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MTH/280 Calculus I The Latest Version A+ Study Guide

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MTH 280 Entire Course http://www.uopstudy.com/mth-280 **********************************************

MTH 280 Wk 1 – Reading & Assignment Complete each section of Ch. 2, “Functions and Graphs.” In Ch. 2, you will learn about basic classes of functions, as well as trigonometric, inverse, exponential, and logarithmic functions. You must access the chapter using this link to earn points.

MTH 280 Wk 2 – Reading & Assignment Complete each section of Ch. 3, “Limits.” In Ch. 3, you will learn about finding limits of functions, limit laws, and continuity. You must access the chapter using this link to earn points.

MTH 280 Wk 3 – Reading & Assignment Complete each section of Ch. 4, “Derivatives (Part 1).” In Ch. 4, you will learn about how to define derivatives,

applications as functions, differentiation rules, and applications to rates of change. You must use this link to earn points.

MTH 280 Wk 4 – Reading & Assignment Complete each section of Ch. 5, “Derivatives (Part 2).” In Ch. 5, you will learn how to take derivatives of trigonometric, inverse, and exponential functions using the chain rule and implicit differentiation. You must use this link to earn points.

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http://www.uopstudy.com/ MTH 280 Wk 4 – Midterm Exam Question 1 A research lab grows a type of bacterium in culture in a circular region. The radius of the circle, measured in centimeters, is given by begin mathsize 12px style r left parenthesis t right parenthesis space equals space 5 space minus space fraction numerator 6 over denominator t squared plus 3 end fraction end style, where t is time measured in hours passed since a circle of a 1cm radius of the bacterium was put into the culture. Express the area, A open parentheses t close parentheses, of the bacteria as a function of time, and find the approximate area of the bacterial culture in 4 hours.

A open parentheses t close parentheses space equals space pi left parenthesis 5 minus fraction numerator 6 over denominator t squared plus 3 end fraction right parenthesis and Area after 4 hours = 14.72 comma space c m squared

A open parentheses t close parentheses space equals space 2 pi left parenthesis 5 minus fraction numerator 6 over denominator t squared plus 3 end fraction right parenthesis and Area after 4 hours = 29.43 comma space c m squared

Question 2 A minivan was purchased for $32,000. If the value of the minivan depreciates by $1,700 per year, find a linear function that models the value V of the car after t years. Use the function and find the value of the car after 5 years.

V(t) = -1,700t + 32,000; value after 5 years = $23,500

V(t) = -1,700 + 32,000; value after 5 years = $30,300

V(t) = 1,700 + 32,000; value after 5 years = $33,700

V(t) = 1,700t + 32,000; value after 5 years = $40,500

Question 3 Simplify tan x left parenthesis c s c x space minus space sin x right parenthesis space.

sin x

cos x

Question 4

A pendulum moving in simple harmonic motion is modelled by the function s open parentheses t close parentheses equals negative 5 cos left parenthesis begin inline style fraction numerator pi t over denominator 4 end fraction end style right parenthesis , where s is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4 inches.

0.82 s

2.6 s

3.2 s

9.9 s

Question 5 Solve 2 log subscript 5 open parentheses square root of x close parentheses minus log subscript 5 open parentheses 6 x minus 1 close parentheses equals 0.

x equals 1 over 6

x equals 1 fifth

x equals 5

x equals 6

Question 6 Estimate the slope of the tangent line to f open parentheses x close parentheses equals x cubed at x equals 2 by finding the slope of the secant line through left parenthesis 2 comma space f left parenthesis 2 right parenthesis right parenthesis and left parenthesis 2.001 comma space f left parenthesis 2.001 right parenthesis right parenthesis.

0.012

4.002

16.012

Question 7 The following image shows the graph of the function f left parenthesis x right parenthesis equals square root of 4 minus x squared end root: Find the area between the x-axis and the graph of f left parenthesis x right parenthesis over the interval of open square brackets negative 2 comma space 2 close square brackets using the shaded squares.

6.27 square units

7.5 square units

12.57 square units

8 square units

Question 8 Evaluate limit as x rightwards arrow 5 to the power of minus of space f left parenthesis x right parenthesis if f left parenthesis x right parenthesis space equals space fraction numerator 1 over denominator left parenthesis x minus 5 right parenthesis to the power of 4 end fraction.

negative infinity

negative 10 to the power of negative 4 end exponent

infinity

Question 9 Consider the function f left parenthesis x right parenthesis shown in the following image:

Evaluate limit as x rightwards arrow 0 to the power of minus of space f left parenthesis x right parenthesis.

negative 1

D o e s space n o t space e x i s t

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http://www.uopstudy.com/ Question 10 Use the Squeeze Theorem to evaluate limit as x rightwards arrow 0 to the power of minus of space f left parenthesis x right parenthesis, where f left parenthesis x right parenthesis equals x squared cos left parenthesis 3 over x right parenthesis.

negative infinity

infinity

Question 11 Evaluate limit as theta rightwards arrow 0 of space f left parenthesis theta right parenthesis, where f left parenthesis theta right parenthesis equals fraction numerator sin theta minus cos 3 theta over denominator theta times left parenthesis 1 plus cos 2 theta right parenthesis end fraction.

1 half

infinity

Question 12 Find the intervals over which the function f left parenthesis x right parenthesis equals fraction numerator 2 x squared plus 3 x plus 1 over denominator x squared minus 5 x end fraction is continuous.

left parenthesis negative infinity comma space 5 right parenthesis space a n d space left parenthesis 5 comma space infinity right parenthesis

left parenthesis negative infinity comma space 0 right parenthesis space a n d space left parenthesis 0 comma space infinity right parenthesis

left parenthesis negative infinity comma space 0 right parenthesis comma space left parenthesis 0 comma space 5 right parenthesis space a n d space left parenthesis 5 comma space infinity right parenthesis

left parenthesis negative infinity comma space minus 5 right parenthesis comma space left parenthesis negative 5 comma space 0 right parenthesis comma space a n d space left parenthesis 0 comma space infinity right parenthesis

Question 13 Find a non-zero value for the constant k that makes the function f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator tan space k x over denominator x end fraction comma space end cell row cell 5 x plus 3 comma space end cell end table close open table attributes columnalign right end attributes row cell x less than 0 end cell row cell x greater or equal than 0 end cell end table close curly brackets continuous at x equals 0.

1 third

infinity

Question 14 Determine the value of delta greater than 0 for which limit as x rightwards arrow 4 of left parenthesis 3 x squared minus 1 right parenthesis equals 47 space.

delta equals m a x left curly bracket negative 4 minus square root of 16 minus epsilon over 3 end root comma space minus 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m i n left curly bracket negative 4 plus square root of 16 minus epsilon over 3 end root comma space minus 4 minus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m a x left curly bracket 4 minus square root of 16 minus epsilon over 3 end root comma space 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

delta equals m i n left curly bracket 4 minus square root of 16 minus epsilon over 3 end root comma space minus 4 plus square root of 16 plus epsilon over 3 end root right curly bracket

Question 15

Determine the value of delta greater than 0 for which limit as x rightwards arrow 2 of fraction numerator 1 over denominator left parenthesis x minus 2 right parenthesis squared space end fraction equals infinity.

delta equals fraction numerator 1 over denominator square root of M end fraction

delta equals 1 over M

delta equals square root of M

delta equals M

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http://www.uopstudy.com/ Question 16 A store determines that the daily profit on pens obtained by charging s dollars per pen is P left parenthesis s right parenthesis equals negative 10 s squared plus 75 s minus 5. If the store currently charges 50 cents for a pen, find the rate of change of profit.

$30

$60

$65

$85

Question 17 A ball is dropped from the top of a building that is 15 m high. The position of the ball after t

seconds is given by the equation s left parenthesis t right parenthesis equals negative 4.9 t squared plus 15. Find the instantaneous acceleration of the ball after t seconds.

negative 4.9 semicolon space m divided by s squared

negative 9.8 semicolon space m divided by s squared

negative 9.8 t semicolon space m divided by s squared

negative 4.9 t semicolon space m divided by s squared

Question 18 Find fraction numerator d over denominator d x end fraction left parenthesis 3 over x squared plus x left parenthesis x minus 1 right parenthesis right parenthesis.

fraction numerator negative 3 over denominator x squared end fraction plus 2 x minus 1

fraction numerator negative 6 over denominator x cubed end fraction plus 1

fraction numerator negative 3 over denominator x squared end fraction plus 1

fraction numerator negative 6 over denominator x cubed end fraction plus 2 x minus 1

Question 19 The price p (in dollars) and the demand x for an item is given by the price-demand function p left parenthesis x right parenthesis equals 15 minus 0.0015 x. Find the marginal revenue at x equals 1 comma 000.

$1.50

$12.00

$13.50

$15.00

Question 20 Find the equation of the tangent line to f left parenthesis x right parenthesis equals 3 c s c x plus x sin x at x equals straight pi over 2.

y equals x minus 3

y equals x plus 3 plus pi over 2

y equals x plus 3

y equals x plus 3 minus pi over 2

Question 21 A bag of sand hanging from a vertical spring is in simple harmonic motion as given by the position function s left parenthesis t right parenthesis equals negative sin left parenthesis straight pi over 2 t plus straight pi over 6 right parenthesis where t is measured in seconds and s is in inches. Find the velocity of the spring at t equals 2 s.

negative fraction numerator square root of 3 straight pi end root over denominator 4 end fraction space i n c h divided by s

straight pi over 4 i n c h divided by s

fraction numerator square root of 3 straight pi end root over denominator 4 end fraction space i n c h divided by s

straight pi over 2 i n c h divided by s

Question 22 Find fraction numerator d y over denominator d x end fraction if y equals tan to the power of negative 1 end exponent left parenthesis square root of x cubed right parenthesis end root.

fraction numerator 1 over denominator 1 plus x cubed end fraction

fraction numerator 1 over denominator 2 square root of x cubed end root left parenthesis 1 plus x cubed right parenthesis end fraction

fraction numerator 3 x squared over denominator left parenthesis 1 plus x cubed right parenthesis end fraction

fraction numerator 3 square root of x over denominator 2 left parenthesis 1 plus x cubed right parenthesis end fraction

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http://www.uopstudy.com/ Question 23 The volume of a right circular cylinder of radius x and height y is given by v equals Ď&#x20AC;x squared straight y. Suppose that the volume of the cylinder is constant at 250 Ď&#x20AC;cm cubed. Find fraction numerator d y over denominator d x end fraction when x equals 5 and y equals 10.

negative 100

negative 4

250

Question 24 Find fraction numerator d y over denominator d x end fraction if y equals 5 x left parenthesis cos x right parenthesis to the power of x over 2 end exponent.

5 over 2 left parenthesis cos x right parenthesis to the power of x over 2 end exponent open square brackets 1 over x plus x over 2 c o t x plus fraction numerator ln left parenthesis cos x right parenthesis over denominator 2 end fraction close square brackets

5 x left parenthesis cos x right parenthesis to the power of x over 2 end exponent open square brackets 1 over x minus x over 2 tan x plus fraction numerator ln left parenthesis cos x right parenthesis over denominator 2 end fraction close square brackets

fraction numerator negative 5 x squared sin x over denominator 2 end fraction left parenthesis cos x right parenthesis to the power of x over 2 end exponent minus 1 plus 5 left parenthesis cos x right parenthesis to the power of x over 2 end exponent

fraction numerator negative 5 x squared over denominator 2 end fraction left parenthesis sin x right parenthesis to the power of x over 2 end exponent minus 1 plus 5 left parenthesis cos x right parenthesis to the power of x over 2 end exponent

Question 25 Find the derivative of fraction numerator e to the power of x minus 1 over denominator e to the power of x plus 1 end fraction.

fraction numerator negative 2 e to the power of x over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator negative e to the power of 2 x end exponent over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator 2 e to the power of x over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

fraction numerator e to the power of 2 x end exponent over denominator left parenthesis e to the power of x plus 1 right parenthesis squared end fraction

Question 26 If f left parenthesis x right parenthesis equals square root of 1 plus 8 x end root, find f space apostrophe left parenthesis a right parenthesis for a equals 3 and the equation of the tangent line to f left parenthesis x right parenthesis at x equals a.

f space apostrophe left parenthesis a right parenthesis equals 5 over 4; equation of the tangent line is y equals fraction numerator 5 x minus 13 over denominator 4 end fraction

f space apostrophe left parenthesis a right parenthesis equals negative 4 over 5; equation of the tangent line is y equals fraction numerator negative 4 x plus 13 over denominator 5 end fraction

f space apostrophe left parenthesis a right parenthesis equals 4 over 5; equation of the tangent line is y equals fraction numerator 4 x plus 13 over denominator 5 end fraction

f space apostrophe left parenthesis a right parenthesis equals negative 5 over 4; equation of the tangent line is y equals fraction numerator negative 5 x minus 13 over denominator 4 end fraction

Question 27 The position of a moving particle as a function of time is given by s left parenthesis t right parenthesis equals 1 third t cubed minus t plus 1, where s is in meters and t is in seconds. Find the time at which the particle is at rest and the acceleration of the particle at t equals 3 s.

Time at which the particle is at rest equals 2 space s; acceleration of the particle equals 2 space m divided by s squared

Time at which the particle is at rest equals 1 space s; acceleration of the particle equals 6 m divided by s squared

Time at which the particle is at rest equals 2 space s; acceleration of the particle equals 1 space m divided by s squared

Time at which the particle is at rest equals 3 space s; acceleration of the particle equals 3 space m divided by s squared

Question 28 A car moves on a straight road. The car's position at time t is given by s left parenthesis t right parenthesis equals t plus sin t, where s is in meters and t is in seconds. Find the acceleration at t equals straight pi over 4 semicolon space s.

negative 1 half semicolon space m divided by s squared

negative fraction numerator 1 over denominator square root of 2 end fraction semicolon space m divided by s squared

fraction numerator 1 over denominator square root of 2 end fraction semicolon space m divided by s squared

1 semicolon space m divided by s squared

Question 29 Find fraction numerator d over denominator d x end fraction open parentheses fraction numerator 1 plus sin squared x over denominator x plus tan cubed 15 x end fraction close parentheses.

fraction numerator 2 sin x cos x over denominator 1 plus 45 tan squared 15 x end fraction

fraction numerator left parenthesis x plus tan cubed 15 x right parenthesis left parenthesis 2 sin x

cos x right parenthesis minus left parenthesis 1 plus sin squared x right parenthesis left parenthesis 1 plus 45 tan squared 15 x s e c squared 15 x right parenthesis over denominator left parenthesis x plus tan cubed 15 x right parenthesis squared end fraction

fraction numerator left parenthesis x plus tan cubed 15 x right parenthesis left parenthesis 2 sin x cos x right parenthesis minus left parenthesis 1 plus sin squared x right parenthesis left parenthesis 1 plus 45 tan squared 15 x right parenthesis over denominator left parenthesis x plus tan cubed 15 x right parenthesis squared end fraction

fraction numerator 1 plus 2 sin x cos x over denominator 1 plus 15 tan squared 15 x end fraction

Question 30 Find fraction numerator d f space to the power of negative 1 end exponent over denominator d x end fraction for f left parenthesis x right parenthesis equals fraction numerator x plus 3 over denominator 2 x plus 5 end fraction at x equals 3.

negative 1 over 25

negative 25

1 over 81

MTH 280 Wk 5 â&#x20AC;&#x201C; Reading & Assignment Complete each section of Ch. 6, â&#x20AC;&#x153;Applications of Derivatives.â&#x20AC;? In Ch. 6, you will learn different methods of applying derivatives using various techniques. You must access the chapter using this link to earn points.

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http://www.uopstudy.com/ MTH 280 Wk 6 – Reading & Assignment Complete each section of Ch. 7, “Integration (Part 1).” In Ch. 7, you will learn about the fundamental theorem of calculus and approximating areas. You must access the chapter using this link to earn points.

MTH 280 Wk 7 – Reading & Assignment Complete each section of Ch. 8, “Integration (Part 1).” In Ch. 8, you will learn how to integrate different functions. You must access the chapter using this link to earn points.

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http://www.uopstudy.com/ MTH 280 Wk 7 – Final Exam Question 1 A car starts from a point at 2:00 p.m. and travels north at 40 mph. Another car starts from the same point at 3:00 p.m. and travels west at 50 mph. After the second car has traveled 1 h, at what rate is the distance between the two cars changing?

60.42 mph

94.34 mph

30.17 mph

64.03 mph

Question 2 A spherical ball is measured to have a radius of 6 space c m with a possible measurement error of plus-or-minus 0.1 rm c m. Use the differentials to estimate the percentage error in computing the volume of the ball.

10%

Question 3 A company determines a cost function of c equals 6 x squared minus 180 x plus 2000, where c is the cost (in dollars) of producing x number of items. How many items should the company manufacture to minimize the cost?

Question 4 The position of an object is given by the equation s left parenthesis t right parenthesis equals 2 x squared plus x minus 6. Find the time t at which the instantaneous velocity of the object equals the average velocity in the interval open square brackets 0 comma 3 close square brackets.

t equals 1.5 semicolon space s

t equals 3 semicolon space s

t equals 2 semicolon space s

t equals 0 semicolon space s

Question 5 Find the locations of local minimum and maximum of x to the power of 9 minus 4 x to the power of 8 using the second derivative test.

Local minimum at x equals 0, local maximum at x equals 32 over 9

Local minimum at x equals 32 over 9, no local maximum

Local minimum at x equals 32 over 9, local maximum at x equals 0

Local minimum at x equals 0, no local maximum

Question 6 In a shop, the revenue and the cost of a product are determined by R left parenthesis x right parenthesis equals 22 x and C left parenthesis x right parenthesis equals 2 x squared plus 2 x plus 1, respectively. If x represents the number of products, how many products should the shop sell to maximize the profit?

Question 7 Evaluate limit as x rightwards arrow 0 of fraction numerator x squared over denominator e to the power of x minus x minus 1 end fractionby applying L'HĂ´pital's rule.

infinity

Question 8 Let f left parenthesis x right parenthesis equals x cubed minus x squared minus 1 and x subscript 0 equals 1. To the nearest three decimal places, find x subscript 5 using Newton's method of approximation.

1.466

1.486

1.625

2.000

Question 9 Evaluate integral sin 2 x cos 2 x comma space d x

1 fourth cos 4 x plus C

negative 1 over 8 cos 4 x plus C

negative 1 fourth cos 4 x plus C

1 over 8 cos 4 x plus C

Question 10 Evaluate integral subscript negative 1 end subscript superscript 1 left parenthesis t squared plus t plus 1 right parenthesis d t using the Fundamental Theorem of Calculus, Part 2.

8 over 3

negative 5 over 6

10 over 6

Question 11 Water is flowing into a tank at a rate of r left parenthesis t right parenthesis equals 3 square root of t over 2 end root cubic meters per minute. How much water entered the tank between 2 and 8 minutes?

3 cubic meters

6 cubic meters

14 cubic meters

28 cubic meters

Question 12 To the nearest two decimal places, calculate R subscript 5 for f left parenthesis x right parenthesis equals x cubed plus 1 on open square brackets 0 comma space 4 close square brackets.

44.96

65.00

33.77

96.16

Question 13 Use substitution to evaluate integral subscript 0 superscript straight pi over 2 end superscript sin 2 x square root of 4 plus 9 sin squared x end root space d x.

1 third left parenthesis 26 square root of 13 minus 16 right parenthesis

1 over 27 left parenthesis 26 square root of 13 minus 16 right parenthesis

2 over 27 square root of straight pi cubed end root

2 over 3 square root of straight pi cubed end root

Question 14 Evaluate the integral integral fraction numerator 1 plus tan x over denominator 1 minus tan x end fraction d x.

ln open vertical bar sin x minus cos x close vertical bar plus C

negative ln open vertical bar cos x plus sin x close vertical bar plus C

negative ln open vertical bar cos x minus sin x close vertical bar plus C

ln open vertical bar cos x plus sin x close vertical bar plus C

Question 15 Evaluate integral subscript 1 superscript 2 fraction numerator 6 over denominator open vertical bar 3 x close vertical bar square root of 9 x squared minus 4 end root end fraction d x.

sin to the power of negative 1 end exponent 3 minus sin to the power of negative 1 end exponent 3 over 2

tan to the power of negative 1 end exponent 3 minus tan to the power of negative 1 end exponent 3 over 2

s e c to the power of negative 1 end exponent 3 minus s e c to the power of negative 1 end exponent 3 over 2

Question 16 Find the area between the curves f left parenthesis x right parenthesis equals 1 minus 2 x and g

left parenthesis x right parenthesis equals negative x minus 1 over the interval open square brackets negative 4 comma space minus 1 close square brackets.

13.5 space u n i t s squared

21 space u n i t s squared

27 space u n i t s squared

28.5 space u n i t s squared

Question 17 If R denotes a region bounded above by the graph of a continuous function f left parenthesis x right parenthesis, below by the x-axis, and on the left and right by the lines x equals a and x equals b, respectively, then which of the following integrals gives the mass of the lamina with density rho?

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared over 2 space d x

m equals rho integral subscript a superscript b x f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b f left parenthesis x right parenthesis space d x

m equals rho integral subscript a superscript b open square brackets f left parenthesis x right parenthesis close square brackets squared space d x

Question 18 Evaluate fraction numerator d over denominator d x end fraction cos h left parenthesis 2 x squared plus 1 right parenthesis.

4 x sin h left parenthesis 2 x squared plus 1 right parenthesis

2 x sin h left parenthesis 2 x squared plus 1 right parenthesis

x sin h left parenthesis 2 x squared plus 1 right parenthesis

left parenthesis 2 x squared plus 1 right parenthesis sin h left parenthesis 2 x squared plus 1 right parenthesis

Question 19 Evaluate integral fraction numerator negative 1 over denominator open vertical bar x close vertical bar square root of 1 plus begin display style x squared over 25 end style end root end fraction space d x.

c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

c s c h to the power of negative 1 end exponent open vertical bar x over 5 close vertical bar plus C

1 fifth c s c h to the power of negative 1 end exponent open vertical bar x close vertical bar plus C

Question 20 Find the vertical and horizontal asymptotes of f left parenthesis x right parenthesis equals x plus sin x.

x equals 0 comma space y equals 0

x equals 1, no vertical asymptote

x equals negative straight pi over 2 comma space y equals straight pi over 2

No vertical and horizontal aymptotes