiating?

Your Turn ... Determine h′( x) if f ( x= ) 3 x − 5 , g ( x= ) 5 x + 2 and h( x) = f ( g ( x))

More Notation for the Chain Rule Given the composite function = h( x) (= f  g )( x) f ( g ( x)) Then: = h′( x) ( f =  g )′( x) f ′( g ( x)) ⋅ g ′( x)

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Example 2: Derivative of a Non-Linear Composite Function ) u 3 + 1 and Given the following information determine the value of f’(x) at x = 1 given f (u= = u g= ( x) x

What to Think About

How to Do It...

What differentiation rule is required for this function?

Which notation would you prefer to use Leibniz or Newton?

What are the advantages of both?

Your Turn ... Determine f’(x) at the given value of x. f (u= ) u5 + 1 u =

x at x = 1

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Example 3: Calculate the Derivative Given a Table of Values Suppose: h( x) = f ( g ( x)) and the following table of values is true: Function f(x) g(x) f’(x) g’(x)

1 5 3 7 5

2 4 3 7 9

3 7 2 8 12

Determine: a)

h′(1)

h′( f ( x))

How to Do It...

h′(3)

What to Think About As h(x) is a composite function, what must be applied to derive it?

How can you write h(f(x)) solely in terms of a composite function in terms of and g(x)

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Your Turn ... Suppose the functions f and g and their derivatives have the following values at x = 2 and x = 3. x

f(x)

g(x)

f’(x)

g’(x)

1 4

−3

−4

3π

Evaluate the derivatives with respect to x of the following combinations at the given value of x. y 3 f ( x) + g ( x) at x = 3 a) =

y = f ( g ( x)) at x = 2

= y f ( x) ⋅ g ( x) at x = 3

f ( x) at x = 2 g ( x)

f ( x) at x = 2

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Example 4: Calculate the Derivative Given Specific Values Suppose h( x) = f ( g ( x)) , g (0) = 1 , f ′(1) = 3 , and h′(0) = 4 . What is g ′(0) ?

How to Do It...

What to Think About Do you need to know what the functions are to solve this type of question?

Your Turn ... a) If h( x) = f ( g ( x)) , g (0) = 2 , f ′(2) = 6 , and h′(0) = 4 , what is g ′(0) ?

b) If h( x) = f ( g 2 ( x)) , g (0) = 2 , f ′(4) = 3 , and h′(0) = 4 , what is g ′(0) ?

Did You Know? In a memoir by Gottfried W. Leibniz, it is believed to be the place that he attempted to use the Chain Rule to differentiate a polynomial inside of a square root. Although the memoir contained various errors, it is believed to the place in which the Chain rule originated. The Chain Rule is a relationship between three rates of changes, stating that the first is a product of the dy dy du other two. In Leibniz notation = is where y is a function ⋅ dx du dx of u, and u is a function of x. 226

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4.2 Practice Determine the value of ( f  g )′ at the given value of x.

f (u ) =u 3 + 1, u =g ( x) = x , x =1

1 1 1− ,u = 2 f (u ) = g ( x) = , x = 1− x u

πu f= (u ) tan = , u g= ( x) 3 = x, x 4 6

1 1 f (u ) = u + 2 ,u = g ( x) = π x, x = sin u 4

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Given that the functions f and g and their derivatives have the following values at x = 1 and x = 2.

f(x)

g(x)

f’(x)

g’(x)

1 3

−3

−4

2π

Calculate the derivative of the following.

2 f ( x) at x = 1

f ( x) + g ( x) at x = 2

f ( x) g ( x) at x = 2

f ( x) at x = 1

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4.3 Implicit Differentiation Warm Up Graph the following two functions on the grids provided. Determine the equations of their tangents at x = 2 and sketch them on the graph.

y = 2x

y = − 2x

What do you notice about the slopes of these two functions?

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Example 1: Implicit Differentiation of a Square Root Relation Given the relation y 2 = x , how would you determine the slope of the tangent at the points (4, 2) and (4, −2)? Graphically support your findings.

How to Do It...

What to Think About Is this equation a function?

What are the obstacles in differentiating a relation?

How can you differentiate a relation that has more than one point at an x value?

Is it possible to simply move left-to-right along the equation and differentiate each term with respect to x?

How is differentiating y 2 with respect to x similar to the chain rule?

What are the advantages of deriving this relation using implicit differentiation?

How would you graph this relation using a graphing calculator?

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Your Turn ... Given the relation y 4 = x , how would you determine the slope of the tangent at the points (16, 2) and (16, −2)? Graphically support your findings.

Example 2: Implicit Differentiation Involving a Circle 41 at the point (4, –5) using implicit differentiation. Find the slope of the circle x 2 + y 2 =

How to Do It...

What to Think About How do you write y explicitly in terms of x in order to derive it?

What does this look like graphically?

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Your Turn ... 25 at the point (−3, −4) using implicit differentiation. Find the slope of the circle x 2 + y 2 =

Note: Implicit differentiation requires differentiating each term with respect to x. Whenever a variable is involved in the equation that is a function of x, you can consider it as a function inside the relation and apply the CHAIN RULE. This is another way to think of implicit differentiation.

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Example 3: Implicit Differentiation Involving an Ellipse 25 at the point (2, 3) Find the tangent and normal to the ellipse x 2 − xy + 3 y 2 =

How to Do It...

What to Think About What differentiation rules must you apply to find the derivative implicitly?

What is the relationship between the slope of the tangent and the slope of the normal?

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Your Turn ... 32 at What are the equation of the tangent lines and normal lines to the ellipse 4 x 2 − 3xy + y 2 + 12 y − 8 = the point (1, 3)?

Example 4: Using Implicit Differentiation in a Proof Show that the slope dy is defined at every point on the graph of 3= y x 2 − sin y . dx

How to Do It...

What to Think About Which differentiation technique must be applied?

What is the link between continuity and differentiability?

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Your Turn ... Find dy of the following relations dx a)

x sin = y cos( x + y )

4 x 2 + 2 xy + x 2 y − 4 y = 0

Example 5: Implicit Differentiation Involving a Hyperbola 5 Finf y′ and y′′ if y 2 − x 2 =

How to Do It...

What to Think About Moving left-to-right and differentiating as you go, which rules do you need to apply to differentiate with respect to x?

What differentiation rule must you apply when taking the second derivative?

How could you simplify the second derivative?

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Your Turn ... 2

8 Find d y if y 2 − xy = dx 2

4.3 Practice For questions 1 to 5, determine dy dx 1.

x3 + y 3 = 6

x 2 y + xy 3 = 4

y2 =

x +1 x −1

x = sin y

x = tan y

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For questions 6 and 7 determine the slope of the curve at the indicated point.

x 2 + y 2 = 41, (−4,5)

x2 + y 2 = 25, (0,5)

For questions 8 to 10, determine the equation of the tangent line to the curve at the indicated point.

x 2 + xy − y 2 = 11, (3, 2)

10.

2 xy + π cos y = 0, (1, π )

x 2 y 2 = 36, (3, 2)

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1 11. Determine d y for the function x3 − y 3 = dx 2

16 , what is the value of d y at the point (4, 0)? 12. On the circle x 2 + y 2 = dx 2

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4.4 Derivatives of Inverse Functions Warm Up What are three characteristics of an inverse function?

2 Graph the linear function= y x + 2 in blue and its inverse in red. What do you notice about the 3 slopes of their tangents?

Suppose (a, b) is a point on the curve y = f (x) or, in other words, f (a) = b. Since an inverse function has the x and y coordinates interchanged, then (b, a) is a point on the curve y = f −1(x) where f −1(x) represents the inverse function.

Note: Use the applet to explore the relationship of the slopes at their inversed points. https://qrs.ly/etgge2w

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Definition: Derivative of inverse functions The relationship between the slope of the inverse functions at their “inversed” points is they are reciprocals of one another. The relationship between the slopes of the tangent line to f at (a, b) and the slope of the tangent line to f −1 at (b, a)? f ′(a ) =

1 ( f )′(b) −1

( f −1 )′(b) =

1 f ′(a )

Example 1 : Determine the Slope of the Inverse at a Point If f(3) = 7 and f’(3) = 9, what are f −1 (7) and ( f −1 )′(7) ?

How to Do It...

What to Think About What is the point on the function?

What is the corresponding point on the inverse function?

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Your Turn ... x

f(x)

f’(x) 1 4 3 5

Given the information in the table above, if g is the inverse of f, what is the slope of the tangent line to g at the point where x = 5?

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Example 2: Determine the Slope of the Inverse at a Given Point Let f ( x) =x3 + 3 x 2 − 4 , and let g be the inverse function. Evaluate g’(0).

How to Do It...

What to Think About What are the two different ways you can solve this problem?

What does it mean to solve by inspection?

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Your Turn ... a) Let f ( x) = x3 − x 2 − 1 , and let g be the inverse function. Evaluate g’(3).

) x3 + x , and let g be the inverse function. Evaluate g’(2). b) Let f ( x=

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4.4 Practice ′(1) =55 and f ′(2) = 3 , find the value of ( f −1 )′(2) . f (1) 2,= 1. Suppose that= , ff '(1)

2. Given the following function values for f, determine g’ (−4).

f(x)

−4

f’(x) 1 3 3 4

For questions 3 to 8, determine the derivative of the inverse at the indicated value for x. 3. y= x + 1 at y = 10 x 3

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= y 3 x − 5 x3 at y = −2

f ( x) = 3 x5 − 2 x3 + 2 x at f ( x) = 3

y= x + x 3 at y = 2

= y 4 x − x3 at y = 3

= y x 3 + x 5 at y = −2

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4.5 Derivatives of Exponential and Logarithmic Functions Warm Up Graph Y1 = e x . Then graph in bold the derivative of this function using the numerical derivative feature of your calculator: Y2 = nDeriv(Y1 , X , X )

What do you notice about the function and its derivative?

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Did You Know? e is an irrational number. This constant, called Euler’s Number, named after Leonhard Euler, to 9 decimal places equals 2.71828128. More formally as:  1 = e lim 1 +  x →∞  x

Chapter 4: Advanced Differentiation Techniques 247

Derivative of base e exponential functions (e x )′ = e x

(eu )=′ eu ⋅ u ′

x) Let u = f ( x) , then (e f (= )′ e f ( x ) ⋅ f ′( x)

Example 1: Derivative of the Exponential Function as Part of a Composite Function Find the derivative of y = e 2 x

How to Do It...

What to Think About What is the embedded function?

Your Turn ... Derive the following functions x

y = e5

y=e x

y = e7 −3x

y = e tan x

y = x 2e x

y = esin x

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Did You Know? e and Compound Interest n  r F P 1 = + Compound Interest is found given the formula    n where F = Future Value, P = Present Value, r = rate, n = number of payments. Substitute x = n and you will get: r xr

 1 F= lim P 1 +  = Pe r x →∞  x

This is why e appears in many banking formulas. It also hints at the idea that a banker, prior to a mathematician, discovered the value of e ! Necessity is the mother of all inventions.

Example 2: How Fast Does a Flu Spread? 100 The spread of a flu virus in a major city is modeled by the equation P(t ) = where P(t) is the 3− t 1 + e total number of citizens infected t days after the flu was first discovered.

a) Graph the function P(t) with the dimensions [−5, 10] by [−25, 120]. b) Estimate the initial number of citizens infected with the flu. c) How fast was the flu spreading after 3 days? d) Graph the derivative with the dimensions [−5, 10] by [−10, 30]; when will the flu spread at its maximum rate? What is this rate?

How to Do It...

What to Think About What point on the original graph identifies the maximum rate of the spread of the flu?

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Example 2: How Fast Does a Flu Spread? - Continued

What are two methods to find the maximum rate of the speed of the spread of the flu?

How fast is the flu spreading by day 30?

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Your Turn ... The growth of the Spanish Influenza (1918), in its first 4 months spread at a rate that can be modelled by the equation P(t ) = 300 where P is the number of citizens and t is the time in months. 1 + e5−t

a) Graph the function P(t) with the dimensions [−1, 6] by [0, 240].

Chapter 4: Advanced Differentiation Techniques 251

b) Estimate the initial number of citizens infected with the flu. c) How fast was the flu spreading after 3 days? d) Graph the derivative with the dimensions [−1, 6] by [0, 240]; w en will the flu spread at its maximum rate? What is this rate?

Did You Know? The Natural Logarithm In Latin, “logarithmus naturali”, denoted by the button on your calculator, is the inverse of e. y = ln x y = log e x

ey = x The Scottish mathematician, John Napier (1550 – 1617), nicknamed the Marvelous Merchiston, was studying the motion of someone covering a distance by dividing the time into short intervals of length. He coined the name for this relationship using the Greek words logos (ratio) and arithmos (number). This term became Latinized into the current version, logarithm. Napier is best known as the discoverer of the concept of logarithms which provided a platform for later scientific advances in astronomy, dynamics, and other branches of physics.

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Example 3: Prove the Derivative of the Natural Log Derive y = ln x

How to Do It...

What to Think About How can we rewrite the natural logarithm into a form that we can differentiate?

As the function y is not written explicitly in terms of x while in exponential form, what type of differentiation must be applied?

SUMMARY (ln x)′ =

1 x

(ln u )=′

1 ⋅ u′ u

Example 4: Application of the Derivative of the Natural Logarithm Find the derivative of y = ln x3

How to Do It...

What to Think About What is the embedded function?

Identify any restrictions

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Your Turn ... Determine the derivative of each function:

5 y = ln   x

= y ln(4 x − 12)

= y ln( x 2 + 9)

y = ln(sin x)

y = (ln x)3

y = x 2 ln x

y = ln(ln x)

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4.5 Practice Determine dy dx

y = e5x

y = 3e x

y=e

−4 x

y=e

= y xe x − x3e x

y = esin x

y=e x

y = ln x 3

y = ln 3 x

10.

y = ln(ln x 2 )

11.

ex y = ln x e +2

−

x 3

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12. At what point on the graph of= y y 3e x − 1 is the tangent line perpendicular to the line =

1 x + 4? 3

13. A line with slope m passes through the origin and is a tangent to the curve y = ln(3x) . What is the value of m?

14. Find the equation of the tangent line to the curve y = e3x and passes through the origin.

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15. Find the equation of the normal line to the curve y = − xe − x that passes through the origin.

Laws of Logarithms Equivalence ln x= y ⇔ _____ Log of 1 ln1 = _____

Recall eln x = _____

Proof:

Log of the Base ln e = _____ Log of a Product ln xy = _____ Log of a Quotient ln

x = _____ y

Log of a Power ln a x = _____

Change of Base log b a = _____

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4.6 Logarithmic Differentiation Warm Up Use the applet to visually identify the graph of the following exponential functions https://qrs.ly/i3gge2x x = y 2= , y 2.73x = & y 3x

All of the graphs have one common coordinate. What is this coordinate?

What happens to the graph of the derivative for the functions mentioned above as you change its base?

Why would the derivatives of their graphs be slightly different?

Definition: Logarithmic Differentiation The process of taking the natural logarithm of both sides of an equation, differentiating, and then solving for the desired derivative. This is applicable when ___________________________.

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Example 1: Both the Base and the Exponent are Variables What is the derivative of y = x x ?

How to Do It...

What to Think About When is logarithmic differentiation applied?

How is this function different from the function y = e x ?

Your Turn ... What is the derivative of y = x ln x ?

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Example 2: The Derivative of an Exponential Function With a Constant Base and a Variable Exponent What is the derivative of y = 5 x ? How to Do It...

What to Think About How does this function differ from the previous example?

Your Turn ... What is the derivative of y = 7 x +3 ?

We now have a new “short-cut” formula for taking the derivative of a function in the form y = a x where a is a constant. SUMMARY

260

(a x )′ = a x ln a

Chapter 4: Advanced Differentiation Techniques

= (a u )′ a u ln a ⋅ u ′

Example 3: Using the Laws of Logarithms At what point on the graph of the function = y 2 x − 5 does the tangent line have a slope of 10? Round all values to the nearest hundredth.

How to Do It...

What to Think About What will we have to equate to each other in order to solve this problem?

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Example 3: Using the Laws of Logarithms - continued

Your Turn ... At what x-value on the graph of the function y = −3x − 2 x does the tangent line have a slope of −8? Round all values to the nearest hundredth.

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SUMMARY (log a x)′ =

1 x ln a

′ (log= a u)

1 ⋅ u′ u ln a

Example 4: Develop the Derivative of a Common Logarithm Develop the derivative of y = log a x

What to Think About

How to Do It...

How can we determine the derivative of a logarithm without being given the formula?

What type of differentiation is required?

Your Turn ... What is the derivative of y = log 5 10 x ?

Did You Know? At a rock concert, the sound pressure can be measured with the logarithmic function 10 logW P= where W represents the sound (in watts) and W0 represents the lowest threshold of W0 sound that humans can detect. The application of the derivative of a common logarithm would determine the change in sound pressure (in decibels) over time.

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Example 5: The Derivative of a Logarithmic Composite Function What is the derivative of y = log 6 x ?

What to Think About

How to Do It...

Which is the embedded function?

Your Turn ... Determine the derivative of each function. a)

y = log a x 3

y = esin x

y = 5cos x

y = x sin x

y = xπ + π x + x x

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Example 6: A Tangent Through the Origin A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m?

How to Do It...

What to Think About What are two methods to determine the slope of the tangent line?

Your Turn ... A line with slope m passes through the origin and is tangent to the graph of the natural logarithm function y = ln2x . What is the value of m?

Chapter 4: Advanced Differentiation Techniques 265

4.6 Practice Determine dy for each equation. dx 1.

y = 5x

y = 3− x

y = 7 cos x

y = log 5 x 2

y = log8 x −2

y = ( log 2 x )

= y ln 5 ⋅ log 4 x

= y log 5 ( x 2 ln 5 + 5 ) 9.

10.

y = ln 7 x

= y xπ + π x

12.

y = x − 3 + x e + x 2+ e

11.

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1 log 3 x

13.

y = xy

14. = y x sin x , x > 0

15. Prove that the graph of y = lnx approaches a horizontal tangent as x → ∞

16. A glass of cold milk from the refrigerator is left on the counter on a warm summer day. Its temperature y (in degrees Fahrenheit) after sitting on the counter t minutes is.

= y 72 − 30(0.98)t a)

What is the temperature of the refrigerator? How can you tell?

What is the temperature of the room? How can you tell?

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4.7 Derivatives of Inverse Trigonometric Functions Warm Up Given the function f ( x) = x3 + x 2 + 1 , find ( f −1 )′(3)

Support your findings graphically using the Draw Inverse function on your calculator.

Determine the equation of the tangent to y = e x at x = 2 .

Determine the equation of the tangent to y = lnx at x = e .

Compare the slopes of the equations and make an inference about the relationship between the two graphs.

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Example 1: Derivative of the Arcsine Function a) Determine the inverse of the sine function written explicitly in terms of x b) Determine the derivative of the inverse of the sine function c) Graph the sine function and its inverse. Identify their domain and range

How to Do It...

What to Think About How do you write the inverse of sine explicitly in terms of x?

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Example 1: Derivative of the Arcsine Function - continued How would you write your derivative solely in terms of x?

How does this relate to the Pythagorean Theorem?

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SUMMARY x )′ (= sin −1 x )′ = ( Arcsin

1 1 − x2

−1 u )′ ( sin u )′ = = ( Arcsin

1 1− u2

⋅ u′

Your Turn ... a) Determine the inverse of the cosine function written explicitly in terms of x

b) Determine the derivative of the inverse of the cosine function c) Graph the cosine function and its inverse. Identify their domain and range

y = cos x

y = arccos x

Domain Range

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SUMMARY x )′ (= cos −1 x )′ = ( Arccos

−1 1 − x2

−1 = = u )′ ( cos u )′ ( Arccos

−1 1− u2

⋅ u′

Example 2: Application of the Derivative of a Trigonometric Inverse Determine:

d ( sin −1 x3 ) dx

How to Do It...

What to Think About Which is the embedded function?

Your Turn ... Determine:

272

d cos −1 x dx

(

)

Chapter 4: Advanced Differentiation Techniques

Example 3: Derivative of the Arctangent Function a) Determine the inverse of the tangent function written explicitly in terms of x. b) Determine the derivative of the inverse of the tangent function. π π c) Graph the restricted tangent function over the domain  − ,  and its inverse.  2 2 d) Identify their domain and range.

How to Do It...

What to Think About How do you write the inverse of tangent explicitly in terms of x?

How would you write your derivative solely in terms of x?

Chapter 4: Advanced Differentiation Techniques 273

Example 3: Derivative of the Arctangent Function - continued How does this relate to the Pythagorean Theorem?

SUMMARY −1 (Arctan = = x)′ (tan x)′

274

1 x +1 2

Chapter 4: Advanced Differentiation Techniques

1 u )′ (tan −= u )′ ( Arctan=

1 ⋅ u′ u +1 2

Your Turn ... a)

Determine the inverse of the cotangent function written explicitly in terms of x.

Determine the derivative of the inverse of the cotangent function.

Chapter 4: Advanced Differentiation Techniques 275

c) Graph the restricted tangent function over the domain ( 0, π ) and its inverse. Identify their domain and range.

y = cot x Domain Range

−

π 2

<x<

y = arccot x

y∈R

π 2

x∈R

−

π 2

<x<

π 2

SUMMARY

cot −1 x )′ x )′ (= = ( Arccot

276

−1 x +1 2

1 u )′ (tan −= u )′ ( Arctan=

Chapter 4: Advanced Differentiation Techniques

1 ⋅ u′ u +1 2

Example 4: Inverse Trigonometric Functions and Rectilinear Motion A particle moves along a line so that its position at any time t ≥ 0 is s (t ) = tan −1 4t . What is the velocity of the particle when t = 2?

How to Do It...

What to Think About Can you use the Chain Rule?

What is the relationship between the position function and the velocity function?

Your Turn ... 1 A particle moves along a line so that its position at any time t ≥ 0 is s (t ) = cos −1  2  . t  What is the velocity of the particle when t = 3?

Chapter 4: Advanced Differentiation Techniques 277

Example 5: Inverse Trigonometric Functions and Their Tangent Lines π  a) Determine an equation for the line tangent to the graph of y = tan x at the point  ,1 . 4   π b) Determine an equation for the line tangent to the graph of y = tan −1 x at the point 1,  .  4

How to Do It...

What to Think About Before you try to solve it algebraically what is the relationship between their slopes?

278

Chapter 4: Advanced Differentiation Techniques

SUMMARY If the equation of the tangent line to f at (a, f (a )) is y − f (a )= f ′(a )( x − a ) then the equation of 1 ( x − f (a )) . f ′(a )

the tangent line to f −1 at ( f (a ), a) is = y−a

Your Turn ... a)

π 3  Determine an equation for the line tangent to the graph of y = sin x at the point  , 

Determine an equation for the line tangent to the graph of y = sin −1 x at the point 

3

2 

 3 π ,   2 3

SUMMARY Derivative of the inverse trigonometric functions. −1 u )′ (cos =

= (sec −1 u )′

−1 1− u

⋅ u′

1 u u2 −1

⋅ u′

1 (cot −= u )′

= (csc −1 u )′

−1 ⋅ u′ u +1 2

−1 u u2 −1

⋅ u′

Chapter 4: Advanced Differentiation Techniques 279

4.7 Practice For questions 10 to 12 determine

dy dx

y = sin −1 ( x 2 )

y = cos −1 ( 2 x )

1 y = sin −1   x

y = cot −1 ( x )

 1  y = tan −1  2  x 

280

Chapter 4: Advanced Differentiation Techniques

1 sin (5 x) −1

7. A particle moves along an x-axis such that its position is designated by the function −1  t  = x(t ) sin =   , t 3 , where t represents the time in seconds. Find the velocity at the time indicated. 4

Chapter 4: Advanced Differentiation Techniques 281

4.8 L’Hôpital’s Rule Imagine you’re trying to figure out what happens when you divide zero by zero, or infinity by infinity. It seems like a mathematical dead end, right? That’s where L’Hôpital’s Rule comes to the rescue! This cool calculus rule helps us evaluate limits that initially look impossible. If you have a fraction where both the top and bottom are heading towards zero or both are growing infinitely large, L’Hôpital’s Rule says you can take the derivative (that is, find the rate of change) of the top and the derivative of the bottom separately, and then try the limit again. It’s like giving the numerator and denominator a little nudge to see which one “wins” the race to zero or infinity. By finding their individual rates of change, we can often determine the limit of the original fraction, turning a seemingly unsolvable problem into something much easier to handle. This rule is a powerful shortcut in calculus and helps us understand the behavior of functions when they approach tricky values. A practical example of L’Hôpital’s Rule can come from problem called the disappearing fraction. Imagine you have the fraction (x² - 1) / (x - 1). If you try to plug in x = 1 directly, you get (1² - 1) / (1 - 1) = 0/0, which is an indeterminate form. It’s like the fraction disappears! But L’Hôpital’s Rule can help. 1. Take the derivative of the top: The derivative of x² - 1 is 2x. 2. Take the derivative of the bottom: The derivative of x - 1 is 1. Now you have a new fraction: 2x / 1, or just 2x. Now try plugging in x = 1: you get 2 * 1 = 2. So, even though the original fraction seemed to disappear at x = 1, L’Hôpital’s Rule shows us that the limit as x approaches 1 is actually 2. This means that if you were to graph the function (x² - 1) / (x - 1), there would be a “hole” at x = 1, but the function would be approaching the y-value of 2. L’Hôpital’s Rule involves taking derivatives of the numerator and denominator. What do these derivatives tell us about the original fraction, and why does comparing them help us find the limit?

282

Chapter 4: Advanced Differentiation Techniques

Example 1: Applying L’Hôpital’s Rule Evaluate lim x →0

sin x x

How to Do It...

What to Think About Do you recall this limit?

Which indeterminate form do you have?

Why do you believe L’Hôpital’s Rule to be true?

Your Turn ... What is lim x→2

7+ x −3 x−2

Chapter 4: Advanced Differentiation Techniques 283

Example 2: A Second Application of L’Hôpital’s Rule e2 x + x 2 x →∞ e x + 4 x

Evaluate lim

How to Do It...

What to Think About Which indeterminate form do you have?

Your Turn ... ex x →∞ x 2

Determine lim

284

Chapter 4: Advanced Differentiation Techniques

Did You Know? For Sale – Mathematical Respect £300! As it turns out, L’Hôpital’s rule was not actually discovered by Guillaume de L’Hôpital, but the Swiss mathematician Johann Bernoulli! In 1691, the acclaimed French mathematician met with the talented, yet widely unknown entity, Bernoulli. L’Hôpital, recognizing the young mathematician’s obvious talents and wanting to learn from him, employed him to give private lessons for an initial retainer of 300 pounds under the caveat that none of their work would be disclosed and exclusive right to any of Bernoulli’s discoveries during that time. Under this business agreement, the rule was discovered in 1964 by Bernoulli, but not published until 1969 in L’Hôpital’s calculus textbook entitled Analise Des Infiniments Petits (which included a translated copy of the arrangement they entered into collegially). L’Hopital did acknowledge the input of Bernoulli in a letter to Leibniz regarding the rule that bears his name!

4.8 Practice Determine each limit

x2 − 9 lim x →3 x − 3

ex − 5 lim x →0 x

lim

ln x x→2 x − 2

lim

sin 2 x x → 0 sin 3 x

Chapter 4: Advanced Differentiation Techniques 285

lim

e x + e− x − 2 x → 0 1 − cos x

lim

2 − x 2 − 2 cos x x →0 x4

1 − cos x x →0 x2

lim

286

a x −1 x →0 x

lim+

x 2 −1 x →1 ln(4 − 3 x )

Chapter 4: Advanced Differentiation Techniques

4.9 Chapter Review For questions 1 to 8, determine

dy dx

y = cos(tan x)

= y sin 1 − x3

y = (sec 3 x) −1

y = tan 2 (sin x)

 sin x  y=   1 − cos x 

y = cot 3 x sin x

 1 + sin x  y=   1 − cos x 

tan 2 x 1 + sec 2 x

Chapter 4: Advanced Differentiation Techniques 287

π  9. Determine the equations of the tangent line to the curve y = 2sin  x  at x = 1 3 

= y sin( x − sin x) have horizontal asymptotes over the interval [−3,3] ? 10. Does the graph of

Use the following information to answer questions 11 to 16. Given that the functions f and g and their derivatives have the following values at x = 3 and x = 4

11.

288

f(x)

g(x)

f’(x)

g’(x)

−6

2 3

−4

2π

−4

−1

−7

1 2

−4

−8

−3

12.

f ( x) − g ( x) at x = 3

3 f ( x) at x = 4

Chapter 4: Advanced Differentiation Techniques

13.

2 f ( x) g ( x) at x = 4

14.

 f ( x)  at x = 3    g ( x) 

15.

g ( x) at x = 3

16.

1 at x = 4 g ( x)

18.

3 x 2 − 4 y + xy 2 = 5

For questions 17 to 19, determine

17.

xy = 7

19.

y2 =

dy dx

x2 x −5

Chapter 4: Advanced Differentiation Techniques 289

For questions 20 and 21, determine

20.

d2y dx 2

x3 + y 3 = 7

21.

y 2 − xy = 8

For questions 22 to 24 determine the equation of the tangent line to the curve at the indicated point.

22.

x2 + 3 y 2 = 12 at (3, 1)

24.

 π 2 xy + π sin y = 2π at 1,   2

290

Chapter 4: Advanced Differentiation Techniques

23.

x + xy = 12 at (9, 1)

f(x)

f’(x)

−1 42 3

Use the table above to answer questions 25 and 26.

25. Given that g is the inverse of f, determine g’(2)

26. Given that g is the inverse of f ( x) = 3 x5 + 2 x + 5 , determine g’(10)

Chapter 4: Advanced Differentiation Techniques 291

Differentiate the following functions:

−2

27.

y = 3e x

29.

y = ln 3 6 x

31.= y x 2 ln( x 2 + 3 x)

33.

y = ln( x + x 2 + a 2 )

35.

y=x x

292

Chapter 4: Advanced Differentiation Techniques

28.

y = esin 2 x

30.= y ln(csc x − cot x)

32.

x  y = ln  tan  2 

34.

y = ln(11x )

36. On what interval(s) is f ′( x) ≥ 0 for f ( x) = x3e 2 x ?

0? 37. For what values of x is y = e 2 x parallel to 12 x − 2 y − 10 =

x 38. What is the equation of the tangent line to the curve y = 3arccos   at the point (1, π ) ? 2

39. You are attempting an inside loop with your new RC model airplane. It is following the route represented 3 2 9 . At what rate is it climbing at the moment represented by the coordinate (1, 2)? by x + 6 xy − y =

Chapter 4: Advanced Differentiation Techniques 293

Evaluate the following limits

40.

lim

5x x →∞ e x

41.

lim

42.

lim

sin 5 x x →0 x

43.

lim

44.

294

e x − e− x x →∞ x

1 − sin x x → 1 + cos 2 x π

ln x x →∞ x

lim

Chapter 4: Advanced Differentiation Techniques

Unit 1 Project - Design a Roller Coaster

You have been hired to design a new roller coaster for Belmont Park in San Diego. Your coaster must 4π satisfy the following safety conditions: no descent can be over , the design must begin with a 9 π incline, never get as high as its previous high point and end at the same height it starts. Write a 4 summary and present your results to the class as if you were trying to convince them to invest in and build your coaster, since your design is the best!

Questions 1. In small groups discuss some possible themes for your design. From your experiences discuss what elements you think make a good roller coaster. Come up with a plan to design the world’s newest roller coaster! You must include the following: • •

Functions you use must be differentiable and continuous at all points along the curve of the roller coaster. Use at least 5 distinct functions. You need to show the graph and a proof that the function you generate is continuous and differentiable at the points where each pair of curves meet. You may use any type of function including linear, trigonometric, exponential, logarithmic, inverse trigonometric, and polynomial.

2. Research the best roller coasters in the world, answer some of the following questions (and get some inspiration for designing your own roller coaster): • • • • • •

What makes them great? Where are they? What kind of names do they have? How much does it cost to ride them? How much did it cost to build them? How fast do they go? If you could ride one roller coaster in the world, which one would you most like to ride and why?

Unit 1 Project: Roller Coaster Design

295

3. Choose a theme for your coaster and transfer your design to a piece of poster paper. It can be drawn freehand, but it should look like what you have drawn on Desmos. Tape or glue several pieces of poster paper together horizontally. You are encouraged to use Desmos or any other graphing application to print your functions. Make sure that you use proper scaling to ensure the shape is accurate when enlarged • • •

Cut out the roller coaster so that the top of it is your track. Then tape or glue one end of the poster paper to the other end. You will end up with a 3-dimensional representation of your roller coaster. Below each of your curves, write its function. Use a heavy dot to show the transition points. On the poster paper itself, show that the curves are continuous and differentiable at those points.

4. Answer the following questions: • Where is the path increasing and decreasing? • Where is it concave up and down? • For each fall, where is the steepest descent and how steep is the angle? • Draw a graph of the slope of the path verses the distance along the ground from the start (graph the first derivative). • Draw a graph of the rate of change of the slope (graph the second derivative). • The thrill of the roller coaster is defined as the sum of the angle of steepest descent in each fall (in radians) plus the number of tops. Calculate the thrill of each coaster. • Approximate how fast your best roller coaster will travel, showing your assumptions and calculations

296

Unit 1 Project: Roller Coaster Design

Unit 2: Applications of Derivatives This unit focuses on the following AP Big Idea from the College Board:

• Big Idea 2: Applications of the derivative include analyzing the graph of a function (for

example, determining whether a function is increasing or decreasing and finding concavity and extreme values), and solving a variety of real-world applications including optimization.

By the end of this unit, you should be able to:

• Use derivatives to analyze properties of a function • Solve problems involving optimization • Understand and apply the Mean Value Theorem • Connect the behavior of a differentiable function over an interval to the behavior of the derivative of

that function at a particular point in the interval through the Mean Value Theorem.

• Increasing • Decreasing • Maximum • Minimum • Horizontal inflection point • Vertical inflection point • Critical point • Concavity

• Extreme Value • Inflection point • Mean Value Theorem • Rolle’s Theorem • Optimization • First derivative test • Second derivative test

At what rate does the surface area of a spherical object d (V ) volume ? dt © Edvantage Interactive 2025

d ( SA) change in relation to the change in its dt

Chapter 5: Analyzing Functions Using Derivatives 297

Chapter 5: Analyzing Functions Using Derivatives Learn how to predict the shape of any graph

We understand our world by observing change. In almost every field of study, an observer can collect data and analyze the change behavior of the situation. Knowing how something is changing over time allows the researcher to make prediction about the future and the past. Mathematicians can create models (or functions) of the rates of change data; and these can be used to discover the conditions that would optimize the situation. Optimization is a key application of the derivative.

EXPLORING THE BIG IDEA

In this chapter you should be able to: • Use derivatives to analyze properties of a function. • Solve problems involving optimization. • Understand and apply the Mean Value Theorem. • Connect the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval through the Mean Value Theorem.

298

Chapter 5: Analyzing Functions Using Derivatives

5.1 The First Derivative Test Warm Up Given the graph of f’

Determine the following characteristics of f

Characteristics of f f is increasing on f is decreasing on f is neither increasing or decreasing at

Interval or values

Characteristics of f‘ Since f’ Since f’ Since f’

Sketch a graph of f that passes through (0, 0) and (2, –4) with zeroes at x = 0 and 3.

Is there a way to determine where the maximum and minimum values of a function occur?

Definition: Maximum, Minimum and Extreme Value

• An extreme value of a function f is called a relative (local) maximum if there is an open interval containing c on which f (c) is a maximum value.

• An extreme value of a function f is called a relative (local) minimum if there is an open interval containing c on which f (c) is a minimum value.

• If f ′(c) = 0 or if f’ is undefined at c, then c is called a critical point of f. © Edvantage Interactive 2025

Chapter 5: Analyzing Functions Using Derivatives 299

Example 1: Determining the Extreme Values of a Function Given f ( x= ) x 3 − 3 x 2 , find the maximum and minimum values of f using the derivative.

How to Do It...

What to Think About What information do we need to determine the behavior of f?

What are the critical points?

What intervals are defined by the critical points?

How do you determine whether f is increasing or decreasing over the interval?

Could you use the graph of to determine the intervals where f is increasing or decreasing?

What are the extreme values of at each critical point?

300

Chapter 5: Analyzing Functions Using Derivatives

Your Turn ... Find the extreme values for each of the following functions. f ( x= ) x4 − 4 x2

f ( x) = x 3

Definition: The First Derivative Test Let c be a critical point of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then f (c) is a: o relative (local) maximum if f ′( x) changes from positive to negative at c. o

relative (local) minimum if f ′( x) changes from negative to positive at c.

Let a be the left end-point of a function f that is continuous on a closed interval [a, b] f (a ) is a relative (local) maximum if lim f ′( x) < 0 . o + x→a

f (a ) is a relative (local) minimum if lim f ′( x) > 0 . x→a+

Let b be the right end-point of a function f that is continuous on a closed interval [a, b] f (b) is a relative (local) maximum if lim f ′( x) > 0 . o − x →b

f (b) is a relative (local) minimum if lim f ′( x) < 0 .

x →b −

Chapter 5: Analyzing Functions Using Derivatives 301

Example 2: Determining the Extreme Values of a Function on a Closed Interval Given f ( x= ) x 4 − 2 x 3 defined on the closed interval [–2, 2] find the extreme values of f.

How to Do It...

What to Think About What are the zeroes (x-intercepts) of f ( x= ) x 4 − 2 x3 ?

What is the y-intercept of f?

What are the endpoints of f?

What does the rate of change information provided by the derivative function indicate about the behavior of f at each value of x ?

302

Chapter 5: Analyzing Functions Using Derivatives

Example 2: Determining the Extreme Values of a Function on a Closed Interval continued Why is there not an extreme value at every critical point?

What happens at x = 0?

Are there any values on [–2, 2]where f has an overall maximum or minimum?

Chapter 5: Analyzing Functions Using Derivatives 303

Definition: Maximum and Minimum An extreme value of a function f is called an absolute (global) maximum if there is a point c in the domain D of f such that f (c) ≥ f ( x) for all x ∈ D . Similarly, an absolute (global) minimum occurs at c if f (c) ≤ f ( x) for all x ∈ D .

Your Turn ... Find the extreme values for each of the following functions. 2

f= ( x) x 3 ( x − 5) 0n [−1,8]

5 − 2 x 2 , f ( x) =   x + 2,

x ≤1 x >1

Theorem: Extreme Value Theorem If a function f is continuous on a closed interval [a, b] then there exists both an absolute maximum and minimum on the interval.

304

Chapter 5: Analyzing Functions Using Derivatives

5.1 Practice In questions 1 to 3, identify each x - value at which any absolute extrema occur. 1.

Did You Know? In 1637 mathematician Pierre de Fermat, in his work Methodus ad Disquirendam Maximam et Minimam developed a method to find local maxima, minima, and tangets to various curves that was equivalent to differential calculus. This was five years before Sir Isaac Newton was born!

Chapter 5: Analyzing Functions Using Derivatives 305

In questions 4 to 7, find the extreme values of the function on the given interval.

= y

1 + ln 3 x, (0, 4] x2

π   3π   6. = y cos  x +  ,  0,  3  2  

= y e 2 x ,[−2, 4]

= y x 3 , (−1,8]

In questions 8 to 12, find the extreme values of the function.

10.

12.

306

y = x2 − 4 x + 7

y = 2 x3 + 3 x 2 − 12 x + 7

x2 − 9

11.

= y

1 4 − x2

y = 3x 4 + 4 x3 − 6 x 2 − 5

Chapter 5: Analyzing Functions Using Derivatives

In questions 13 to 15, identify all critical points and determine local extreme values.

13.

15.

= y x 2 (6 − x)

14.

= y x 9 − x2

4 − x 2 , −3 ≤ x < 0 y= 2 x − 1, 0 ≤ x ≤ 2

16. If f (c) is a local minimum on the continuous function f over the open interval [a,b] then f ′(c) = 0 . Determine a counter example to this statement.

17. Find the absolute maximum on the function y = ( x 2 − 1)3 ,[−1, 2] .

Chapter 5: Analyzing Functions Using Derivatives 307

18. Which of the following functions contains exactly two extreme values?

y= x − 5

y = x3 − 9 x + 7

y = x3 + 9 x + 7

y = tan 2 x

y= x + ln 3 x

308

Chapter 5: Analyzing Functions Using Derivatives

5.2 Modelling and Optimization Warm Up A rancher has 200 m of fencing with which to enclose three adjacent rectangular pens. They border a straight river and do not require fencing along the river. What dimensions should be used so that the enclosed area is a maximum?

Draw a diagram and label the quantities relevant to the problem.

What are the possible quantities for the variables used for the dimensions? (What are the restrictions on the variables?)

Write a system of equations to describe the situation and the quantity to be maximized or minimized.

Reduce the system to one equation involving the quantity to be maximized or minimized and one independent variable.

What kind of equation is the reduced system?

Change the equation from standard form to vertex form.

How else could you find the extreme value of the situation?

Chapter 5: Analyzing Functions Using Derivatives 309

Example 1: Optimization Problems An open topped box is created by taking a rectangular piece of cardboard measuring 16cm by 30cm and cutting out congruent squares from each corner so that the sides can be folded upwards to create the box. What size should the squares be to maximize the volume of the box?

How to Do It...

What to Think About What does this situation look like?

What are the variables?

What equation can be used to determine the volume?

The first derivative test allows us to determine extreme values.

Are there any restrictions on the variables?

What happens at the remaining critical point?

310

Chapter 5: Analyzing Functions Using Derivatives

Example 1: Optimization Problems - continued

Your Turn ... The yearbook pages will have an area of 672 cm2 with top and side margins of 3 cm and bottom margin of 4 cm. What page dimensions will create the greatest possible area available for pictures.

Chapter 5: Analyzing Functions Using Derivatives 311

Example 2: Optimizing Using the Coordinate Plane Find the area of the largest rectangle that can be inscribed in a semicircle of radius 2 cm.

How to Do It...

What to Think About How can a coordinate plane be used to help model the situation?

What equations can be used to model the situation?

What are the restrictions?

Your Turn ... Find the area of the largest rectangle that can be inscribed between the x-axis and under the curve y= 4 − x 2 .

312

Chapter 5: Analyzing Functions Using Derivatives

5.3 Practice 1. A rectangular plot is to be fenced using two kinds of fencing. The opposite sides will use chain-link fencing selling for $3 a metre, while the other two sides will use wood panels that cost $2 a metre. What are the dimensions of the rectangular plot of greatest area that can be fenced at a cost of $6,000?

2. A rectangular plot of land is bounded on one side by a river and on the other three sides by barbedwire fence made with two strands of wire. If you have 2400 m of barbed wire what is the largest area that can be enclosed? What are the dimensions of the largest rectangular plot of land in this situation?

3. Your company has obtained a contract to build a square-based, open-topped rectangular tank for storing water. The contract states that the tank must have a volume of 500 ft3; be made from stainless steel; and, be as light as possible. What are the dimensions of the tank that meet these requirements?

Chapter 5: Analyzing Functions Using Derivatives 313

4. Two sides of a triangle have lengths x and y. The angle, in radians, between the two sides is ϑ . What is 1 the measure of the angle that results in the largest possible area of the triangle? (NB : A = ab sin C ) 2

5. A rectangle has its two lower corners on the x-axis and it two upper corners on the curve y= 9 − x 2 . What are the dimensions that create a rectangle with the largest area?

6. An open box is to be made from a 3 cm by 8 cm piece of metal by cutting squares of equal size from each corner and bending up the sides to make a box. What is the maximum volume of the box?

314

Chapter 5: Analyzing Functions Using Derivatives

7. Find the lengths of the sides of an isosceles triangle that has a perimeter of 12 inches and has the maximum area.

8. What are two positive numbers with a product of 200 such that the sum of one number and twice the second number is as small as possible?

9. Challenge: The trough in the figure is to be made to the dimensions shown. Only the angle θ can be varied. What value of θ will maximize the trough’s volume?

Chapter 5: Analyzing Functions Using Derivatives 315

5.3 The Second Derivative Test Warm Up a) Given g ( x= ) 4 x − x 2 , graph g’ and use the first derivative test to sketch g(x).

b) Let g ( x) = f '( x) . If f (0) = −2 , use the first derivative test to sketch f.

At which point does f change from curving up to curving down?

Definition: Inflection Point Concavity is the word used to describe the curvature of a function. Intervals can be described as concave up (“smile”) or concave down (“frown”). The concavity of a function changes at a point of inflection.

316

Chapter 5: Analyzing Functions Using Derivatives

Example 1: Determining the Inflection Points of a Function ) x3 − 3 x 2 , find the point(s) of inflection of f. Given f ( x=

How to Do It...

What to Think About How does the graph of f behave?

What function tells us how the slope of f changes overtime?

The second derivative of f is the first derivative of f’. We can find the critical points of f’ to find the extreme rates of change of f.

What characteristics on f do the critical points of f’ indicate?

What connection can you find between the second derivative and the behavior of f?

Chapter 5: Analyzing Functions Using Derivatives 317

Definition: Concavity, Points of Inflection, and the Second Derivative Let be a critical point of a function f’ that corresponds to a differentiable function f that is continuous on an open interval containing c. If f ′′(c) > 0 , then f’ is increasing and f is concave up. ° If f ′′(c) < 0 , then f’ is decreasing and f is concave down. ° If f ′′(c) = 0 , then f’ has a critical point and f may have a point of inflection at c. °

f ′′(c) > 0 and lim f ′′(c) < 0 , or vice versa, then f’’ changes sign at c. • If xlim →c x →c −

Thus, is a point of inflection on f. • Note: when f’’ changes sign, f’ changes from increasing to decreasing or vice versa and f changes from concave up to concave down or vice versa.

Your Turn ... Find the points of inflection and intervals of concavity for each of the following functions.

= f ( x) 2sin x + cos 2 x for 0 ≤ x ≤

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f ( x) = xe x

Example 2: Determining the Extreme Values of a Function Using the Second Derivative Test 2 −x

a) Given f ( x) = x e , find the extreme values of f using the second derivative test. b) Find the inflection points of f.

How to Do It...

What to Think About How can we use concavity to determine the behavior of f at each critical point?

If f ′′ > 0 , then how are f’ and f behaving?

If f ′′ < 0 , then how are f’ and f behaving?

What about at the critical point?

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Example 2: Determining the Extreme Values of a Function Using the Second Derivative Test - continued

How do you know if f will have an inflection point?

Theorem: Second Derivative Test for finding Local Extrema Let f be a function such that f ′(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f ′′(c) > 0 , then f(c) is a relative minimum. 2. If f ′′(c) < 0 , then f(c) is a relative minimum. If f ′′(c) = 0 , then then tests fails. In such cases, use the first derivative test.

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Your Turn ...

−3 x + 5 x using the second derivative test. Justify your Find and identify each extreme value of f ( x) = reasoning.

5.3 Practice In questions 1 and 2, use the First Derivative Test to determine the extreme values of the function. 1.

= y x 6 − x2

 x 2 + 2, x ≥ 0 y= 5 − x, x < 0

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In questions 3 and 4, use the second derivative to determine the intervals on which the function is concave up and concave down.

x3 = y − x2 6

1 4

= y 3x + 5

= y x 3 ( x − 5)

Find the points of inflection for the following functions 2

y =15 + 2 x − x

For questions 7 to 9, use the graph to estimate where f ′′ > 0 , f ′′ < 0 and f ′′ = 0 . 7.

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Chapter 5: Analyzing Functions Using Derivatives 323

10. The graphs of the first and second derivative of a function y = f ( x) are shown below. On the same diagram, sketch a possible graph of the function f.

y = f ′′( x) y = f ′( x)

11. If a differentiable function f has an interior point c such that f ′(c) = 0 in its domain, must f have a local maximum or minimum at x = c? Explain.

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12. Sketch the following function on [2, 5] x f f’ f’’

2 0 4 0

3 3 0 −2

4 0 DNE DNE

5 −4 −5 0

13. If f ′′(c) = 0 , must there be a point of inflection at x = c? Explain.

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14. If a > 0, find the interval on which the function y = ax3 + 3 x 2 − 8 x + 7 is concave up.

15. How many inflection points does the function y = x 5 − 5 x 4 + 3 x − 9 have?

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5.4 Curve Sketching Warm Up Use the following information to sketch a possible graph of g. a) g is an even function and is continuous on [−3, 3]. x g g’ g’’ x g g’ g’’

0<x<1 + − +

1<x<2 − − −

0 2 DNE DNE

1 0 0 0

2 −1 DNE DNE

2<x<3 − − + 3 −3 −1 0

lim g ′(c) = −∞

x → 2−

lim g ′(c) = −∞

x → 2+

b) Determine the absolute extrema on [−3, 3]. Justify your reasoning. What is happening at each point of inflection?

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Definition: Horizontal and vertical inflection point A horizontal inflection point occurs when f’’ changes sign at a point c and both f ′′(c) = 0 and f ′(c) = 0 . A vertical inflection point is a special case where f (c) exists but both f ′′(c) and f ′(c) do not exist, where f has a vertical tangent line at point c. Note f’’ must also change sign at c.

Example 1: Determining the Inflection Points of a Function Given the graph of f’ to the right a) Over what intervals is the graph of f increasing? b) At what value(s) of x should f have a relative maximum? c) Over what interval(s) is f concave up? d) Sketch a possible graph for f .

How to Do It...

What to Think About What are the critical points of f? What does the behaviour of f’ tell us about f? What do the extreme values of f’ tell us about f’’?

What does the behaviour of f’ tell us about f’’? What function tells us how the slope of f changes over time? What assumptions was made when sketching the graph f?

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Your Turn ... Given the graph of f’ to the right: a) Over what intervals is the graph of f increasing?

b) At what value(s) of x should f have a relative minimum?

c) What behavior does f have at x = 0?

d) Over what interval(s) is f concave up?

e) What are the inflection points of f?

f ) Sketch a rough graph of a possible f.

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Example 2: Sketch the Curve Without a Calculator 2 −x

Given f ( x) = x e , from example 2 in section 5.3, sketch the curve by including all characteristics of the function.

How to Do It...

What to Think About What are the critical points?

What does the first derivative tell you about f?

What does the second derivative tell you about f?

Where do possible inflection points occur?

How do you know if f will have an inflection point?

Which term of f dominates the behavior as x → −∞ ?

Which term of f dominates the behavior as x → ∞ ?

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Your Turn ...

Sketch the curve of f ( x) = 4 xe 2 by including all characteristics of the function.

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5.4 Practice 1. Sketch a continuous curve y = f ( x) with the following properties. Remember to label appropriate coordinates.

•

f (−3) = 7

•

f ′( x) > 0 for x > 3

•

f (0) = 3

•

f ′( x) < 0 for x < 3

•

f (3) = −1

•

f ′′( x) < 0 for x < 0

•

′(3) f= ′(3) 0 f=

•

f ′′( x) > 0 for x > 0

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2. Sketch a continuous curve y = f ( x) with the following properties. Remember to label appropriate coordinates. x x<3 3 3<x<6 6 6<x<8 8 x>8

y 2 5 8

Curve Decreasing, concave up Horizontal tangent Increasing, concave up Inflection point Increasing, concave down Horizontal tangent Decreasing, concave down

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3. Use the graph of f’, where the domain is [0,3) ∪ (4,5] , to estimate the intervals on which the continuous function f is: a. increasing

b. decreasing

c. estimate the x-coordinates of all local extreme values.

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4. f is continuous on [1, 4] and satisfies the following: x f f’ f’

1 0 2 0 x f f’ f’

2 2 0 −1 1<x<2 + + −

3 0 Does not exist Does not exist 2<x<3 + − −

4 −3 −1 0 3<x<4 − − −

a. What are the absolute extrema of f and where do they occur?

b. What are the points of inflection?

c. Sketch a possible graph of f.

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5.5 The Mean Value Theorem Warm Up Sketch a differentiable function that passes through the points (–2, 4) and (5, 4). Is there at least one point on the graph where the derivative is zero? What is the average rate of change on the interval? Is it possible to graph a differentiable function that passes through both points without at least one point where the derivative is zero?

Sketch a differentiable function that passes through the points (–9, 8) and (7, –4). Is there at least one point on the graph where the derivative is zero? What is the average rate of change on the interval? Is there at least one point on the graph where the derivative is equal to the slope of the secant line? Is it possible to graph a differentiable function that passes through both points without at least one point where the slope of the curve is equal to the slope of the secant line? Draw the secant line between both endpoints.

How could you write these two situations using a mathematical equation?

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Definition: Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then f (b) − f (a ) there exists at least one x-value c ∈ (a, b) such that f ′(c) = . b−a

Definition: Rolle’s Theorem (a special case of the MVT) If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and f (a ) = f (b) then there exists at least one x-value c ∈ (a, b) such that f ′(c) = 0 .

Did You Know? Joseph Louis Lagrange lived from 1736 – 1813. By the age of 19, he was appointed professor of mathematics at the Royal Artillery School in Turin. Lagrange’s mean value theorem has a clear physical interpretation. If we assume that f(t) represents the position of a body moving along a line, f (b) − f (a ) depending on time t, then the ratio of is the average b−a velocity of the body in the period of time b – a. Since f ′(t ) is the instantaneous velocity, this theorem means that there exists a moment of time in which the instantaneous speed is equal to the average speed. © Edvantage Interactive 2025

Chapter 5: Analyzing Functions Using Derivatives 337

Example 1: Using the Mean Value Theorem ) x 3 + 1 on the interval [1, 2], there exists at least one point such that f ′(c) = 7 . Show that for f ( x= Justify your reasoning.

How to Do It...

What to Think About What conditions must be met to use the Mean Value Theorem?

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Your Turn ... Two stationary police vehicles equipped with radar are 8 kilometers apart on a highway. As a logging truck passes the first police car, its speed is clocked at 90 km/hr. Four minutes later, when the truck passes the second police car, its speed is clocked at 80 km/hr. Prove that the truck driver must have exceeded the speed limit of 110 km/hr at some point during the 4 minutes.

Chapter 5: Analyzing Functions Using Derivatives 339

Example 2: Necessary Conditions for the Mean Value Theorem and Rolle’s Theorem “If the graph of a function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal”. Is this statement always true, sometimes true or never true?

How to Do It...

What to Think About What conditions results in the Mean Value Theorem not applying?

Your Turn ... True or False? (justify your answer) a) The Mean Value Theorem can be applied to f ( x) =

1 on the interval [−1, 1]. x

b) If the graph of a polynomial function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal.

c) If f ′( x) = 0 for all x in the domain of f, then f is a constant function.

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5.5 Practice For questions 1 to 4, state whether the mean value theorem is applicable on the given interval and the value of c for which it applies.

y = x 2 + 4 x − 7 on [0, 2]

y = sin −1 x on [−1, 1]

y = 3 x on [−8, 8]

1 on [−5, −1] x2

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5.6 Chapter Review 1. Find the local extrema of f using the first derivative test. Sketch the graph and indicate where the function is increasing and decreasing.

y= − x3 + 4 x 2 − 3x

= y 43 x − x3

2. The gas mileage in your car can be modeled by the function y= 15 + 1.2 x − .02 x 2 in which y is the gas mileage depending on the car’s speed, x, measured in km/hr. This function is applicable for speeds 0 < x < 60. What is the maximum gas mileage and what is the speed that maximizes it?

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3. Find the local extrema of f using the second derivative test. Sketch the graph and indicate the points of inflection as well as intervals of concavity.

1 x +1 2

y =x 4 + 4 x 3 − 2 x 2 − 12 x + 6

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4. A man in a wingsuit jumps out of a plane from 15000 feet as part of his “flying” experience. Initially he keeps his body horizontal to slow his descent before maneuvering to a more vertical position so that he may increase his  15000  − + 15000 . As he speed. His altitude is modeled by h = −2.5t   1 + 200e  approaches the ground, he once again flattens his body position to utilize the functionality of the wingsuit and slow his speed prior to landing. The rate of his descent, as it relates dh −7500000e 2.5t height (feet) to time (minutes), is given by the differential equation . At what point = 2.5t dt (e + 200) 2 was his rate of descent at a maximum?

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5. A farmer wants to put a fence around a rectangular field and then sub-divide it into three smaller rectangular fields by placing two parallel fences to one of the sides. If he has a total of 1000 yards of fencing, what dimensions will give him the maximum area?

6. Find the height of a cylinder of maximum surface area that can be inscribed in a sphere of radius x.

Chapter 5: Analyzing Functions Using Derivatives 345

7. The interior of a 400-meter track consists of a rectangle with semi-circles at two opposite ends. Finds the dimensions that will maximize the area of the rectangle.

3 2 8. If f ( x) = x + x + x + 1 , find the number c that satisfies the Mean Value Theorem on the interval of [0, 4]

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Chapter 6: Solving Problems Using Derivatives

Figure out how fast things are changing, find the best way to do something, and even predict future values In the previous chapter we looked at using derivatives to determine information about graphs of functions and optimization problems. These are not the only applications for derivatives. We will also see how derivative can be used to solve the following types of problems:

Rates of Change – In this section you will take a look at problems that involve two quantities or variables. If the rate of change of one quantity, such as the height of water in a conical tank, changes with respect to time, how do you find the rate of change of the second quantity, the radius of the water in the tank, with respect to time?

Linear Approximations – In this section you will explore how to use the derivatives to compute a linear approximation to a function. You can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do when you have the function, it is much easier to solve certain problems using a simpler “approximation”.

Differentials – In this section you will compute the differential for a function. You will use differentials to explore how much a change in one variable affects the amount of change in a second variable. For example Newton’s second law states that, for a rocket, the thrust is prodv dm . portional to the exhaust velocity and fuel burn rate, or m =u dt dt

EXPLORING THE BIG IDEA

In this chapter you should be able to: 1. Use derivatives to analyze properties of a function. 2. Solve problems involving optimization.

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6.1 Related Rates Involving Shape and Space Warm Up You throw a rock into a calm pond. A circular ripple pattern emerges with the outermost circle’s radius growing at a rate of 8 cm per second. Using Leibniz notation, write the rate of change of the radius as a derivative.

What is the formula for the area of the outermost circle?

Write an expression for the rate of change of the area?

Steps to solve problems involving related rates: 1. Draw a picture and label and name the variables 2. Consider which variables change with time. Consider which aspects are constant. 3. Consider any restrictions or constraints on your variables. 4. Write an equation (a mathematical model) that relates the variables. 5. Simplify the expression by substituting any constant values. 6. Differentiate with respect to time. You will likely need to use implicit differentiation. 7. Substitute the given rate information and solve for the unknown. 8. Ensure that your answer is reasonable and eliminate any extraneous solutions.

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Example 1: Determine the Rate of Change in Volume Problems a) A spherical balloon is being filled with air. The radius of the balloon is increasing at 2 cm/s. What is the rate of increase in the volume of air when the radius is 3 cm? b) Water is being poured into a cylindrical water tower in Shaunavon, Saskatchewan that is 8 m tall and 1 m in radius at a rate of 0.6m3 / s . At what rate does the surface of the water rise?

How to Do It...

What to Think About What equation can be used as a mathematical model for this situation?

Which quantities are changing, and which are not?

With respect to which variable should we differentiate?

Which equation can be used as a mathematical model for this situation?

Chapter 6: Solving Problems Using Derivatives 349

Example 1: Determine the Rate of Change in Volume Problems - continued Which quantities are changing, and which are not?

Your Turn ... You are working at a local outdoor recreation center. To get ready for the summer, your boss had you clean the bottom of the swimming pool. The pool opens tomorrow, and you need to fill the pool back 1 m3 . up. The hose you are using lets water in at a rate of 4 min If the pool is in the shape of a rectangular prism, that is 12 m long, 6 m wide and always 2 m deep, what is the rate at which the height of the water is changing when the water is 0.5 m deep? Would the rate change when the height is at 1.5 m? Explain.

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Example 2: Determine the Rate of Change in Volume Problems A conical vat in Charlie’s Chocolate Factory is being filled with liquid chocolate at a constant rate of 6 ft 3 /min . If the vat stands point down with a height of 8 ft and a radius of 4 ft, find the rate at which the level of chocolate is rising when the depth is 2 ft? What about when it is at 4 ft?

How to Do It...

What to Think About What mathematical model can be used to represent this situation?

Which quantities are changing, and which are not?

Which quantities rates of change are unknown?

Why do we need to write the radius in terms of the height?

Could we solve the given problem by substituting for height in terms of the radius? Why or why not?

Does it make sense that the rate that the height increases is slowing down as the cone fills up?

Chapter 6: Solving Problems Using Derivatives 351

Your Turn ... A trough is 20 feet long and 4 feet across the top. Its ends are isosceles triangles with heights of 3 feet. Water is running into the trough at a rate of 2.1 ft 3 /min . What is rate at which the height of the water in the trough is increasing when the water is 1.5 ft deep? What is the rate change when the height is at 2 ft deep? Explain.

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6.1 Practice (t ) 4 4 t + 8 , where 1. As a spherical balloon is being inflated, its radius (cm) after t minutes is given by r= 0 ≤ t ≤ 10 . Calculate the rate of change with respect to t of each of the following at t = 8. a. r(t)

b. The volume of the balloon c. The surface area

2. A stone is dropped into the lake, causing water waves to create concentric circles. If, after t seconds, the radius of the waves is 35 cm, find the rate of change with respect to t of the area of the circle caused by the wave at: a. t = 1 sec b. t = 2 sec c. t = 3 sec

3. A circular metal pancake griddle is being heated, its radius changes at a rate of 0.02 cm/min. When the radius is 10 cm at what rate is the area changing?

Chapter 6: Solving Problems Using Derivatives 353

4. A balloon is being filled up with gas at a rate of cm3/min. Find the rate at which the radius is changing when the diameter is 16 cm.

5. Salt leaks out of a hole in a container and forms a conical pile whose height is always double the radius. If the height of the pile is increasing at a rate of 5 cm/min, what is the rate at which the salt is leaking out of the container when the height is 8 cm.

6. A spherical snowball is melting, and the radius is decreasing at a constant rate, changing from 11 inches to 7 inches in 45 minutes. How fast was the volume changing when the radius was 9 inches.

7. The ends of a water trough 8 ft long are equilateral triangles whose sides are 3 feet long. If water is being pumped into the trough at a rate of 4 ft3/min, find the rate at which the water level is rising when the depth is 8 inches.

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8. A point P(x,y) moves on the graph of the equation y = x 3 + x 2 + 5 , the x-coordinate is changing at a rate of 3 units/second. How fast is the y coordinate changing at the point (1, 7)?

9. A spherical balloon is deflating at a rate of 9 ft3/hr. At what rate is the radius changing when the volume is 375 ft3?

10. A stone is dropped into the pond and it creates circular waves whose radii increases at a rate of 0.4 m/sec. At what rate is the circumference of a wave changing when the radius is 5 m?

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6.2 Related Rates Involving Motion Warm Up Consider a 13-foot ladder leaning against a vertical wall. If the bottom of the ladder begins to slide along the ground away from the wall at a rate of 0.5 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 5 feet from the ground? Draw a diagram that represents this situation. Label the quantities.

Which quantities are changing? Which quantities are constant?

Can you create an equation that models this situation?

What would the derivative of this equation be if this situation is changing over time?

Can you answer the question posed?

What are the units of your solution? Why is your solution negative?

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Example 1: Solving Related Rates Problems When One Quantity is Constant A taxi leaves Walmart and begins travelling east for 3 km at a speed of 60 km per hour. The driver turns north at an intersection and travels at a rate of 90 km per hour. When the taxi is 4 km away from the intersection, how fast is the distance between the taxi and Walmart increasing?

How to Do It...

What to Think About Which quantities are changing, and which are constant?

How is the distance the taxi has travelled east changing when driving north?

What mathematical model can be used to represent how the distance between the taxi and Walmart changes over time?

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Your Turn ... a) An armadillo is walking away from a 20 m tall tree at a rate of 3 m/min. At what rate is the distance from the top of the tree changing when it is 3 m from the base of the tree?

b) A boat is being pulled into a boat launch that is 2 m above the level of the surface of the lake. Assume the rope is attached to the bow of the boat and is 50 cm above the water. If the rope is pulled in at a rate 0.2 m/sec, how fast is the boat approaching the launch when it is 10 m from the dock? Note: this picture is not an accurate depiction of the situation. Draw your own diagram. Don’t launch your boat this way!

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Example 2: Solving Related Rates Problems When All Quantities are Changing In a high-speed chase, a police cruiser approaches a right-angled intersection from the north. The officers are chasing an escaped fugitive on a speeding motorcycle that has turned the corner and is now heading east. When the cruiser is 0.6 miles north of the intersection and the motorcycle is 0.8 miles east, the police officers determine with radar that the distance between them and the motorcycle is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the fugitive?

How to Do It...

What to Think About Which quantities are changing in this situation?

At the instant of measurement, which quantities are known, and which others can be determined?

Why is the rate of change of the distance between the intersection and the police cruiser considered negative?

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Your Turn ... You are the captain of a small sailboat drifting at sea at midday and spot a ferry 15 km west of you. You are travelling at 6 km/h north and the ferry is travelling east at 8 km/h. At 2 pm, how fast is the distance between the boats changing?

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6.2 Practice 1. A ladder 25 feet long leans against a vertical building. If the top of the ladder slides down the wall at a rate of 4 ft/sec, how fast is the ladder sliding away from the wall when the top of the ladder is 7 feet from the ground?

2. Sebastian starts at point A and runs east at a rate of 12 ft/sec. One minute later Orla runs north at a rate of 10 ft/sec. At what rate is the distance between them changing 1 minute after Orla starts.

3. A man 6 feet tall walks at a rate of 6 ft/sec away from a lamp that is 18 feet above the ground. When he is 10 feet from the base of the lamp at what rate is his shadow changing?

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4. Jake is driving east at 60 km/h and Samantha is driving north at 75 km/h. Both cars are approaching the intersection of the two roads. At what rate is the distance changing between them when Jake’s car is 0.5 km from the intersection and Samantha’s car is 0.4 km from the intersection?

dy dx 5. A point P(x,y) moves along the graph y 2 = 2 x 3 such that at the = x , where t is time. Find dt dt point (2,4).

6. Sebi is flying a kite at cattle point and holds the string 4ft above ground level and lets the string out at a rate of 2 ft/sec as the kite moves horizontally at height of 110 ft. Assuming the string is taut, find the rate at which the kite is moving when 150 ft of string has been let out.

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7. A hot air balloon is rising vertically at a rate of 2 ft/sec. An observer waiting for their turn is sitting 100 yards from a point directly below the balloon. At what rate is the distance between the balloon and the observer changing when the balloon is 500 feet high.

8. When two electrical resistors R1 and R 2 are connected in parallel the total resistance R is given by 1 1 1 = + . If R1 and R 2 are increasing at rates of 0.02 ohm/sec and 0.01 ohm/sec, respectively, at R R1 R2 what rate is R changing at the instant when R1 = 90 ohms and R2 = 30 ohms?

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6.3 Related Rates Involving Periodic Functions Warm Up The minute and hour hand are moving at different rates as they travel around the clock.

What is the rate of change of the minute hand as it travels around the clock?

What is the rate of change of the hour hand as it travels around the clock?

What is the rate of change of the angle between the minute hand and the hour hand as they travel around the clock at 4:42pm?

What is the rate of change of the angle between the minute hand and the hour hand as they travel around the clock at 3:00pm?

At what time(s) does the rate of change switch from negative to positive or vice versa?

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Example 1: Related Rates Involving Changing Angles The London Eye ferris wheel has a radius of 60 m and rotates at about 2 revolutions per hour. How fast is the rider rising or falling when the rider is 70 m above the ground?

How to Do It...

What to Think About What equation can you create that represents a mathematical model of the situation?

How fast is the ferris wheel rotating?

Your Turn ... An airplane approaches at an altitude of 5 miles towards a point directly overhead of an observer. The speed of the plane 550 mph. Find the rate at which the angle of elevation is changing when the angle is 40 ?

Chapter 6: Solving Problems Using Derivatives 365

Example 2: Determine Rate of the Hour and Minute Hands A clock has a minute hand 20 cm long and an hour hand of 12 cm. Determine the rate the distance between the hour and minute hands are changing at 4:42 pm.

How to Do It...

What to Think About What equation can be used to relate the changing quantities?

How can you find the angle between the minute and hour hands?

Why must the rate of change of θ be in terms of radians for the answer to make sense?

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Your Turn ... You spot two freighters moving towards each other offshore. From your perspective you measure the angle between the ships as 130 with the first ship anchored in the harbor 500 m away from you. The second ship is travelling on a circular arc that keeps it always 900 m away from you. If the angle between the ships is changing at 30 per minute, how fast is the distance between the ships changing?

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6.3 Practice 1. A balloon rises at a rate of 4 m/s from a point on the ground 45 m from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 m above the ground.

2. Joshua is standing on the Johnston street bridge and is reeling in a fish at a rate of 1.5 ft/sec from a point 18 ft above the water. At what rate is the angle between the fishing line and the water changing when there is 30 feet of line out.

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3. The beacon of a lighthouse located at a perpendicular distance of 300 m from point N on a straight shoreline revolves 5 rev/min and shines a spotlight on the shore.

300m

How fast is the spot of light sweeping along the shore at a point 400 m from N?

Chapter 6: Solving Problems Using Derivatives 369

4. Two sides of a triangle have lengths 15 m and 25 m. The angle between them is increasing at π π rad/sec . How fast is the length of the third side changing when the angle between the sides is ? 6 75

5. As the sun rises, the shadow cast by a 20 m tree is decreasing at a rate of 50 cm/h. At what rate is the angle of elevation from the shadow to the sun increasing when the shadow is 12 m in length?

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6.4 Linearization and Differentials Warm Up 2 ) 2 x − 1. Let f ( x) = x . Show that the line tangent to the graph of f at (1, 1) the point is g ( x=

2 2 x − 1 . Zoom in on the two graphs at (1, 1). Sketch what you see. Set y1 = x and y= 2

2 ) 2 x − 1 . Find f (1.1) and g (1.1) . Consider f ( x) = x and g ( x= How good of an approximation is g (1.1) for f (1.1) ?

Is g (1.1) an underestimation or an overestimation? Explain.

What happens to the accuracy of the approximations as one moves further away from the point of tangency?

Chapter 6: Solving Problems Using Derivatives 371

Definition: Differential and linearization

• The differential, dy = f ′( x)dx is the principal change in a function dy relative to an infinitesimal change in the independent variable dx. • The linearization is a linear approximation model used to estimate a past or future value of an unknown function. The linearization uses a specific point on the function and the rate of change information at that point; thus, it is equivalent to the tangent line at the specific point. x) f (a ) + f '(a )(∆x) such that L(a + ∆x) ≈ f (a + ∆x) . This is • In practice, we often find L(a + ∆= generally a reasonable approximation provided ∆x is sufficiently small.

Example 1: Determining the Linearization and Using it to Approximate a Function ) 2 − x at x = 1. a) Find the linear approximation L(x) (or linearization) of f ( x= b) How accurate is the approximation L(1 + 0.1) ≈ f (1 + 0.1) for values of x near 1? c) Sketch what the linearization looks like at values close to 1.

How to Do It...

What to Think About What is the linearization equivalent to?

How well does the linearization approximate the curve near the point of tangency?

372

Chapter 6: Solving Problems Using Derivatives

Example 1: Determining the Linearization and Using it to Approximate a Function - continued

What happens to the approximation at x = 1.5? At x = 2 ?

Chapter 6: Solving Problems Using Derivatives 373

Your Turn ... a) What is the linearization to the graph of y = 2 cos x at x =

b) How good is the approximation for x=

374

π 2

Chapter 6: Solving Problems Using Derivatives

π 2

+ 0.1 ?

Example 2: Using Differentials Find and evaluate dy for each of the following functions at the given point. a) y = x 3 at x = 1 and dx = 0.01 x

 1 b) y= 1 +  at x = 10 and dx = 1  x

How to Do It...

What to Think About What does the differential dy represent?

Your Turn ... For f ( x) = 1 + sin x , find dy when x = π and dx =

π 6

Chapter 6: Solving Problems Using Derivatives 375

6.4 Practice 1. Find and evaluate dy for each of the following functions at the given point. a. y = 2 x3 − 5 x + 3 at x = 1 and dx = −0.01

b. y = x + 2 x − 8 at x = −1 and dx = 0.1

c. y =

1 at x = 1 and dx = −0.03 3+ x

) 2. Find the linear approximation to the function f ( x= for x = 0 + 0.1?

376

Chapter 6: Solving Problems Using Derivatives

1 + x at x = 0? How good is the approximation

3. Find the linearization to the graph of y = 3sin x at x = ? How good is the approximation for 6 π x= + 0.1 ? 6

2 4. Find the linear approximation L(x) (or linearization) of f ( x)= (1 + x) at x = 1. a. How accurate is the approximation L(1 + 0.1) ≈ f (1 + 0.1) for values of x near 1?

b. Sketch what the linearization looks like at values close to 1.

x) ln(1 + x) at x = 1. How accurate is the 5. Find the linear approximation L(x) (or linearization) of f (= approximation L(1 + 0.1) ≈ f (1 + 0.1) for values of x near 1?

Chapter 6: Solving Problems Using Derivatives 377

6.5 Chapter Review 1. Water is being emptied out of a spherical fishbowl of radius 10 cm. If the depth of the water in the tank is 6 cm and is decreasing at a rate of 3 cm/sec, at what rate is the radius of the top of the surface of the water decreasing?

10 cm

6 cm

2. A water tank has the shape of a right circular cone of altitude 12 ft and base radius 4 ft, with the vertex at the bottom of the tank. If water is being taken out of the tank at a rate of 10 ft3/min, for fast is the water level falling when the depth is 5 ft?

3. At what rate is the area of an equilateral triangle increasing if its base is 12 cm long and is increasing at a rate of 0.4 cm/sec?

378

Chapter 6: Solving Problems Using Derivatives

4. A balloon is being filled at a rate of 121 π cm /sec . At what rate is the balloon increasing when its radius is 12 cm. 3

5. The base of a rectangle is increasing at 4 cm/s while the height is decreasing at 3 cm/s. At what rate is the area changing when its base is 20 cm and its height is 12 cm?

6. A square is expanding. When each edge is 16 cm its area is increasing at a rate of 80 cm3/sec. At what rate is the length of each ledge changing?

Chapter 6: Solving Problems Using Derivatives 379

7. Two roads intersect at right angles. At 9 am a car passes through the intersection headed due west at 60 km/h. At 10 am a truck heading south at 75 km/h passes through the same intersection. Assume they maintain their speeds and directions. At what rate are they separating at 1 pm?

8. A circle is inscribed in a square as shown. The circumference of the circle is increasing at a constant rate of 8 cm/sec. As the circle expands the square expands to maintain its tangency.

a. Find the rate at which the perimeter of the square is increasing.

b. At the instant when the area of the circle is 36π square inches, find the rate of increase of the area between the circle and the square.

380

Chapter 6: Solving Problems Using Derivatives

9. Water is draining from a conical tank with a height of 10 feet and diameter 8 feet into a cylindrical tank. The depth of the water in the conical tank is changing at a rate of (h – 8) feet per minute.

1 a. Volume of a cone V = π r 2 h , write an expression for the volume of water in the conical tank in terms 3 of h.

b. At what rate is the volume of the water in the conical tank changing when h = 4?

Chapter 6: Solving Problems Using Derivatives 381

10. The radius of a basketball is increasing a rate of 0.01 cm/sec. a. When the basketball has a radius of 10 cm, what is the rate of increase of its volume?

b. At the time when the volume of the basketball is 48π cubic centimeters, what is the rate of increase of the area cross section through the center of the basketball?

c. When the radius and the basketball are increasing at the same rate, what will the radius be?

382

Chapter 6: Solving Problems Using Derivatives

Unit 2 Project: Shrinking Lollipop This project compares the predicted rate of change of the radius of a lollipop to the actual rate of change. Test the hypothesis that the rate of change of the volume of a lollipop is proportional to its surface area. It seems reasonable that the rate of change of the volume of a lollipop would be proportional to its surface area, since the candy dissolves from the surface as you suck the lollipop. In mathematical terms, we predict that where k is a constant of proportionality. But is this prediction true? What can we learn, by mathematical deduction, about k, and how can we test this prediction experimentally?

dV = kA dt 1. 2. 3. 4.

5. 6. 7. 8. 9.

Prepare a table with time in the first column, radius in the second column. Measure (mm) and record the radius (diameter/2) of a lollipop (Tootsie Roll Pops work well). Suck on the lollipop for one minute and measure the radius. Record your data. Repeat step 3 until the lollipop is gone or you reach candy of a different consistency (e.g. the inner nougat of the Tootsie Roll Pop). Note: it is important to suck on the lollipop the same from one trial to the next. If you suck especially vigorously on one trial but only lightly on the next, you won’t be able to interpret your results. Graph your results: radius (y-axis) vs time (x-axis). What kind of function do you see: linear? parabolic? Use the regression function on your calculator or Desmos to determine an equation. dV Now consider the prediction. Use a chain rule decomposition of to help find k. dt dV dV ? ⋅ kA ) (Hint: = dt ? dt Do the experimental results support your prediction?

Questions 1. What happens to the shape of a square lollipop (or other shape with edges and corners) as it dissolves? Why? 2. How does the theoretical compare to the calculus you have performed?

Unit 2 Project: Shrinking Lollipop 383

384

Unit 2 Project: Shrinking Lollipop

Unit 3: Integral Calculus Unit focuses on the following AP Big Idea from the College Board: Big Idea 3: Integrals are used in a wide variety of practical and theoretical applications. It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus By the end of this unit, you should be able to:

• Recognize antiderivatives of basic functions. • Interpret the definite integral as the limit of a Riemann sum in integral notation. • Express the limit of a Reimann sum in integral notation. • Approximate a definite integral. • Calculate a definite integral using areas and properties of definite integrals. • Analyze functions defined by an integral. • Calculate antiderivatives. • Evaluate definite integrals. • Interpret the meaning of a definite integral within a problem. • Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume. • Analyze differential equations to obtain general and specific solutions. • Interpret, create, and solve differential equations from problems in context. By the end of this unit, you should know the meaning of these key terms:

• Antidifferentiation • Area under a curve • Average value of a function • Definite integral • Differential equation

• Fundamental Theorem of Calculus • Indefinite integral • Integral • Riemann sum • Rotational volume

ds =v dt

dv =a dt

→ ←

Chapter 7: Antidifferentiation 385

Chapter 7: Antidifferentiation

If derivatives tell you how fast something is changing, antiderivatives help you figure out where it started. Differentiation allows us to determine how a function is changing. Most of the time, in real-world experiments, we collect data that measures change. We then try to determine a function, which is a derivative that represents the data. Once the derivative is known, we can use antidifferentiation to find the original function. Integration is like filling a tank from a tap. The input (before integration) is the flow rate from the tap. Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank. Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap). As the flow rate increases, the tank fills up faster and faster. With a flow rate of 2x, the tank fills up at x 2 . We have integrated the flow to get the volume. Example: with the flow rate in litres per minute, and the tank starting at t = 0 After 3 minutes x = 3:

• the flow rate has reached 2 x 2 3 6 liters/min, • and the volume has reached x=2 3=2 9 liters And after 5 minutes x = 5:

• the flow rate has reached 2 x 2 5 10 liters/min, • and the volume has reached x=2 5=2 25 liters EXPLORING THE BIG IDEA

In this chapter you should be able to: • Recognize antiderivatives of basic functions. • Calculate antiderivatives.

386

Chapter 7: Antidifferentiation

7.1 The Antiderivative – Working Backwards Warm Up Work backwards to find the original Function. f’(x)

What to Think About

f(x)

3 2x

What pattern do you observe?

3x 2 4x3

What happens when the coefficient does not fit the pattern?

x2 7x 4 xn kx

Generalized pattern

Are there any other functions that you could Differentiate to get f ′( x) = 2 x ?

Can you determine a general formula to follow?

Definition: Integral/Antidifferentiation Finding the Integral (the antiderivative) using the “Reverse Power Rule”

( x) If f ′( x) = x then f= n

x n +1 x n +1 n dx + C and we state ∫ x= +C . n +1 n +1

Integral notation, ∫ dx , asks you to find the antiderivative. The constant of integration C must always be added when anti-differentiating because the derivative of a constant is zero. Without more information about the original function we cannot know the value of the constant term. © Edvantage Interactive 2025

Chapter 7: Antidifferentiation 387

Example 1: Finding the Antiderivative Using the Reverse Power Rule. Find the antiderivative of the following: f ′'(( x)) = x 2 − 12 x 3 +

x5 +3 2

f ′( x) =

How to Do It...

2 x2

What to Think About How do you apply the reverse power rule?

Why must you include a constant?

Can you write the derivative as a power term?

Your Turn ... a)

388

∫ (3x + x − 2 x + 3)dx = 2

Chapter 7: Antidifferentiation

∫ ( x + 1) dx = 2

Example 2: Determining the Antiderivative of a Function Find the antiderivative of the following: a)

f ′( x) = x

f ′( x= ) x4 −

1 x3

How to Do It...

What to Think About How can you write the function with an exponent?

Your Turn ... a)

2  4 ∫  x − 3x + x 2 dx =

dx ∫ x=

Derivative Notation

Differential Notation

Integral Notation

dy = f ′( x) dx

dy = f ′( x)dx

y = ∫ f ′( x)dx

Chapter 7: Antidifferentiation 389

7.1 Practice 1.

∫ 3x dx

∫ x dx

∫ (4 x + x )dx

390

Chapter 7: Antidifferentiation

∫ x dx

∫ 6t

∫ y (2 y − 3)dy

tdt

∫ (8 x + 4 x − 6 x − 4 x + 5)dx

∫ (3 − 2t + t )dt

∫ ( x 2 − x)dx

10.

∫  x + x + 5  dx

11.

∫ x ( x + 1)dx

12.

∫ (2 x − 1) dx

 2



Chapter 7: Antidifferentiation 391

7.2 Antiderivative of Trigonometric Functions Warm Up Consider all the primary trigonometric differentiation formulas: (sin x)′ =

(csc x)′ =

(cos x)′ =

(sec x)′ =

(tan x)′ =

(cot x)′ =

These differentiation formulas lead to the following antiderivatives − cos x + C ∫ sin xdx =

xdx sin x + C ∫ cos =

xdx sec x + C ∫ sec x tan =

− csc x + C ∫ csc x cot xdx =

xdx tan x + C ∫ sec =

− cot x + C ∫ csc xdx =

392

Chapter 7: Antidifferentiation

Example 1: Find the Antiderivative by Reversing Trigonometric Differentiation Solve the following:

a) ∫ (2 sin x + 3 cos x)dx

b) ∫ (4 cos x + sec 2 x)dx

How to Do It...

∫ (3 sec x tan x − csc x)dx 2

What to Think About What would you differentiate to get 2sin x + 3cos x ?

If you remember the derivatives can you find the antiderivatives?

Your Turn ... a)

∫ csc x(csc x + cot x)dx =

x 2 sin x + 2sin x dx = ∫ 2 + x2

Chapter 7: Antidifferentiation 393

Example 2: Using Trigonometric Identities Before Finding the Antiderivative Solve the following: sin 2 x a) ∫ cos x dx

dx ∫ sec x =

How to Do It...

What to Think About Can you rewrite the function using a trig identity?

Why does using a trigonometric identity help you to antidifferentiate?

Your Turn ...

∫ tan xdx 2

394

Chapter 7: Antidifferentiation

sin x

∫ cos x dx 2

7.2 Practice 1.

∫ 1 + sin xdx

∫ 4 csc x cot x + 2sec xdx

∫ (2 cot x − 3 tan x)dx

∫ tan x + 1dx

∫ sec xdx

∫ 2 − csc xdx

∫ 2 x + cos xdx

1 − sin 2 x ∫ cos2 x dx

∫ sin x tan xdx

10.

∫ (4sec x tan x − 2sec x)dx

Chapter 7: Antidifferentiation 395

7.3 Antidifferentiation Using Substitution Warm Up Take the Derivative of the following Functions:

d (5 x + 3) 4 dx d (sin x + 2)3 dx d 2 ( x + 3)9 dx d n (u ) dx After reviewing the derivatives using the chain rule how can you use this idea to find the antiderivative?

∫ 9( x + 3) ⋅ 2 xdx 2

∫ 3(sin x + 2) ⋅ cos xdx 2

∫ nu

n −1

⋅ u ′dx

How else can u’dx be written?

396

Chapter 7: Antidifferentiation

Definition: “Reverse chain rule” n dx ∫ x=

x n +1 +C n +1

n n ∫ u ⋅ u′dx = ∫ u du =

u n +1 +C n +1

Example 1: Reversing the Chain Rule Using “u/du” Substitution Solve the following: a)

∫ ( x + 1) 2 xdx 2

How to Do It...

∫ ( x + 1) dx 2

What to Think About Are you trying to find the antiderivative of a composite function?

Is the derivative of the embedded function present?

Why do you need to expand this function first?

Which rule did you use to find the antiderivative?

Chapter 7: Antidifferentiation 397

Your Turn ...

∫ sin x cos xdx = 3

398

Chapter 7: Antidifferentiation

∫ 3x

x 3 + 1dx =

Example 2: Finding the Antiderivative Using “u/du” Substitution Solve the following: a)

∫ ( x + 4) 3x dx 3

How to Do It...

∫ ( x + 1) dx = 2

What to Think About Can you simplify the expression using a u/du substitution?

Why is a u/du substitution not possible for this function?

What would we need to do to find the antiderivative?

Have you ever heard of the binomial expansion theorem?

Chapter 7: Antidifferentiation 399

Your Turn ...

∫ ( x − 2) x dx =

∫ ( x − 1) dx =

∫ ( x − 1) x dx = 4

Did You Know? Pascal’s Triangle and the Binomial Expansion Theorem Unexpanded Form

Expanded Form

( a + b) 0 =

(a + b)1 =

a+b

( a + b) 2 =

a 2 + 2ab + b 2

( a + b)3 =

a 3 + 3a 2b + 3ab 2 + b3

( a + b) 4 =

a 4 + 4a 3b + 6a 2b 2 + 4ab3 + b 4

( a + b)5 =

a 5 + 5a 4b + 10a 3b 2 + 10a 2b3 + 5ab 4 + b5

The binomial expansion theorem is a quick way to expand a binomial raised to a large integer exponent. Pascal’s triangle provides a quick way to determine the coefficients. 400

Chapter 7: Antidifferentiation

7.3 Practice Evaluate each integral

∫ (5 x − 1) 7 x dx =

∫ x + 5 (3x)dx =

∫ (2 x + 1) 3dx =

∫ ( x + 2 x + 4) dx =

∫ cos x sin xdx =

∫ tan x sec xdx =

x +1

Chapter 7: Antidifferentiation 401

(1 + x ) 4 ∫ x dx =

∫ 4x +

11.

2 ∫ x sin x dx =

402

4x 16 − x 2

dx =

Chapter 7: Antidifferentiation

x 2 + 3x + 7 dx = ∫ x

10.

 1  1  ∫ 1 + t   t 2  dt =

12.

csc 2 x ∫ cot 3 x dx =

7.4 Advanced Antidifferentiation Techniques Warm Up Differentiate each expression

d sin 3 x = dx d cos( x 2 + 5) = dx d tan x5 = dx d sec(sin x) = dx (sin u )′ =

(csc u )′ =

(cos u )′ =

(sec u )′ =

(tan u )′ =

(cot u )′ =

Trigonometric Integration Formulas − cos u + C ∫ (sin u )du =

− csc u + C ∫ (csc u cot u )du =

)du sin u + C ∫ (cos u=

)du sec u + C ∫ (sec u tan u=

)du tan u + C ∫ (sec u=

− cot u + C ∫ (csc u )du =

Chapter 7: Antidifferentiation 403

Example 1: Evaluate Integrals Involving Trigonometric Functions and u/du Substitution Evaluate the following integrals: a)

∫ cos 5xdx

∫ sin( x + 9) xdx 2

∫ sin(sin x) cos xdx

What to Think About

How to Do It...

Why is the u/du helpful for this question?

Is a u/du always necessary? Sometimes, it is best to ask the question, “What would I differentiate to get this function?”

Your Turn ... 4sin x

∫ (1 + cos x) dx =

404

Chapter 7: Antidifferentiation

∫ cos 3 xdx =

Example 2: Evaluate the Indefinite Integral Solve the following: a)

2 ∫ sec π xdx

cos x ∫ x dx

How to Do It...

What to Think About What would you need to 2 differentiate to get sec x ?

How does the constant π effect the final answer?

Why do you not have to use a reverse of the quotient rule?

Your Turn ... Determine each indefinite integral: 2 a) ∫ csc 2θ dθ =

∫

sec x tan x dx = 5 x

Chapter 7: Antidifferentiation 405

Example 3: Evaluating an Integral Using Reverse Substitution Sometimes the “u/du” method appears not to work. We can use a “sneaky” u/du reverse substitution method. In these cases, a u/du substitution is unavoidable. Solve the following: a) ∫ x x + 1 dx =

How to Do It...

b) ∫

x dx 1− x

What to Think About There is a problem after using u/du. What is it?

What needs to be substituted to write the integral in terms of u?

406

Chapter 7: Antidifferentiation

Your Turn ... Evaluate the integral

∫x

x − 1 dx =

7.4 Practice 1.

x2 − 2 x + 3 ∫ x 4 dx =

∫ ( x + 1) x dx =

∫ sin(3x )5 x dx =

∫ cos x sin x dx =

Chapter 7: Antidifferentiation 407

∫ sec (4 x + 1) dx =

∫ ( x − 2) x dx =

∫ x + 3 dx =

∫ x  1 + x  dx =

∫x

10.

∫ sec(1 − x) tan(1 − x)dx =

11.

∫ tan x sec xdx =

2x −1

408

x 3 + 1 dx =

1 



Chapter 7: Antidifferentiation

7.5 Antidifferentiation Involving Exponential and Logarithmic Functions Warm Up Integrate the following:

∫ x dx = 2

∫ x dx =

∫ dx =

∫ x dx = 2

∫ x dx =

What did you notice? What about the last one? Did it follow the rule? SUMMARY: The antiderivative of the reciprocal function:

dx ln x + C ∫ x=

∫ u ⋅ u′dx = ∫ u du = ln u + C Chapter 7: Antidifferentiation 409

Example 1: Removing Constants Before Finding the Antiderivative Solve the following: dx a) ∫ 5x =

How to Do It...

∫ x dx = What to Think About It is often beneficial to remove a constant before finding the antiderivative.

∫ kf ( x)dx = k ∫ f ( x)dx Why do we have to include the absolute value?

Your Turn ... Evaluate each integral

5dx

∫ 3x − 2 =

∫ (3x − 2) =

2dx

410

Chapter 7: Antidifferentiation

Example 2: Finding the Antiderivative of a Reciprocal Function Solve the following: dx a) ∫ x +1 =

4 x3 ∫ x 4 + 5 dx =

How to Do It...

What to Think About What would you differentiate to 1 get ? x +1

Why may we omit the absolute value in this case?

Your Turn ... Evaluate each integral a)

(ln x) 2 ∫ x dx =

∫ x ln x =

∫

sin(ln x) dx = x

Chapter 7: Antidifferentiation 411

Example 3: Evaluate the Indefinite Integral Solve the following: a)

∫ e dx 2x

∫e

sin π x

cos π xdx

How to Do It...

e 2 x + 2e x + 1 ∫ e x dx

What to Think About What would you differentiate to 2x get e ?

What would you differentiate to get the exponential component of this function?

Can you use u/du substitution to integrate?

Why must you first break this up into three separate fractions first?

Your Turn ...

∫ 5e dx = x

412

Chapter 7: Antidifferentiation

∫e

tan x

sec 2 xdx =

7.5 Practice 1

∫ (ln x) x dx

∫ xe dx

∫ e dx =

∫ e cos e dx =

sec 2 x ∫ tan x dx =

∫ sin x + 1 dx =

1 + ex ∫ x + 1 + e x dx

∫ e sec e dx =

e2 x ∫ (e2 x + 1)3 dx =

cos x

−x

Chapter 7: Antidifferentiation 413

10.

∫e

tan 2 x

sec 2 2 xdx

3 x

12.

e ∫ x 2 dx =

14.

5 − ex ∫ e2 x dx

414

Chapter 7: Antidifferentiation

11.

∫e

13.

∫ ln(e

sin π x

cos π xdx =

2 x −1

)dx

7.6 Antidifferentiation Involving Inverse Trigonometric Functions Warm Up

∫ e dx = 3x

∫ sin 5xdx = xdx

∫ x +1 = 2

If you figured out it is not the same as the rest, you are correct. 1 What can you differentiate to get 2 ?

x +1

d (arctan 3 x) = dx d (arctan e x ) = dx d (arctan(cos x)) = dx Did

∫ 1 + x follow the same rule? 2

SUMMARY: The Corresponding Integrals

arctan x + C ∫= 1+ x 2

arctan u + C ∫= 1+ u 2

Chapter 7: Antidifferentiation 415

Example 1: Recognizing an Inverse Trigonometric Derivative Evaluate the following integrals: 5 a) ∫ x 2 + 1 dx

How to Do It...

∫ 1+ 4x

What to Think About Is there a constant to be factored out?

Do you recognize the function?

Why can’t we use u/du?

How is this different than the previous example?

How can you recognize this in the future?

Your Turn ... cos x

∫ 1 + sin x dx =

416

Chapter 7: Antidifferentiation

sin 2 x

∫ 1 + sin x dx = 2

Example 2: Using u/du and Recognition to Find the Antiderivative Evaluate the following integrals: a)

ex ∫ 1 + e2 x dx

e2 x ∫ 1 + e2 x dx

What to Think About

How to Do It...

What would you differentiate to get this function?

Can you use a u/du? If so, what u is most appropriate so that du simplifies the function?

What function can you differentiate to get the simplified function?

Your Turn ... a)

∫ 1− x = 2

xdx

∫ 1− x = 2

Chapter 7: Antidifferentiation 417

7.6 Practice 1.

∫ r sec r dr =

∫ sin 5 cos 5 dx =

e x dx

∫ sec x tan x cos(sec x)dx =

∫ 1− 9x =

sec 2 xdx

∫ 1− e

∫ 1 − tan x =

∫ 3x + 5 dx

∫ x − 3 dx

418

Chapter 7: Antidifferentiation

ex ∫ x 2 dx

10.

∫ 16 + x dx

11.

ex ∫ e x − 1 dx

12.

∫ (2 x + 4) dx

13.

∫ 3x ( x − 5) dx

14.

∫ x cos  x  dx

15.

∫ 5 p + 6 dp

1

Chapter 7: Antidifferentiation 419

16.

∫ 3 y 6 − 3 y dy

18.

∫ x + 3 dx

20.

∫ cos 5 x sin 5 xdx

420

2x −1

Chapter 7: Antidifferentiation

17.

∫ (1 + x ) dx

19.

∫ (1 − sin 2t ) 3 cos 2tdt

2 2

21.

∫ (1 − cos x)dx 2

Other Useful Formulas:

kxdx sin kx + C ∫ cos= k 1

− cos kx + C ∫ sin kxdx = k 1

kxdx tan kx + C ∫ sec= k 2

1 kx

kxdx sec kx + C ∫ sec kx tan= k 1

− csc kx + C ∫ csc kx cot kxdx = k 1

− cot kx + C ∫ csc kxdx = k 2

dx e +C ∫ e= k

dx ln x + C ∫= kx k

du 1 u = ∫ a 2 + u 2 a arctan a + C

arcsin + C ∫ a= a +u

−du

arcsec + C ∫ u= a a u −a

arccos + C ∫ = a a −u

∫ tan udu =ln sec u + C =−ln cos u + C

∫ sec udu= ln sec u + tan u + C

∫ cot udu =ln sin u + C =−ln csc u + C

∫ csc udu= ln csc u − cot u + C

∫ e du= e + C u

u du ∫a =

au +C ln a

Chapter 7: Antidifferentiation 421

7.7 Chapter Review Evaluate each indefinite integral.

x +1

∫ x dx

∫ x + 1 dx

x −1 ∫ 3x 2 − 6 x + 2 dx

x 2 + 3x + 1 ∫ x dx

e x − e− x ∫ e x + e− x dx

∫ x(3

422

Chapter 7: Antidifferentiation

− x2

)dx

10 x ∫ x dx

∫5

−2 x

∫ xe

dx x

10.

(1 + e x ) 2 ∫ e2 x dx

11.

∫ cos(3 − 2 x)dx

12.

∫ csc 8xdx

Chapter 7: Antidifferentiation 423

13.

∫ e tan e dx

14.

∫ cos x 8 + 2sin x dx

15.

cos x ∫ 1 + sin 2 x dx

16.

∫ 36 x − 25 dx

424

Chapter 7: Antidifferentiation

1 2

Chapter 8: Solving Differential Equations Add up infinitely small pieces to find the whole

There are many ways to model real-world situations mathematically. Differential equations describe how phenomena change. A differential equation is a mathematical model used to measure the change of a quantity that is determined by another quantity. There are a variety of solution techniques for these types of equations depending on the complexity of the equations involved. Differential equations are composed of several terms, just as conventional algebraic equations are. Consider the example of Yellow Fever in the body.

In vaccinology, scientists are interested in the changing of cells, molecules and virus concentrations with respect to time. The following equation is one that represents a model of the virus (V) for Yellow Fever. dV = π v (V ) H * −cvV − kvVAt dt Each of the variables represents a specific quantity that is being measured related to the Yellow Fever antibody.

EXPLORING THE BIG IDEA

In this chapter you should be able to: • Interpret the definite integral as the limit of a Riemann sum in integral notation. • Express the limit of a Reimann sum in integral notation. • Approximate a definite integral. • Calculate a definite integral using areas and properties of definite integrals. • Analyze differential equations to obtain general and specific solutions. • Interpret, create, and solve differential equations from problems in context.

Chapter 8: The Integral 425

8.1 Solving Differential Equations Analytically Warm Up Consider:

dy =x dx

What does

dy mean? dx

Multiply both sides by dx How can you solve for y ? Draw a possible graph of y.

Definition: Differential equation An equation involving a derivative is called a differential equation.

426

Chapter 8: The Integral

There is one differential equation that you probably know. It is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us.

F = ma To see that this is in fact a differential equation we need to rewrite it. First, remember that acceleration, a, can be written in one of two ways. a=

dv dt

d 2s dt 2

Where v is the velocity of the object and s is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position. Now Newton’s Second Law can be written as a differential equation in terms of either the velocity, v, or the position, s, of the object as follows. 2 dv Or F  t , s ds  = m ds F (t , v) = m   dt 2  dt  dt

Example 1: Evaluate a Differential Equation by Separating the Variables Solve for the following: dy a) = x3 dx

How to Do It...

dy = sin x dx

What to Think About How can you separate the variables so that all x terms are on one side of the equation?

How can you solve for y? Why is there no constant of integration on the left side of the equation?

What does the constant of integration determine about the behavior of y?

What do you need to know to find C?

Chapter 8: The Integral 427

Your Turn ... Solve the differential equation

dy =x 4 − 2 x 2 + 1 dx

Example 2: Evaluate Differential Equations Implicitly to Solve for “y” Solve for the following: a)

dy x = dx y

How to Do It...

dy = xy dx

What to Think About How can you separate the variables so that all x terms are on one side and all y terms are on the other?

What do you need to do to remove the differentials from the equation?

Why is there no constant of integration on the left side of the equation?

What does C1 equal?

428

Chapter 8: The Integral

Example 2: Evaluate Differential Equations Implicitly to Solve for “y” - continued How can you separate the variables so that all x terms are on one side and all y terms are on the other?

Why do we not need to use the absolute value sign anymore?

Your Turn ... Solve each differential equation. a)

dy 1 = dx y

dy = x2 y dx

Chapter 8: The Integral 429

Example 3: Interpret the Solution to a Differential Equation Graphically If we go back to the differential equation

dy = 2 x , what does the solution mean graphically? dx

How to Do It...

What to Think About What does the x 2 represent

What does the C represent?

How can you know which is the correct graph?

Your Turn ... Graph the solutions to the differential equations and create at least two different curves. dy = 3x 2 + 2 dx

430

Chapter 8: The Integral

dy = cos x dx

Example 4: Find the Particular Solution Given an Initial Value for the Function: Suppose the slope of the tangent line to each point on the curve y = f ( x) is x and the curve passes through the point (2, 3). Find the specific curve.

How to Do It...

What to Think About How can you separate the variables?

How does the initial value help us find the exact function y? correct graph?

Your Turn ... Given that

dy 1 = and the solution curve passes through the point (e, −3) , what is y? dx x

Chapter 8: The Integral 431

8.1 Practice dy = x3 3 y dx

dy = sin π x dx

dy 1 − y 3.= = , y (1) 0 dx x

dy = 6 x 2 − 7 x + 3, y (−1)= 0 dx

dy 12 − x, y (0) = = −1 dx

dy =+ cos x sin x, y (π ) = 1 dx

2 = , y (4) 0 3 x

dy = e − x + e x , y (0) = 0 dx

dy dx

7.=

432

Chapter 8: The Integral

9. A particle moves along a line with an acceleration of 3+5t at a time t. When t = 0, its velocity is 3 and its position s = 2. When t = 1, what is the position s.

10. An automobile accelerates from a standing start with a constant acceleration of 0.6 m/s2. How far does it travel in the first 10 s?

11. Suppose that the acceleration a(t) of a particle at time t, is given by a(t) = 6t − 3, v(2) = 4 and s(3) = 6 where s(t) is the position function. Find v(t) and s(t).

Chapter 8: The Integral 433

12. A tree farm sells a bush after 7 years of growth. The growth rate of the tree during the first seven dh years is approximated by = 1.5t + 5 , where t is the time in years and h is the height in cm. The seeddt lings are 10 cm tall when planted t = 0.

a. Find the height after t years

b. How tall are the bushes when they are sold?

434

Chapter 8: The Integral

8.2 Solving Differential Equations Graphically Warm Up dy Given that = 2 x + 1 , find the slope at each point (x, y). dx Use each point (x,y) to calculate dy dx

dy dx ( x , y )

(−3,1), (−3, 2), (−3, −1), (−3, −2) (−2,1), (−2, 2), (−2, −1), (−2, −2) (−1,1), (−1, 2), (−1, −1), (−1, −2) (0,1), (0, 2), (0, −1), (0, −2) (1,1), (1, 2), (1, −1), (1, −2) (3,1), (3, 2), (3, -1), (3, -2)

At each of the points (x, y) draw a short line through the point with slope dy . dx ( x , y )

Can you sketch a graph of y through the point (0, 1)?

What does function y?

dy tell us about the dx

Chapter 8: The Integral 435

Definition: Slope Field A slope field uses the idea of local linearity; that is, if a function is differentiable at a point, then the tangent line approximates the function close to that point. By drawing a slope field, we can graphically “see” the family of functions or the general solution to the given differential equation. The slope field is like the wind blowing through a field of grass. You can see the change behavior that the differential equation is describing. Then by choosing a starting point, an initial value, one can “determine” the particular solution. Some differential equations can only be solved by considering the slope field.

Example 1: Sketching a Slope Field dy Consider the differential equation = x − 2 through the point (4, 2). What does the slope field look dx like?

What curve passes through the initial value?

How to Do It...

What to Think About dy tell us about dx the slope at each point?

What does

How can we use the slope field to draw a curve through the initial point?

436

Chapter 8: The Integral

Your Turn ... Sketch a graph of the slope field for

dy = x2 dx

Example 2: Determine a Specific Solution to a Differential Equation Determine the slope field for

dy 1 = . Sketch a solution function that passes through (0, 1). dx y

Confirm algebraically the solution to the differential equation.

How to Do It...

What to Think About dy tell us about dx the slope at each point?

What does

How can we use the slope field to draw a curve through the initial point?

Why does the solution show only “half” of the curve?

Chapter 8: The Integral 437

Example 2: Determine a Specific Solution to a Differential Equation - continued

Why must we choose between the positive and negative versions of the y function

How do we decide which is the solution?

Your Turn ... a) What is the slope field for the differential equation

438

Chapter 8: The Integral

dy x = − dx y

b) Draw the solution to this differential equation that passes through the point (0, −2), then solve the differential equation algebraically given this initial condition.

Example 3: Solving a Non-Separable Differential Equation Graphically: dy Draw solution curves for the differential equation = x − y that pass through each of the following dx points: (0, −2), (0, 1), (0, 2)

How to Do It...

What to Think About Where is the slope zero? Why?

Can you find an algebraic solution to this differential equation?

Chapter 8: The Integral 439

Your Turn ... Sketch the slope field for

440

Chapter 8: The Integral

dy = x + y then sketch the curve through (0, 1). dx

8.2 Practice 1. Draw slope fields for the following differential equations

dy = x+2 dx

dy = y2 dx

dy = sin x dx

dy = x− y dx

Chapter 8: The Integral 441

2. a) Draw the slope field for the differential equation

dy 1 = . dx x

b) Draw the unique curve that passes through the point (2, 1).

c) Solve the differential equation with this initial condition.

442

Chapter 8: The Integral

3. Match the slope field (a – d) to one of the differential equations (i – iv). a)

dy = 2y dx

ii.

dy = x+ y dx

iii.

dy =x dx

iv.

dy = x 2 (1 − y ) dx

4. Where would the slope field for the differential equation

dy 1 = have vertical segments? dx x − y

Chapter 8: The Integral 443

5. Draw the solution of each differential equation. Then, determine the corresponding differential equation below each function.

dy = sin x dx

(π , 0)

dy = cos dx

π   ,2 2 

dy 1 = dx x

(e,3)

dy = x2 dx

(3, 4)

444

Chapter 8: The Integral

6. Consider the differential equation

dy = x − 2y dx

a. On the axis provided, sketch a graph of the slope field for the given differential equation at the 6 points indicated.

b. Find d y in terms of x and y. Determine the concavity of all solution curves in Quadrant I. Explain. dx 2

c. Let y = f(x) be the particular solution to the differential equation with the initial condition f(3) = 2. Does f have a relative minimum, relative maximum, or neither at this point? Explain.

Chapter 8: The Integral 445

8.3 Approximating Area Under a Curve Using Riemann Sums Warm Up Consider a car travelling with the following velocity-time information. a. How far has the car travelled between t = 60s and t = 80s

b. How far has the car travelled between t = 0s and t = 20s

Can you determine the exact distance travelled when the function is a curve?

Definition: Definite integral The process of evaluating a product in which one factor varies is called finding the definite integral. The definite integral will give you the area under the curve. We can approximate the definite integral in several ways.

446

Chapter 8: The Integral

Example 1: Estimating the Area Under a Curve Geometrically. Graph the function y =

How to Do It...

1 x on the grid and calculate the area under the curve on [2, 8]. 2

What to Think About Can you determine an exact answer in this case?

Is there another way other than counting we can find the area under the curve?

What shapes can we use to determine the area?

What other shape can be used to represent the area?

Chapter 8: The Integral 447

Your Turn ... Graph y = 3x and calculate the area under the graph from x = 1 to x = 4

448

Chapter 8: The Integral

Example 2: Rectangular Approximation Methods to Determine the Area Under a Curve −0.5 x 2 + 8 and make a table of values. Approximate the area under the Graph the function f ( x) = curve from x = 0 to x = 4 by using 4 quadrilaterals of equal width.

How to Do It...

What to Think About How do you determine the width of each sub interval? What does i represent? What does n represent?

What height should we choose for each rectangle?

What does the ai represent in the area equation?

How do you determine f (ai ) for each rectangle?

Does the left rectangular approximation method overestimate or underestimate the area in this example?

Chapter 8: The Integral 449

Example 2: Rectangular Approximation Methods to Determine the Area Under a Curve - continued

What height should we choose for each rectangle?

How do you determine f (bi ) for each rectangle?

Does the right rectangular approximation method overestimate or underestimate the area in this example?

Provide any counterexamples where this is not always true What do a and b represent?

Is the trapezoidal approximation an overestimation or an underestimation for this function?

450

Chapter 8: The Integral

Example 2: Rectangular Approximation Methods to Determine the Area Under a Curve - continued Compared to the trapezoidal approximation, what is different about the average used in the midpoint approximation?

Which is the most accurate approximation, LRAM, RRAM, Trapezoidal, or MRAM?

Rule

Formula Note: These formulas work only for n equal sub intervals.

Left Endpoint x0 = a

n −1

Right Endpoint xn = b x1 = a + ∆x

b−a

∫ f ( x)dx ≈ n [ f ( x ) + f ( x ) + f ( x ) + ... + f ( x )] 1

Midpoint

b − a   x0 + x1   xn −1 + xn    x1 + x2   + ... f  + f   2  2  2   

∫ f ( x)dx ≈ n  f  a

Trapezoidal

b−a

∫ f ( x)dx ≈ n [ f ( x ) + f ( x ) + f ( x ) + ... + f ( x )]

 b − a  1 

∫ f ( x)dx ≈  n   2  [ f ( x ) + 2 f ( x ) + 2 f ( x ) + ... + 2 f ( x ) + f ( x )] 0

n −1

Chapter 8: The Integral 451

Your Turn ... A particle starts at x = 0 and moves along the x-axis with velocity v(t = ) t 2 − 1 for time t ≥ 0 . Where is the particle at t = 4? Approximate the area under the curve using four quadrilaterals of equal width. a. Calculate the left-hand area (LRAM):

b. Calculate the right-hand area (RRAM):

c. Calculate the trapezoid area:

d. Calculate the midpoint area (MRAM):

) t 2 − 1 that passes through the point x(0) = 0. e. Solve the differential equation x′(t = Then, determine x(4).

452

Chapter 8: The Integral

Example 3: Evaluate A machine fills a milk carton with milk at a constant rate. The rates (in cases per hour) are recorded at hourly intervals during a 11-hour period, from 6:00 am to 5:00 pm. Use the trapezoidal approximation method with n = 11 to determine approximately how many cases of milk are filled by the machine over the 11-hour period.

How to Do It...

What to Think About Why is the value 17 used for 5:00 pm?

Chapter 8: The Integral 453

Your Turn ... The economy is continuously changing, we can analyze it with certain measurements. The following table records the annual inflation rate as measured each month for 13 consecutive months. Use the trapezoidal rule with n = 12 to find the overall inflation rate for the year.

454

Chapter 8: The Integral

January February March April May June July August September October November December January

0.04 0.04 0.05 0.06 0.05 0.04 0.04 0.05 0.04 0.06 0.06 0.05 0.05

8.3 Practice 1. The function f is continuous on the interval [2, 8] and has values that are given in the table. Using the 8 subintervals [2, 5], [5, 7] and [7, 8], what is the trapezoidal approximation of ∫ f ( x)dx ? 2

x f(x)

2 10

5 30

7 40

8 20

2. The following table shows the speedometer readings of a truck, taken at ten-minute intervals during one hour of the trip. Use the table and the midpoint rule to estimate the distance that the truck traveled in the hour. Watch your units! Time (min) Speed (km/h)

0 40

10 45

20 50

30 60

40 70

50 65

60 60

Chapter 8: The Integral 455

e− x dx ? 3. If three equal subdivisions of [− 4, 2] are used, what is the trapezoidal approximation of ∫ 2 −4

1 4. Use the Midpoint Rule with n = 5 to approximate ∫ dx . x 1

5. Use the Trapezoidal Rule with n = 2 to approximate the integral ∫ x dx . 3

1 dx . x +1 0

6. Use the Right Endpoint Rule with n = 4 to approximate the integral ∫

456

Chapter 8: The Integral

7. The temperature, in degrees Celsius ( C ) , of the water in a pond is a differentiable function W of time t.

t days

W(t) ( C )

0 3 6 9 12 15

20 31 28 24 22 21

The table above shows the water temperature as recorded every 3 days over a 15-day period.

a) Use the data from the table to find an approximation for W’(12). Show the computations that lead to your answer. Indicate units of measure.

b) Determine the LRAM over the time interval 0 ≤ t ≤ 15 days with 5 subintervals.

c) Determine the RRAM over the time interval 0 ≤ t ≤ 15 days with 5 subintervals.

d) Determine the MRAM over the time interval 0 ≤ t ≤ 12 days with 2 subintervals .

Chapter 8: The Integral 457

8.4 Calculating Area Under Functions Graphically Warm Up 2 x + 4, −4 < x < 1 Graph the function y =  and make a table of values. Find the area under the curve 1< x < 4 6, from x = −4 to x = 4.

What happens when part of the graph is below the x-axis?

What is the difference between net area, and total area?

458

Chapter 8: The Integral

Example 1: Determining the Area Under Piece-Wise Functions Geometrically Evaluate the following integral given the graph of f(x) below:

How to Do It...

What to Think About Where should you start?

Do we need to find the equation of each linear section?

What happens when some of the area is below the x-axis?

Does the integral ask you to determine the net area or the total area?

What is the difference between the net area and the total area?

Chapter 8: The Integral 459

Your Turn ... Given the piece-wise function below, write an integral statement and evaluate.

a. Find the net area on [0, 7].

b. Find the total area on [0, 7].

460

Chapter 8: The Integral

Example 2: Finding the Area Under a Curve of Known Shape: 3

2 Evaluate ∫ 9 − x dx −3

What to Think About

How to Do It...

What shape is the curve of the function?

What formula can be used to determine the area under the curve over the interval [−3, 3]?

Your Turn ... 4

2 Using the graph calculate the integral ∫ 16 − x dx . −4

Chapter 8: The Integral 461

Example 3: Solving Real-World Integration Problems using Geometry A car moves along the highway at a constant rate of 65 miles per hour from 6:00 am to 8:00 am. Express the total distance travelled as an integral and evaluate.

How to Do It...

What to Think About Under which type of function willthe shape under the curve be a rectangle?

Consider the units that determine the area function, why is the result in miles?

Your Turn ... Find the output from a pump producing 30 gallons per minute during the first 2 hours of operation. Express your answer using correct units.

462

Chapter 8: The Integral

Example 4: Evaluating Definite Integrals Using a TI Calculator 3

2 Evaluate ∫ 9 − x dx −3

How to Do It...

What to Think About Which buttons on your calculator should you press to evaluate a definite integral?

Your Turn ... 4

2 Use your calculator to evaluate ∫ 16 − x dx . −4

Chapter 8: The Integral 463

Example 5: Solving Real-World Problems using a Calculator The rate of consumption of oil in Canada during the 1990’s (in billions of barrels per year) is modt

eled by the function C = 22.08e 30 , where t is the number of years after January 1, 1990. Find the Total consumption of oil in Canada from January 1, 1990 to January 1, 2000.

How to Do It...

What to Think About Why are the constants of integration 0 and 10?

Could you use different constants to get the same result?

What are the units obtained after integrating?

Your Turn ... The rate at which our homes consume electricity is measured in kilowatts. Most homes consume electricity at a rate of 1 kilowatt for 1 hour. Suppose that the average consumption rate of your home is  πt  modeled by the function C (= t ) 3.6 − 2.4sin   , where C(t) is measured in kilowatts and t is measured  12  in hours past midnight. Find the average daily consumption for your home, measured in kilowatt hours.

464

Chapter 8: The Integral

8.4 Practice 1. Determine the area under each graph

2. Evaluate the integral using geometrical shapes.

∫ 5dx

−3

∫

2dx

−1

∫ 2 dx

−4

∫ −15dx 3

Chapter 8: The Integral 465

3. Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. 4

x ∫2 2 + 4dx

∫ (5 − x)dx 0

∫ xdx 0

∫ (1 − x )dx

−1

∫ r − x dx

−r

466

Chapter 8: The Integral

4. The graph of f below consists of line segments and a semicircle. Evaluate each definite integral using geometric formulas.

∫ f ( x)dx

∫ f ( x) dx

−4

∫ f ( x)dx

−4

∫ f ( x)dx

−4

∫ [ f ( x) + 2] dx

−4

Chapter 8: The Integral 467

8.5 Chapter Review 1. Consider the curve given by x 2 y − xy 3 = 12 a) Show that

dy 2 xy − y 3 = dx 3 xy 2 − x 2

b) Find all points on the curve whose y-coordinate is 1. Write an equation for the line tangent to the curve at each of the points.

2. Consider the differential equation

468

Chapter 8: The Integral

dy 2 x3 . Find the solution to the differential equation f(0) = 1. = dx e y

3. A function is differentiable for all real numbers. The point (2, 1) is on the graph of y = f(x) and the dy slope at each point is given by = y 3 (3 + x 2 ) . dx a. Find the second derivative and evaluate it at the point (2, 1).

dy b. Find y = f(x) by solving = y 3 (3 + x 2 ) using the initial condition (2, 1). dx

Chapter 8: The Integral 469

dy dx

= 4. Consider the differential equation

1− x , y

y≠0

a. On the axis provided, sketch a possible slope field.

b. Find the solution given the initial condition (2, 1).

5. For the following use the LRAM, RRAM, Trapezoidal, and MRAM to approximate the definite integral for the stated value of n. 4

∫ xdx, n = 6

470

Chapter 8: The Integral

∫ x + 2dx,

n=8

∫ x + 1dx, 4

n=4

6. Evaluate the definite integral using geometrical shapes.

∫ 8dx

∫ −2 x + 5dx 2

∫ 2 x − 8 dx

∫ 16 − x dx 2

−4

7. Given ∫ f ( x)dx = 8 and ∫ g ( x)dx = −4 find:

∫ f ( x) + g ( x)dx 2

∫ f ( x) − 2 g ( x)dx 2

∫ f ( x) − 5 f ( x)dx 2

Chapter 8: The Integral 471

8. The graph of g(x) is below. Evaluate each definite integral using geometric formulas.

∫ f ( x)dx

−10

∫ f ( x)dx

−5

∫ f ( x) + 2dx

−10

∫ f ( x)dx

−10

472

Chapter 8: The Integral

Chapter 9: Integrals and the Fundamental Theorem of Calculus The secret connection between how fast something is changing (derivatives) and how much of it you have (integrals).

n=5

n = 15

n = 50

n= ∞

EXPLORING THE BIG IDEA

In this chapter you should be able to: • Analyze functions defined by an integral. • Evaluate definite integrals. • Interpret the meaning of a definite integral within a problem. • Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume.

Chapter 9: Integral and the Fundamental Theorem of Calculus 473

9.1 Integrating Using the Fundamental Theorem Warm Up 13

1 Approximate ∫ dx using a left rectangular approximation method with: x 1

a) 3 equal subintervals x

1 x

1 2 3 4 5 6 7 8 9 10 11 12 13

b) 4 equal subintervals

c) 6 equal subintervals

d) 12 equal subintervals

1 e) Use your calculator to determine the exact value of ∫ dx x 1

What happens to the approximations as we increase the number of rectangles?

474

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 1: Making the Connection Between the Area Under a Curve and the Antiderivative 8

a) Write the area ∫ f ( x)dx in terms of ∫ f ( x)dx and ∫ f ( x)dx given f ( x) = x , then calculate the value of the integral.

b) Determine ∫ f ( x) = dx

How to Do It...

What to Think About How can you split up the area under the curve from 0 to 8 at x = 2 into two known shapes?

What is the connection b

between ∫ f ( x)dx and a

∫ f ( x) = dx ?

Chapter 9: Integral and the Fundamental Theorem of Calculus 475

Your Turn ...

Write the area ∫ f ( x)dx in terms of ∫ f ( x)dx and ∫ f ( x)dx given f ( x) = 2 x then calculate the value of the integral over the interval [2, 4].

476

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: The Definite Integral (more formally)

Let f be a continuous function defined on a closed interval [a, b]. b−a Let the interval be subdivided into n intervals of equal length ∆x = . n As we increase the number of subdivisions, n → ∞ , and ∆x → 0 , we can use an infinite number of infinitely thin rectangular slices to calculate a more accurate approximation for the area under a curve by using a limit. The Riemann sum of all these rectangles calculates the area under the curve as follows, n

lim ∑ f (ci )∆x n →∞

i =1

where ci is an x-value chosen arbitrarily in the i th subinterval. Mathematicians chose new notation using the differential, ∆x ≈ dx , to write the definite integral of f over [a,b] as: n

i =1

lim ∑ f (ci )∆x =∫ f ( x)dx n →∞

If y = f(x) is integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b, b

Net Area = ∫ f ( x)dx a

The above statement is read “integral of f from a to b”. a lower limit of integration, b upper limit of integration, f(x) is the function of the integrand. ∫ the integral sign, dx is the variable of integration. Note: Use the app to explore this concept more. https://qrs.ly/t5gge2z

Chapter 9: Integral and the Fundamental Theorem of Calculus 477

Defiinition: The Fundamental Theorem of Calculus If a function f(x) is continuous on a closed interval [a, b] and F(x) is the antiderivative of f(x) on the interval [a, b], then b

)dx F (= x) F (b) − F (a ) ∫ f ( x= a

Note: It is not necessary to include the constant of integration C in the antiderivative because b

)dx F ( x) + C ∫ f ( x= a

a b

= [ F ( x) + C ]ba

= [ F (b) + C ] − [ F (a) + C ] = F (b) − F (a )

478

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 2: Using the Fundamental Theorem of Calculus Evaluate the following: 4

2 ∫ ( x + 5)dx

∫ x dx

−1

What to Think About

How to Do It...

How do you use the antiderivative and the fundamental of theorem of calculus to solve? Why is the answer in b) negative? How does this relate to the graph? What conclusion can you make?

How do you know before you start if the answer will be negative or positive?

Your Turn ... Evaluate the following: a)

∫ dx 4

∫ cos xdx

−π 2

Chapter 9: Integral and the Fundamental Theorem of Calculus 479

Example 3: Connecting the FTC to the Net Area Under the Curve π 2

Find ∫ sin xdx graphically and confirm your answer using the FTC. −

How to Do It...

What to Think About What do you notice about the shape?

Is there any symmetry?

Can you guess what the net area will be?

Compare the areas on either side of the x-axis, what conclusion can you make?

Does your answer make sense?

How do you know before you start if the answer will be negative or positive or zero?

480

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: Even Function and Odd Function If f is a function whose domain contains –x whenever it contains x, then a. f is even if f(–x) = f(x) for all x in the domain of f a

−a

∫ even dx = 2∫ even dx b. f is odd if f(–x) = –f(x) for all x in the domain of f a

∫ odd dx = 0

−a

Your Turn ... Evaluate the following:

3 ∫ x dx =

−1

∫π − cos xdx =

−

Chapter 9: Integral and the Fundamental Theorem of Calculus 481

SUMMARY – Graphical connections to the Fundamental Theorem of Calculus The observations made can be summarized as follows: b

1. If f(x) is above the x-axis between a and b then ∫ f ( x)dx = F (b) − F (a) > 0 . a

2. If f(x) is below the x-axis between a and b then ∫ f ( x)dx = F (b) − F (a) < 0 . a

3. If f(x) is above and below the x-axis between a and b then b

∫ f ( x)dx = A − A + A which is the net area. 1

4. ∫ f ( x)dx + ∫ f ( x)dx = ∫ f ( x)dx

5. ∫ f ( x)dx = 0

6. ∫ f ( x)dx = − ∫ f ( x)dx

Note: a

When b > a, and we calculate the integral from b to a, dx < 0.

482

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 4: Changing the Limits of Integration When Using the Substitution Method π 4

Solve ∫ tan x sec2 xdx by substitution. 0

How to Do It...

What to Think About How would you integrate using substitution if this was an indefinite integral?

What values will u take, when π integrating over x ∈ 0,  ? 

4

Why must you change the constants of integration when using substitution?

Your Turn ... 2

3 2

Solve ∫ ( x + 1) x dx 3

Chapter 9: Integral and the Fundamental Theorem of Calculus 483

9.1 Practice Evaluate each integral: 2

∫ 2xdx

1   ∫−1  u − u 2  du

 1 − sin 2 x  ∫0  cos2 x  dx

2

∫  v  dv 1

∫ (4sec x tan x)dx

−π 3

−1

484

∫ 5 x + 6 dx

∫ (−3x + 2)dx

Chapter 9: Integral and the Fundamental Theorem of Calculus

∫ sin xdx 0

π 6

9. Given: ∫ sin 2 x cos 2 xdx . Let u = sin 2 x . Change the limits of integration and write the new integral in 0

terms of u, then solve.

2 x3 + 1 dx and leave the answer in exact form. 4 + x 2 x 1

10. Evaluate the integral ∫

11. Suppose ∫ f (2 x)dx = 10 , then evaluate ∫ f (u )du =

Chapter 9: Integral and the Fundamental Theorem of Calculus 485

12. Given ∫ f ( x)dx = 8, ∫ g ( x)dx = −3 , and ∫ f ( x)dx = 3 find

∫ f [( x) + g ( x)]dx

∫ 2 g ( x)dx 2

∫ f ( x)dx

−2

13. Given ∫ f ( x)dx = 0 and ∫ f ( x)dx = 5 , find

∫ f ( x)dx =

−2

486

Chapter 9: Integral and the Fundamental Theorem of Calculus

∫ 4 f ( x)dx =

−2

9.2 Derivative of a Definite Integral Warm Up Now we are going to analyze the Fundamental Theorem of Calculus more closely. Consider what the theorem states: b

)dx [ F ( x= ) ] F (b) − F (a ) ∫ f ( x= a

What is the relationship between f(x) and F(x) ?

For the integral ∫ 2xdx identify f(x) and F(x) ? 0

Can you state the Fundamental Theorem of Calculus in an alternative way?

Chapter 9: Integral and the Fundamental Theorem of Calculus 487

Definition: Fundamental Theorem of Calculus – stated another way b

)dx f (= x) f (b) − f (a ) ∫ f ′( x= a

This represents the notion that the integral of a rate of change function gives us the net change of the integral function over the interval of integration.

Example 1: Evaluate the Derivative of a Definite Integral a)

d tdt dx ∫1

How to Do It...

d (3t 2 − 2t )dt ∫ dx 5

What to Think About What does ∫ dx ask you to do?

What does

d ask you to do? dx

How does your answer relate to the original integrand?

What does ∫ dx ask you to do?

What does

d ask you to do? dx

How does your answer relate to the original integrand?

488

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: Second Fundamental Theorem of Calculus There is one other form of the FTC often called the Second Fundamental Theorem of Calculus: x

d f (t )dt = f ( x) dx ∫a

If the function g is defined as = g ( x)

∫ f (t )dt a ≤ x ≤ b then g’(x) = f(x) for a < x < b. a

In other words, g is an antiderivative of f and therefore f is the derivative of g. The second fundamental theorem of calculus is sometimes referred to as the Newton-Leibniz axiom

Your Turn ... What does each expression represent? a)

d 1 − sin 2 t dt dx ∫3

2 d e −5t dt ∫ dx 1

Chapter 9: Integral and the Fundamental Theorem of Calculus 489

Example 2: Evaluate Using the Second Fundamental Theorem with the Chain Rule a)

d cos t dt dx ∫1

How to Do It...

d f (t )dt , where u = g(x) dx ∫a

What to Think About What is different about this expression?

When differentiating which rule must you follow when there is a function in the place of a constant of integration?

What part of the expression indicates that the chain rule must be used?

490

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: Extension of the Second Fundamental Theorem of Calculus u ( x)

d f (t )dt= f (u ) ⋅ u ′ − f (v) ⋅ v′ dx v (∫x )

Your Turn ... What does each expression represent? a)

d sin tdt dx ∫1

d 1 − t 2 dt ∫ dx 1

Chapter 9: Integral and the Fundamental Theorem of Calculus 491

Example 3: Connecting the FTC to the Area Under the Curve x

Consider the Area Function A( x) ∫ t 2 dt 1

Evaluate the following

A(1)

A(2)

A(−1)

A’(x)

A’(3)

A’(−2)

How to Do It...

What to Think About What do the constants of integration mean graphically?

What does A(2) represent graphically?

492

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 3: Connecting the FTC to the Area Under the Curve - continued Why is the answer negative?

What does the derivative of the area function represent graphically?

How is the value of this expression represented graphically?

Why is this solution not negative?

Chapter 9: Integral and the Fundamental Theorem of Calculus 493

Your Turn ... x

Let g ( x) = ∫ f (t )dt where f(t) is the function graphed here. −2

a) Evaluate g(−2), g(2).

b) Estimate g(−3) g(−1), g(0).

c) On what interval is g(x) increasing?

d) Where does g have its maximum value?

494

Chapter 9: Integral and the Fundamental Theorem of Calculus

9.2 Practice Evaluate each expression: t

d − x2 e dx dt ∫0

d cos t dt dx ∫1 t

d 1 + t 4 dt ∫ dx tan x

For g ( x) = ∫ cos tdt , find g ′( x) . 1

d tan 3 a − a da ∫ dp 7

d dx ∫0

For g (= x)

cos x

1 − t 3 dt

∫ 1 + t dt , find g ′( x) . 2

For g ( x) = ∫ 3t sin tdt , find g ′( x) . x

1 dt , find g ′( x) . 2 + et 2x

For g ( x) = ∫

Chapter 9: Integral and the Fundamental Theorem of Calculus 495

t −3 dt for −∞ < x < ∞ . t +7 0

10. F ( x) = ∫ 2

a) Find the value of x where F attains its minimum value.

b) Find open intervals over which F is only increasing or only decreasing.

c) Find open intervals over which F is only concave up or only concave down.

496

Chapter 9: Integral and the Fundamental Theorem of Calculus

11. Let g ( x) = ∫ f (t )dt where f(t) is the function graphed to the right. 0

a) At what values of x do the local maximum and minimum of g occur?

b) Where does g attain its absolute maximum?

c) On what intervals is g concave downward?

Chapter 9: Integral and the Fundamental Theorem of Calculus 497

9.3 The Average Value of a Function Warm Up The area of the two shaded regions are equal. a) Find the area under the curve g(x) on the interval [1, 3].

g ( x) = x 2

b) Find the value of g(c), such that the area of the rectangle on the interval [1,3] is equal to the area under the curve of g(x) on the interval [1, 3]

c) How is g(c) related to the area of g(x)? g (c )

c d) Is c the midpoint of the interval?

The area under any curve, ∫ g ( x)dx , can be repa

resented as a rectangle with height g(c) over the interval [a, b]. Can you write this relationship as an equation?

498

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: Average Value of a function 1

If f is integrable on the interval [a, b] then f (c) = f ( x)dx is the average value of the function b − a ∫a on the interval.

Example 1: Finding the Average Value of a Function on an Interval a) Find the average value of the function f ( x) = x 2 − 2 x + 1 on the interval [0, 2]. b) Find c such that f(c) is the average value of on the interval [0, 2].

How to Do It...

What to Think About Graphically, what does the average value represent?

Why does dividing the integral by the width of the interval gives us the average value?

Chapter 9: Integral and the Fundamental Theorem of Calculus 499

Example 1: Finding the Average Value of a Function on an Interval - continued What value will the function have at c?

Your Turn ... a) What is the average value of the function f ( x) = 4 x on the interval [0, 4]?

b) Find c such that f(c) is the average value of on the interval [0, 4].

500

Chapter 9: Integral and the Fundamental Theorem of Calculus

Definition: Mean Value Theorem for Integrals If f is a continuous function on [a, b], then there exists a number c in [a, b] such that: b

( x)dx f (c)(b − a ) ∫ f= a

The area under the graph of f from a to b is equal to the product of the interval length (b−a)and the average value f(c). The average value is the height of the area represented as a rectangle. The average value (mean value) of the rate of change function can be written and shown in two ways: f ′(c) =

1 f ′( x)dx b − a ∫a

f ′(c) =

f (b) − f (a ) b−a

You will recognize the first as the average value of f’ on [a, b], otherwise known as the average rate of change of f on [a, b]. The second, you will recognize from the mean value theorem as the average rate of change of f on [a, b]. Using the fundamental theorem of calculus, we can see how these two expressions are equivalent. b

1 1 = f ′( x)dx [ f (b) − f (a )] ∫ b−a a b−a =

f (b) − f (a ) b−a

Thus, if we are looking for the average rate of change for a function and we are given the derivative function, we use the mean value theorem for integrals f ′(c) =

1 f ′( x)dx b − a ∫a

If we are looking for the average rate of change for a function and we are given the original function, we use the mean value theorem for derivatives f ′(c) =

f (b) − f (a ) b−a

Chapter 9: Integral and the Fundamental Theorem of Calculus 501

Example 2: Finding the Average Rate of Change Find the average velocity for 2 ≤ t ≤ 4 given each situation below a) The position function x(t= )

t3 2 −t 3

b) The velocity function v(t = ) t 2 − 2t

How to Do It...

What to Think About What formula should you use to calculate the average value?

Why do you not use an integral?

What formula should you use to calculate the average value?

What do you get when you integrate a rate of change?

How do you determine the average value given the net change over an interval?

502

Chapter 9: Integral and the Fundamental Theorem of Calculus

Your Turn ... Show that the average velocity of a car over a time interval [a, b] is the same as the average of its velocities during the trip.

9.3 Practice 3

1. Let f be a continuous function on [0, 3]. If, 1 ≤ f ( x) ≤ 5 then the greatest possible value of ∫ f ( x)dx is. 0

) x 3 − 3 x 2 on the interval [−2, 1]. 2. Find the average value of the function f ( x=

3. Let be a continuous function on [0, 3]. If 6 ≤ f ( x) ≤ 12 , what are the greatest and smallest possible 3

values of ∫ f ( x)dx ? 0

Chapter 9: Integral and the Fundamental Theorem of Calculus 503

4. Find the average value of the function on the given interval f ( x) = x 2 + 3 x − 1,

[−1, 2]

f ( x= ) x 3 + 1, [2, 4]

5. Given the graph of f 8

a. Evaluate ∫ f ( x)dx 0

b. Determine the average value of f on the interval [0, 8]

c. Determine the answer to a, b is the graph if translated 3 units upwards

504

Chapter 9: Integral and the Fundamental Theorem of Calculus

9.4 The Integral as Net Change Warm Up Given the graph of the velocity of a particle moving on the x-axis. The particle starts at x = 2 when t = 0.

a) What is the net distance traveled by the particle during the trip?

b) Find where the particle is at the end of the trip.

c) Find the total distance traveled by the particle.

Chapter 9: Integral and the Fundamental Theorem of Calculus 505

Example 1: Motion Along a Line The velocity of a particle is given by v(t = ) t2 − horizontal x-axis for [0, 4],

5 cm/sec. If the particle is moving along a (t + 2) 2

a) Draw the graph of the velocity function and describe the motion of the particle. b) Find the net distance traveled by the particle in the first 4 seconds. c) Suppose the initial position of the particle is s(0) = 4. What is the particle’s position at t = 1, t = 4 . d) Use your calculator to find the total distance traveled by the particle in the first 4 seconds.

How to Do It...

What to Think About How do we know from the graph when it is moving left or right? What is the point where the direction changes?

When you integrate the rate of change function what do you get?

What does s(0) = 4 represent?

506

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 1: Motion Along a Line - continued Why must we consider this when determining the particle’s position at a different time?

Which buttons do you need to press to do this on your calculator?

Your Turn ... The velocity of a particle is given by = v(t ) 3cos 2t − zontal x-axis for [0, 6]

sin t 2 cm/sec. If the particle is moving along a hori0.2t

a) What is the net distance traveled by the particle in the first 6 seconds?

b) Suppose the initial position of the particle is s(0) = 2. What is the particle’s position at t = 2, t = 5.

Chapter 9: Integral and the Fundamental Theorem of Calculus 507

c) What is the total distance traveled by the particle in the first 4 seconds?

Making Predictions Using the FTC Since the integral of a rate of change function gives us the net change of the original function, we b b dt ∫ v(t )= dt x(t ) + C and ∫ v(t= use the facts that ∫ x′(t )= )dt x(= t ) a x(b) − x(a ) to determine future or a past values of the function. We can rearrange the fundamental theorem of calculus to create the following formula given the velocity function and one value, x(a), to find a future value x(b). In other words, the value of x at b is the value of x at a plus the net change of x from a to b. b

x= (b) x(a ) + ∫ v(t )dt a

or given x(b), to find a past value x(a) b

x= (a ) x(b) − ∫ v(t )dt a

508

Chapter 9: Integral and the Fundamental Theorem of Calculus

Did You Know? Three hundred years before Newton and Leibniz first developed calculus Nicole Oresme (1323 – 1382), in his Treatise on the Configuration of Qualities and Motions proved geometrically that under uniform acceleration, the distance traveled is equal to the distance traveled at constant average velocity.

Example 2: Using your Calculator to Solve Real-World Applications using Net Change People enter a line for a turnstile at a rate modeled by the function r(t) given by 7   t 3  t  1− , 0 ≤ t ≤ 300 10 r (t ) =   60   300  0, t > 300 

where r(t) is measured in people per second and t is measured in seconds. As people go through the turnstile the exit the line at a constant rate of 0.85 person per second. There are 22 people in line at a time t = 0. a) How many people enter the line for the turnstile during the first 5 minutes? b) During the first 5 minutes, there are always people in line. How many people are in line at t = 300? c) After 5 minutes, what is the first time that there are no people in line? d) During the first 5 minutes, at what time will the number of people in line be a minimum? Find the number of people in line to the nearest whole number at this time. Justify your answer.

How to Do It...

What to Think About How do you determine the net change of people entering the line in the first 5 minutes?

Chapter 9: Integral and the Fundamental Theorem of Calculus 509

Example 2: Using your Calculator to Solve Real-World Applications using Net Change continued How many people are in line to begin with?

How many people enter the line in the first 5 minutes?

How many people exit the line in the first 5 minutes?

What equation can be used to model this situation?

Are there other strategies to get this answer?

Can we determine this by starting at t = 0?

How can the rate of change function P′(t ) be represented as a piecewise function?

Where do minimums occur?

How can you justify that a minimum has occurred?

How do you determine the net change in people in line?

510

Chapter 9: Integral and the Fundamental Theorem of Calculus

Example 2: Using your Calculator to Solve Real-World Applications using Net Change continued Can you write P(t) be as an equation?

Chapter 9: Integral and the Fundamental Theorem of Calculus 511

Your Turn ... Oil is pumped into an underground tank at a constant rate of 5 liters per second. Oil leaks out of the tank at a rate of

t + 9 liters per second, for 0 ≤ t ≤ 2 hours. At t = 0, the tank contains 115 liters of oil.

a) How many liters of oil leak out of the tank in the first 40 seconds?

b) How many liters of oil are in the tank at t = 40 seconds?

c) Write an expression for the total amount of oil in the tank at any time t.

d) At what time t for 0 ≤ t ≤ 100 seconds, is the amount of oil in the tank a maximum. Justify your answer.

512

Chapter 9: Integral and the Fundamental Theorem of Calculus

9.4 Practice

 π t −  4 1. A particle moves along the x-axis so that its velocity at time t is oven by v(t )= (t + 2) cos  . At 2 time t = 0, the particle is at a position x = 2.

a) Find the acceleration of the particle at time t = 1. Is the speed of the particle increasing at t = 1? Explain your answer.

b) Find all times in the open interval (0, 5) when the particle changes direction. Justify your answer.

c) What is the total distance traveled by the particle from t = 0 to t = 5?

d) What is the particles position at t = 4?

Chapter 9: Integral and the Fundamental Theorem of Calculus 513

2. A 100-gallon aquarium in a dentist’s office contains a rare collection of fish at t = 0. During a 12-hour period from 8:00 am t = 0 to 8:00pm t = 12 the doctor is cleaning the tank by refilling the tank at a rate ( − ( t +1)) of h(t )= 5 + 15e and during the same time interval there is water being removed from the tank at (t − 4) 2 . a rate of r (t ) = 10 cos 15

a) How many gallons of water is being pumped into the tank during the time interval 0 ≤ t ≤ 12 hours?

b) Is the amount of water in the tank rising or falling at time t = 8 hours? Give a reason for your solution.

c) How much water is in the tank at t = 12 hours?

d) At what time t between 8 am and 8 pm is the volume of the water in the fish tank th least?

514

Chapter 9: Integral and the Fundamental Theorem of Calculus

9.5 Chapter Review Find the average value over each interval.

x2 4 − , [0,3] 4

∫ (3x )dx = 27 on [−1, 2] 2

−1

2 + 3 x,

[0, 4]

4. Show that the average value of f’ is equal to the average rate of change of f over the interval [a, b]

Chapter 9: Integral and the Fundamental Theorem of Calculus 515

Evaluate the following definite integrals 3

2 ∫ (5 + x − 6 x )dx

−2

 x −3 ∫9  x  dx

−1

 x 3 + 27  ∫0  x + 3  dx

 5 

∫  8 x  dx 6

−1

∫ ( x + 2 ) dx 8

−8

10.

−2

∫ (8 + 3x − 6 x ) dx 2 25

−2

∫ (5 + x)( x + 2)( x + 1)dx

11.

−1

Evaluate the derivative of the following. x

d (5t + t 3 − 6t 2 )dt dx −∫2

12.

516

13.

Chapter 9: Integral and the Fundamental Theorem of Calculus

x d  1    dt dx ∫0  1 − t 2 

14.

x d  1    dt dx ∫0  t 3 − 3 

15.

d (t 3 + 1)10 dt ∫ dx 3 x

Change the limits of integration and then evaluate the function. 4

16.

 x  ∫0  x 2 + 9  dx

17.

18.

 1 − x −1  ∫1  x 2  dx  

19.

  1  ∫1  x ( x + 1)3  dx

∫ (4 x + 1)(4 x + 2 x − 3) dx 2

Chapter 9: Integral and the Fundamental Theorem of Calculus 517

20. Use the function f in the figure and the function g defined by to g ( x) = ∫ f (t )dt answer the following: 0

a) Complete the chart. x g(x)

b) Plot the points from the table.

c) What are the absolute extreme(s) on [0, 8]? Explain.

518

Chapter 9: Integral and the Fundamental Theorem of Calculus

21. The rate of consumption of an electric vehicle was recorded during a road trip along the coastal Chapman Peak’s Drive in Cape Town, South Africa. The table of selected values represents a twicedifferentiable function E(t) over time t for the interval of 0 ≤ t ≤ 40 minutes.

t (minutes) 0 10 15 25 30 40

kWh (per minute) .21 .28 .3 .34 .3 .26

a) Determine E’(20) and explain its meaning as it relates to E.

b) What can you state about the rate of consumption on the interval 15 ≤ t ≤ 30 ?

c) Determine the value of ∫ E (t )dt using a trapezoidal sum of 5 subintervals. Explain the significance of

1 E (t )dt 40 ∫0

Chapter 9: Integral and the Fundamental Theorem of Calculus 519

22. The rate of growth of the radius of a tree is given by the function πd  r= (d ) 2.25 + sin   where the radius r is written in terms of the # of  5  days, d. This is often due to environmental changes both seasonal and those due to natural disasters. a) Is the rate of the radial growth of the tree increasing or decreasing at d = 62?

b) How much has the tree’s radial distance grown between days 30 and 80?

c) Suppose that a forest ranger measures the tree’s radius to be 7cm at the one-month point and then determines that this function represents the tree’s radial growth starting after its first month. What is the tree’s expected radius after 4 years? What was the average rate of growth over this time frame? Determine the age of the tree at this point given the following process: • Find the tree’s radius • Subtract ¼”−1” to account for bark (dependent upon the species of tree) • Research the average ring width online

• Age=

520

radius-bark average ring width

Chapter 9: Integral and the Fundamental Theorem of Calculus

Chapter 10: Area and Volume

It’s like having a superpower to measure anything Every day you see examples of how integration is used in a wide variety of contexts. For example: Consumer Testing A car magazine tests two kinds of engine. One engine 2 had acceleration modeled by f (t )= 6 + .7t , ft/s , t seconds after starting from rest. The acceleration of a turbocharged model could be 6 1.6t + 0.05t 2 , ft/s 2 , t seconds after starting approximated by g (t ) =+ from rest. How much faster is the turbocharged model at the end of a 10 second test run?

Population Predictions A government estimates that the population of the country (in 1000 people per year) will grow at the rate of P = 60e0.02t . If an education program is instituted, they believe the population growth will change to P =−t 2 + 60 over a 5-year period. How many fewer people will be in the country if the education program is implemented and is successful?

EXPLORING THE BIG IDEA In this chapter you should be able to: • Evaluate definite integrals. • Interpret the meaning of a definite integral within a problem. • Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume.

Chapter 10: Area and Volume 521

10.1 Area Between Two Curves Warm Up Find the area between the x-axis and the curve y = x 2 from x = 1 to x = 4

Find the area under the curve y =

1 from x = 1 to x = 4 x

What would you do to find the area between the line, y = –1 and the functions above?

522

Chapter 10: Area and Volume

Example 1: Area Between Two Functions Over a Closed Interval Write an expression that represents the area over the interval [a, b]: a) Between f(x) and the x-axis

b) Between g(x) and the x-axis

c) Between f(x) and g(x)

How to Do It...

What to Think About What integral can you use to calculate the area under the curve?

How can you combine the first two areas to determine the area between these two curves?

How can you combine the difference of the areas into one integral expression?

Chapter 10: Area and Volume 523

Area of the Region Between Two Curves To determine the area between two curves, partition (slice) the interval into n equal sub intervals of ∆x . Use a rectangular area for each sub-interval with ∆x as the width and f ( xi ) − g ( xi ) as the th

height of the i sub-interval. By taking the limit as n → ∞ , we can find the exact area using the definite integral. Note, the region will be bounded on an interval between two x-values where x ∈ [ a, b] .

= Area lim ∑ ( f ( xi ) − g ( xi ) )∆x n →∞

i =1

∫ ( f ( x) − g ( x) )dx a

= ∫ ( TOP-BOTTOM )dx a

Your Turn ... Find the area bounded by each region.

= y e x= , x 0,= x ln 4,= y 0

524

Chapter 10: Area and Volume

y =x 3 − 2 x 2 − 3 x

Example 2: Finding the Area Between Two Curves Find the area of the region between = y 3 x − x 2 and = y x2 − 5x .

How to Do It...

What to Think About What do the graphs look like?

What interval contains the region?

How do you determine the intersection points?

Sketch a rectangular slice with width ∆x . What is the area of this rectangular partition?

Does it matter that one curve is below the x-axis?

What are the limits of integration for this region?

Chapter 10: Area and Volume 525

Your Turn ... 1. Find the area between = y 6 x − x 2 and = y x2 − 2x .

−x + 2x + 3 2. Determine the area of the region enclosed by f ( x) = x − 4 x + 3 and g ( x) =

526

Chapter 10: Area and Volume

Example 3: Finding the Area Between Two Curves that Intersect at More than Two Points Find the area of the region between = y x3 + 3 x 2 and y= x + 3 .

How to Do It...

What to Think About What do the graphs look like?

At which x-values do the graphs intersect?

Sketch a slice of the region on the interval [−3, −1]. What is the area of this rectangular partition with width ∆x ?

Sketch a slice of the region on the interval [−1, 1]. What is the area of this rectangular partition?

What integrals can be used to find the area of each region?

Chapter 10: Area and Volume 527

Your Turn ... 3

1. Find the area between f ( x) = 3 x − x − 10 x + 8 and g ( x) = − x2 + 2x + 8 .

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Chapter 10: Area and Volume

10.1 Practice Sketch the region bounded by the graphs of the given equations, show a typical vertical slice and find the area of the region. 1. y =

1 2 and f ( x) = − x between x = 1, x = 2 2 x

2. x = y, y = 3x and x + y = 4

2 3

3. x = y and x = y 2

Chapter 10: Area and Volume 529

4. = y x 3 + 8 and = y 4x + 8

5. y =

4 and y = − x2 + 5 2 x

 π 5π  6. y = sinx and y = cos2x from  ,  6

530

Chapter 10: Area and Volume

6 

7. f ( x) =x − 3 x + 3 x, g ( x) =x

8. y =− x4 2 x2 , y = 2 x2

9. y = x

5− x , x-axis 3+ x

Chapter 10: Area and Volume 531

10. f ( x)= 2sin x + sin(2 x), y= 0, 0 ≤ x ≤ π and x = 0

x) 11. f (=

532

xe x= , y 0,= x 0,= x 1

Chapter 10: Area and Volume

10.2 Area Between Two Curves Using Horizontal Slices Warm Up Use your graphing calculator to find the area bounded by the graphs of f ( x) = 4 cos x and g ( x= ) x 2 − 4 . Sketch the region and a sample rectangular partition.

Is there a way to determine where maximum and minimum values of a function occur?

Chapter 10: Area and Volume 533

Example 1: Finding the Area Bounded by a Top Curve and Two Different Bottom Curves Find the area bounded f ( x) = x , the x-axis and g ( x)= x − 2 .

How to Do It...

What to Think About Does the top curve stay the same for the entirety of the bounded region? Does the bottom curve stay the same for the entirety of the bounded region?

What is the area of a slice in each region with width ∆x ?

What integral can be used todetermine the area of the region defined by each bottom curve?

534

Chapter 10: Area and Volume

Your Turn ... 3

x + 6, g ( x) = x and h( x) = Find the area of the region bounded by f ( x) =

x . 2

Chapter 10: Area and Volume 535

Example 2: Finding Area Using Horizontal Slices Find the area bounded f ( x) = x , the x-axis and g ( x)= x − 2 .

How to Do It...

What to Think About What would a horizontal rectangular slice look like?

What is the vertical width of the slice?

What is the horizontal length of the slice?

What is the area of a rectangular slice?

What integral can be used to determine the area of the region between the right curve and the left curve?

What are the limits of integration?

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Chapter 10: Area and Volume

Area of the Region Between Two Curves using Horizontal Partitioning To determine the area between two curves, using horizontal partitions (slices) the interval into n equal sub-intervals of ∆y . Use a rectangular area for each sub-interval with ∆y as the width and f ( yi ) − g ( yi ) as the length of the ith sub-interval. By taking the limit as n → ∞ , we can find the

exact area using the definite integral. Note, the region will be bounded on an interval that ranges between two y-values where y ∈ [c, d ] . n

AREA == lim ∑ ( f ( yi ) − g ( yi ) ) ∆y AREA n →∞

i =1

∫ ( f ( y) − g ( y) ) dy

∫ ( RIGHT − LEFT ) dy c

Your Turn ... x 2 y 2 − 4 and x = y 2 Find the area bounded=

Chapter 10: Area and Volume 537

Example 3: Finding Area with a Graphing Calculator Find the area bounded y = x 3 and = x y2 − 2 .

How to Do It...

What to Think About How can you graph the given relations on your calculator?

Which slicing method, horizontal or vertical, makes the most sense to use in this case?

What integral can be used to determine the area of the region between the right curve and the left curve?

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Chapter 10: Area and Volume

Example 3: Finding Area with a Graphing Calculator - continued

What are the limits of integration?

Chapter 10: Area and Volume 539

Your Turn ... Find the area of the region bounded by the following curves x = siny, y = 0, y = cosx and x ≥ 0 using horizontal slices.

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Chapter 10: Area and Volume

10.2 Practice Find the area of the enclosed region. 2

1. x = y and y − x = 2 between y = −2, y = 3

2. f ( x) =

1 x2 and g x = ( ) 1 + x2 4

2 3. f ( y ) = y and g ( y )= y + 2

y ) y (2 − y ) and g ( y ) = − y 4. f (=

Chapter 10: Area and Volume 541

5. f ( y ) =

y 16 − y 2

and g ( y ) = 0 and y = 3.5

6. y 2= 4 + x and y 2 + x = 2

7. f ( y ) = y 2 and g ( y )= y + 12

542

Chapter 10: Area and Volume

10.3 Volumes of Solids Warm Up You are given a roll of sushi and want to determine the volume of the entire roll. a. Cut the sushi up into slices of width ∆x . Sketch the shape of each slice.

b. Write an expression for the volume of one slice with width ∆x and radius f(x)

c. Imagine the top of the sushi roll can be represented by the line f ( x) = 2 . The 3-D sushi roll is a cylinder that is created by rotating around the x-axis so that the axis goes through the center of the roll. Sketch the roll over the interval [a, b] and label one slice with ∆x , f ( x)

d. Create an equation to find the volume of the entire roll.

How can you determine the volume using infinitely thin compact discs (cylinders)?

Chapter 10: Area and Volume 543

Definition: Rotational volume – solid of revolution Let f(x) be a continuous function on [a, b]. The volume V of the solid of revolution generated by x a= , x b and the x-axis around the x-axis revolving the region bounded by the graphs of f ( x),= is: b n b−a 2 2 = V lim ∑ π = f x ∆ x f= ( ) π ( x) ) dx, ∆x ( [ i ] ∫ n →∞ n i =1 a Note: f(x) is the radius R of the solid of revolution from the x-axis to the outer-most function.

Example 1: Calculating the Volume of a Solid Revolution About the x-axis ) x 2 + 1 , find the volume of the solid generated by revolving the region under the graph of f If f ( x= from x = −1 to x = 1 about the x-axis.

How to Do It...

What to Think About What does this shape look like?

What is the shape a slice that is perpendicular to the x-axis?

What is the width of the slice?

What other dimension do I need to determine the volume of the slice?

What is the expression for the volume of a slice?

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Chapter 10: Area and Volume

Your Turn ... Find the volume of the solid generated by revolving the region under the graph of f about the x-axis.

f f( x( x) )== sin sinxx[ 0, [0,ππ] ]

f ( x)= 4 − x 2

Chapter 10: Area and Volume 545

Example 2: Calculating the Volume of a Solid of Revolution About the y-axis Find the volume of the solid generated by revolving the region between the graph of x = 3 y 2 , y = −1, y = 1 and x = 0 about the y-axis.

How to Do It...

What to Think About What is the shape a slice that is perpendicular to the y-axis?

What is the width of the slice?

What is the radius of the slice?

What is the expression for the volume of each slice?

What are the limits of integration?

Definition: Rotational volume – revolving a region The volume V of the solid of revolution generated by revolving the region bounded by the graphs of g ( y ),= y c= , y d and the y-axis around the y-axis is: n

= V lim ∑ π = y ) ) dy, ∆y [ g ( yi )] ∆y π ∫ ( g (= n →∞

i =1

d −c n

Note: g(y) is the radius R of the solid of revolution from the y-axis to the outer-most function. 546

Chapter 10: Area and Volume

Your Turn ... What is the volume of region bounded by x = 1 − y , y =0, x =0 , if the region is rotated around the y-axis?

Chapter 10: Area and Volume 547

Example 3: Calculating the Volume of a Solid of Revolution About a Line Find the volume of the solid generated by revolving the region between the graph of y= 2 − x 2 and y = 1, around the line y = 1.

How to Do It...

What to Think About Draw the radius for one slice of the solid of revolution. What expression can be used to find the distance between the axis of rotation and the function to be rotated?

What is the radius of each slice?

Is it possible to write the radius in another way? Does it make a difference to the solution?

548

Chapter 10: Area and Volume

Your Turn ... The region, bounded by = y each volume.

the line y = 3

x + 3 , y = 3 and x = 9 , is rotated about each of the following lines. Find

the line x = 9

Chapter 10: Area and Volume 549

10.3 Practice Set up and evaluate the integral that gives the solid formed by revolving about the indicated axis or line.

y =− x + 1 [0, 1] and the x-axis

y = x [1, 4] and the line y = 2

y= 4 −

x 2 , about the line y = 2 4

= y x= , y 0,= y 4 , about the line x = 6

550

Chapter 10: Area and Volume

10.4 Finding Volume Using the Washer Method Warm Up You are building an electric go-kart for a competition and your car is over the allowable weight. To make it lighter, you decide to hollow out the cylindrical steel frame. Consider a steel bar of length 1.4 meters and radius 20mm that weighs 9.87 kg/m. Determine the volume of the steel bar, its weight and then its density in kg/m3. Show the integral and calculations that lead to your solution. Include units in your answer.

The bar must weigh no more than 10 kg. You hollow out a cylindrical hole down the middle of the bar. What is the minimum radius required to remove enough material? Write an integral expression for the volume of the cylindrical hole created.

Use two integrals and the radii above to write an expression for the volume of the steel remaining in the bar.

Is there a way to determine the volume of a solid that has a hole in the middle?

Chapter 10: Area and Volume 551

Definition: Rotational volume – washer method (x-axis) The volume V of the solid of revolution generated by revolving the region bounded by the graphs of b

(

)

two functions R = f ( x) and r = g ( x) around the x-axis is V = π ∫ ( R 2 − r 2 )dx = π ∫ [ f ( x) ] − [ g ( x) ] dx 2

Note: There are two radii that must be considered in this situation. The larger radius, R, is determined as the difference between the outermost function and the axis of rotation. The smaller radius, r, is determined as the difference between the innermost function and the axis of rotation.

552

Chapter 10: Area and Volume

Example 1: Finding the Volume of a Solid with a Hole in the Middle – the Washer Method Find the volume of the solid generated by the region bounded by the graphs of = y x 2 + 4 , the line 1 = y x + 2 , x = 0, x = 2 rotated around the x-axis. 3

How to Do It...

What to Think About What shape best describes a slice?

What is the volume of the whole slice if there was no hole in the middle?

What is the volume of the hole in the middle of each slice?

What is the expression for the volume of each slice?

Chapter 10: Area and Volume 553

Your Turn ... Find the volume of the solid generated by the region bounded by the graphs of f ( x) = x 2 − 4 x + 4 the line = y 2 x + 4 , x = 0, x = 6 rotated about the x-axis.

Example 2: The Washer Method – Rotations Around the y-axis.

Find the volume of the solid generated by the region bounded by the graphs of x = 4 y 3 , 1 the line x = y rotated about the y-axis. 4

How to Do It...

What to Think About What is the outer radius of each slice?

What is the inner radius of each slice?

What is the expression for the volume of a slice?

554

Chapter 10: Area and Volume

Definition: Rotational volume – washer method (y-axis) The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions R = f ( y ) and r = g ( y ) around the y-axis is d

(

)

V = π ∫ ( R 2 − r 2 )dx = π ∫ [ f ( y ) ] − [ g ( y ) ] dy 2

Your Turn ... Find the volume of the solid generated by the region bounded by the graphs of x= y 2 , y= x − 2 about the y-axis. HINT: Find the intersection points first.

Chapter 10: Area and Volume 555

Example 3: Finding the Volume of Revolution by Rotating Around a Line Find the volume of the solid generated by the region bounded by the graphs of = y x 2 + 4 , the line 1 = y x + 2 , x = 0, x = 2 rotated around the line y = −2. 3

How to Do It...

What to Think About What is the outer radius of each slice?

What is the inner radius of each slice?

What is the expression for the volume of a slice?

556

Chapter 10: Area and Volume

SUMMARY The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions R = f ( x) and r = g ( x) around the x = k is b

= V π ∫ ( ( R − k ) 2 − (r − k ) 2 )= dx a

∫ ([ f ( x) − k ] − [ g ( x) − k ] )dx 2

The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions R = f ( y ) and r = g ( y ) around the y = k is d

= V π ∫ ( ( R − k ) 2 − (r − k ) 2 )= dy c

∫ ([ f ( y) − k ] − [ g ( y) − k ] )dy 2

Note: There are two radii that must be considered in this situation. The larger radius, R − k , is determined as the difference between the outermost function and the axis of rotation. The smaller radius, r − k , is determined as the difference between the innermost function and the axis of rotation.

Your Turn ... 3

Find the volume of the solid generated by the region bounded by the graphs of x = y , the line y = 0 , 1 y = x − 2, x = 8 rotated around the line x = 8 . Write the integral used and evaluate using a calculator. 2

Chapter 10: Area and Volume 557

10.4 Practice Find the area of the enclosed region. 1. y = 6 − x, x = 0, y = 0, y = 4 about the line x = 6

2. = y

1 ,= y 1,= x 1 about the line x = 4 x

3. y= sec x, y= 0, 0 ≤ x ≤

558

π 3

about the line y = 3

Chapter 10: Area and Volume

4.= y

x= , y 0,= x 4

a) About the x-axis

b) About the y-axis

c) About the line x = 4

d) About the line y = 2

Chapter 10: Area and Volume 559

10.5 Finding Volume Using Cross-Sectional Area Warm Up So far, we have been considering solids created by rotating a region around a line. Imagine taking a cross sectional slice of one of these solids, what does one of the slices look like? Write an expression for its volume.

Imagine another type of solid that is formed with a known cross-section. How can we find the volume of one of these solids? For example, consider a solid with a flat circular base, that has cross-sections perpendicular to the x-axis, that are squares of side length equal to the distance across the circular base at that point. Draw or build a solid that has these features.

What would this shape look like if the cross-sectional area was a semicircle, or an isosceles right triangle, or some irregular area?

Note: Use this app to help with this activity https://qrs.ly/otgge30

Definition: The volume V of the solid with cross sectional area A(x) taken perpendicular to the x-axis b

V = ∫ A( x)dx a

560

Chapter 10: Area and Volume

Example 1: Finding the Volume of a Solid With Known Cross-Section Perpendicular to the x-axis Find the volume of a solid with a circular base of radius 3, that has cross-sectional areas perpendicular to the x-axis that are isosceles right triangles with one leg in the plane of the base.

How to Do It...

What to Think About What is the equation of the base region?

What is the area expression for the cross-sectional area of this shape?

Why does the integrand have the constant 1 ? 2

Why does the integrand have the constant 2?

Why does the integrand have an exponent of 2?

Chapter 10: Area and Volume 561

Your Turn ... Find the volume of a solid with a circular base of radius 3, that has cross-sectional areas perpendicular to the x-axis that are squares.

Definition: Rotational volume – cross sectional area The volume V of the solid with cross sectional area A(y)taken perpendicular to the y-axis, d

V = ∫ A( y )dy c

562

Chapter 10: Area and Volume

Example 2: Finding the Volume of a Solid of Known Cross-Section Perpendicular to the y-axis Find the volume of a solid with base region bounded by y = cos x , x = 0 and y = 0 , that has cross-sectional areas perpendicular to the y-axis that are semi-circles.

How to Do It...

What to Think About What does this solid look like?

What is the equation of the base region when considering cross-sections perpendicular to the y-axis?

What is the area expression for the cross-sectional area of this shape?

Why does the integrand have the constant π ? 2

Why is the function divided by 2?

Why does the integrand have an exponent of 2?

Chapter 10: Area and Volume 563

Your Turn ... Find the volume of a solid with base region bounded by y = cos x , x = 0 and y = 0 , that has cross-sectional areas perpendicular to the y-axis that are equilateral triangles with one leg in the plane of the base region.

Example 3: Finding the Volume of a Solid of Known Cross-Section Between Two Curves Find the volume of a solid bounded above by y = e x and below by y = e − x with cross-sectional areas perpendicular to the x-axis that are squares where the sidelength is equal to the distance between the two boundary curves over the interval [0, ln 2] .

How to Do It...

What to Think About What is the area expression for the cross-sectional area of this shape?

Why is the function not divided by 2?

564

Chapter 10: Area and Volume

Your Turn ... Find the volume of a solid bounded above by y = e x and below by y = e − x with crosssectional areas perpendicular to the x-axis that are semi-circles where the side-length is equal to the distance between the two boundary curves over the interval [0, ln 3] .

Chapter 10: Area and Volume 565

10.5 Practice 2

16 . 1. The base of a solid is the circular region bounded by the graph of x + y = Find the volume of the solid: a) If every cross-section perpendicular to the x-axis is a square.

b) If every cross-section perpendicular to the x-axis is an isosceles triangle and the height equal to the base.

c) If every cross-section perpendicular to the x-axis is a semicircle.

d) If every cross-section perpendicular to the x-axis is an equilateral triangle.

566

Chapter 10: Area and Volume

2. A solid has as its base the region bounded by y = 9 and y = x 2 . Find the volume of the solid if every cross-section is perpendicular to the x-axis. a) Is an isosceles right triangle with hypotenuse in the xy–plane

b) Is a square

3. The base of the solid is bounded by the graphs of y = 2 x and y 2 = 4 x . Find the volume of the solid if every cross-section perpendicular to the y-axis is a semicircle with diameter in the xy–plane.

Chapter 10: Area and Volume 567

x π  2 + cos  x −  and h( x) = e 4 and the y-axis. 4. The base of the solid is bounded by the graphs of g ( x) = 2  Find the volume of the solid if every cross-section perpendicular to the x-axis is an isosceles right

triangle with one leg in the xy-plane.

−2 x 5. The base of the solid is bounded by the graphs of g ( x) = 2 x and h( x) = e . Find the volume of the solid if every cross-section perpendicular to the x-axis is a square with side extending from g(x) to h(x).

568

Chapter 10: Area and Volume

6. The base of the solid is bounded by the graphs of r ( x) = 6sin

( x) and h( x) = x 2 − 3 x + 4 . Find the

3 volume of the solid if every cross-section perpendicular to the x-axis is an isosceles right triangle with one of the legs extending from r(x) to h(x).

7. The base of the solid is a region in the fourth quadrant bounded by the x-axis and the y-axis, and the 1 y x − 4 . If the cross-section of the solid is perpendicular to the x-axis are semicircles. What is line= 2 the volume?

Chapter 10: Area and Volume 569

10.6 Chapter Review 1. Let R be the region in the first quadrant enclosed by the graphs of y = 2 x and y =

1 2 x . 2

a) Find the area of R.

b) The region R is the base of a solid. The cross sections perpendicular to the y-axis are squares. Find the volume of the solid.

c) The region R is rotated about the line y = 8. Find the volume of the solid.

570

Chapter 10: Area and Volume

π   1 + cos π  x −  . Let R be the region 2. Let h and k be the functions given by h( x) = 3− x and k ( x) =  12  in the first quadrant enclosed by the y-axis and the graphs of h and k, and S be the region in the first quadrant enclosed by the graphs h and k.

k(x)

h(x)

a) Calculate the area of R.

b) Calculate the area of S.

c) Determine the volume of the solid when S is revolved about the horizontal line y = –2 .

Chapter 10: Area and Volume 571

2 π  3. Let R be the region in the first quadrant enclosed by the graphs of h( x) = 2− x and h( x) = 1 + sin  x −  2  and the y-axis.

a) Find the area of the region R.

b) The volume of the solid generated when the region R is revolved about the x-axis.

c) Find the volume of the solid generated when the region R is revolved about the line y = 2.

d) The region R is the base of a solid. For the solid, each cross-section perpendicular to the x-axis is a semi-circle. Find the volume of the solid.

572

Chapter 10: Area and Volume

4. Let R be the region bounded by the x-axis, the graphs of h( x) = 2 x and the line y = 6 a) Find the area of region R.

b) Find the volume of the solid generated when R is evolved about the y-axis.

c) The region R is the base of a solid. For the solid, each cross-section perpendicular to the x-axis is an isosceles triangle with one of its legs bound by the two graphs. Find the volume of the solid.

Chapter 10: Area and Volume 573

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Chapter 10: Area and Volume