Chapter 3,4,5

Chapter 3,4,5,6,7

Differentiation

Chapter 3 • What Can be Done With a Derivative?

– Find Slope Of A Tangent Line.

– Maximum And Minimum On Graph Of A Function.

– Analyze Rates Of Change.

– Analyze Motion On Object.

– Optimize Word Problems.

Derivative

by Definition • FIRST FORM OF THE DEFINITION The derivative of a function f at a number c, denoted by f '(c), is given by

Example • Using the definition above, find the derivative of f(x) = x2 – 5x + 3 at x = 2; that is, find f '(2).

SECOND FORM OF THE DEFINITION • The derivative of a function f at a number c, denoted by f '(c) is given by:

Example • Using the definition above, find the derivative of f(x) = x2 – 5x + 3 at x = 2, that is, find f '(2).

Example • For f(x) = 3x2 – 12x + 9, find f '(x), the derivative at any point.

Find the Equation of a Line Tangent to a Curve • to find an equation of the line tangent to the graph of f(x) = x3 – 6x2 + 9x – 13 at the point with x-coordinate 2.

Horizontal Tangents • Find Points on a Curve at Which Tangent Line is Horizontal • Example : Find the coordinates of each point on the graph of f(x) = x3 – 6x2 + 9x – 13 at which the tangent line is horizontal.

Alternate Notations for a Derivative

Differentiability and Continuity

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Chapter 4

Example

Rolle’s Theorem Then, there exists at least one number c in • (a,b) for which f '(c) = 0.

Then, there exists at least one number c in (a,b) for which f '(c) = 0.

Example For the function f(x) = x2 – 8x + 19, find the value of c in the open interval (2, 6) that is mentioned in Rolle’s Theorem.

The Mean Value Theorem

Example For f(x) = x3 – x2 – 2x, find the value of c in the interval (–1,1), which is mentioned in the Mean Value Theorem.

Solution :

Limits: Indeterminate Forms and L’Hopital’s Rule

Example 1

Example 2

Chapter 5

Applications of Differentiation Tangent Line to Graph ofa Function at a Point Example : Find the equation of the line tangent to the graph of f(x) = x3ln x at the point with x coordinate e.

Horizontal Tangents Find the coordinates of each point on the graph of f(x) = x3 – 12x2 + 45x – 55 at which the tangent line is horizontal.

Critical Numbers DEFINITION OF CRITICAL NUMBERS The number c is a critical number for f(x) if and only if f '(c) = 0 or if f '(c) is undefined.

Example Find the critical numbers for f(x) = 2x + 3x2 – 6x + 4.

Increasing and Decreasing Functions Definition Of Increasing/Decreasing Function On An Interval

1.The function f is increasing on an open interval (a,b), if for any two numbers c and d in (a,b) with c < d, then f(c) < f(d).

2. The function f is decreasing on an open interval (a,b), if for any two numbers c and d in (a,b), with c < d, then f(c) > f(d).

Example Find the intervals over which f(x) is increasing/decreasing for the function f(x) = 2x3 + 3x2 â&#x20AC;&#x201C; 12x.

Extrema of a Function on a Closed Interval DEFINITION OF EXTREMA ON AN INTERVAL If f is a function defined on an interval containing c, then: 1. f(c) is a minimum of f on that interval, if f(c) â&#x2030;¤ f(x) for all x in that interval.

2. f(c) is a maximum of f on that interval, if f(c) â&#x2030;Ľ f(x) for all x in that interval.

Example • Find the extrema of f(x) = 5x4 – 4x3 on the interval [–1,2].

RELATIVE EXTREMA DEFINITION OF RELATIVE EXTREMA • f(c) is called a relative maximum of f if there is an interval (a,b) containing c in which f(c) is a maximum.

• f(c) is called a relative minimum of f if there is an interval (a,b) containing c in which f(c) is a minimum.

Example Find the relative extrema of f(x) = 3x4 â&#x20AC;&#x201C; 28x3 + 60x2.

Concavity and Point of Inflection â&#x20AC;˘ If the graph of f lies above all its tangents on an interval (a,b), then f is said to be concave upward on (a,b). â&#x20AC;˘ If the graph of f lies below all its tangents on an interval (a,b), then f is said to be concave downward on (a,b).

TEST FOR CONCAVITY • If f"(x) > 0 for all x in (a,b), then the graph of f is concave upward on (a,b).

• If f"(x) = < 0 for all x in (a,b), then the graph of f is concave downward on (a,b).

Example For the function f(x) = x4 + 2x3 â&#x20AC;&#x201C; 12x2 â&#x20AC;&#x201C; 15x +22, find the intervals over which its graph is concave upward or downward.

Inflection point The point P is called a point of inflection for the graph of f if the concavity changes at the point P.

Example Find the intervals over which the graph of f(x) = 4x3 â&#x20AC;&#x201C; x4 is concave upward and downward, and find any inflection points.

Optimization Let’s say you are cutting equal squares from each corner of a rectangular piece of aluminum that is 16 inches by 21 inches. You will then fold up the “flaps” to create a box with no top. Find the size of the square that must be cut from each corner in order to produce a box having maximum volume.