1

JOHNSON COUNTY COMMUNITY COLLEGE

Calculus I (MATH 241) Final Review Spring 2013

This final review will be a useful starting point as you study for your final exam.

You should also study your exams and homework. ALL topics listed in the Course Objectives OR the Content Outline listed below are possible topics for Final Exam questions.

The entire course outline is available at

www.jccc.edu/catalog/credit/spring-2012/outlines/math/math-241.html

Course Objectives:

Upon successful completion of this course the student should be able to:

1. Evaluate the limits of functions. 2. State whether a function is continuous or discontinuous based on both the graph

and the definition of continuity. 3. Use limits to describe instantaneous rate of change, the slope of the tangent line

and the velocity and acceleration of a moving particle. 4. Differentiate algebraic, trigonometric, and transcendental functions explicitly and,

where appropriate, implicitly. 5. Use derivatives for curve sketching. 6. Use and interpret the derivatives of functions to solve problems from a variety of

fields, including physics and geometry. 7. Integrate algebraic, trigonometric, and transcendental functions. 8. Compute definite integrals by the Fundamental Theorem of Calculus, by numerical

techniques, and by substitution. 9. Use integration results to calculate areas, volumes, and mean values.

Content Outline & Competencies:

I. Using Limits

A. Evaluation of limits

1. Evaluate the limit of a function at a point both algebraically and graphically.

2. Evaluate the limit of a function at infinity both algebraically and graphically.

3. Use the definition of a limit to verify a value of the limit of a function.

B. Use of limits

1. Use the limit to determine the continuity of a function.

2. Use the limit to determine differentiability of a function.

C. Limiting process

1. Use the limiting process to find the derivative of a function.

http://www.jccc.edu/catalog/credit/spring-2012/outlines/math/math-241.html

2

II. Finding Derivatives

A. Find derivatives involving powers, exponents, and sums.

B. Find derivatives involving products and quotients.

C. Find derivatives involving the chain rule.

D. Find derivatives involving exponential and logarithmic functions.

E. Find derivatives involving trigonometric and inverse trigonometric functions.

F. Find derivatives involving implicit differentiation.

G. Use the derivative to find velocity, acceleration, and other rates of change.

H. Use the derivative to find the equation of a line tangent to a curve at a given point.

III. Using Derivatives

A. Curve sketching

1. Use the first derivative to find critical points.

2. Apply the Mean-Value Theorem for derivatives.

3. Determine the behavior of a function using the first derivative.

4. Use the second derivative to find inflection points.

5. Determine the concavity of a function using the second derivative.

6. Sketch the graph of the function using information gathered from the first and

second derivatives.

7. Interpret graphs of functions.

B. Applications of the derivative

1. Solve related rates problems.

2. Use optimization techniques in economics, the physical sciences, and geometry.

3. Use differentials to estimate change.

4. Use Newton’s Method.

IV. Finding Integrals

A. Find area using Riemann sums.

B. Express the limit of a Riemann sum as a definite integral.

C. Evaluate the definite integral using geometry.

D. Integrate definite integral using numerical approximation.

E. Evaluate definite integrals using the Fundamental Theorem of Calculus.

F. Integrate algebraic, natural exponential, natural logarithm, trigonometric, and inverse

trigonometric functions.

G. Integrate indefinite integrals.

H. Integration by substitution.

I. Integration by parts.

V. Using the Integral

A. Utilize the Mean-Value Theorem for Integrals.

B. Calculate the area between curves using integration.

C. Calculate the volume of a solid of revolution by the disk method.

D. Calculate the volume of a solid of revolution by the washer method.

E. Calculate the volume of a solid of revolution by the cylindrical shells method.

F. Calculate the arc length and surface area using integration.

3

JOHNSON COUNTY COMMUNITY COLLEGE

Calculus I (MATH 241) Final Review

This final review will be a useful starting point as you study for your final exam. You should also study your tests and homework from this semester.

1. Refer to the graph of the function y = f(x) to evaluate or approximate each of the following expressions or state that the value does not exist.

a. )(lim3

xfx

b. )(lim3

xfx

c. )(lim3

xfx

d.

f 3 e.

f 3 f. )(lim

2

xfx

g. )(lim2

xfx

h. )(lim2

xfx

i.

f 2 j.

f 2 k. )(lim

2

xfx

l. )(lim2

xfx

m. )(lim2

xfx

n.

f 2 o.

f 2 p.

f 1 q.

f 1

2. Evaluate the following limits or state that the limit does not exist. Justify your answers.

a. 3

3 2

5 3 2lim

4 6 1x

x x

x x

b. 2

5lim

3 4x

x

x

c. 2

22

6lim

4x

x x

x

d. 0

sinlim

5x

x

x

e. 3

22

lim4x

x x

x

f. 2

1lim

2x x

g. 2

1lim

2x x

h. 0

cos tanlimx

x x

x

i. 1

lim ln( 1)x

x

j. 0

3 4lim

2x

x

x

k. 0

5(1 cos )limx

x

x

l. 9

3lim

9x

x

x

m. 0

sin 3lim

4y

y

y

n. 2

3lim 9x

x

o. 1lim tan

xx

p. 1 cos

limx

x

x

q. 2

| 2 |lim ( 3)

2x

xx

x

r. lim7

x

xx

e

e

s. 2 10

lim sinxx

x

4

3. Suppose that the inequalities 2

1cos1

242

12

2

x

xx hold for values of x close to zero. What, if

anything, does this tell you about 20

1 coslimx

x

x

? Give reasons for your answer.

4. Prove the following limit statements using the epsilon-delta definition of limit.

a. 2

lim (2 1) 3x

x

b. 4

lim 2x

x

5. For each function ( )f x below, state the intervals on which the function is continuous. For each

discontinuity, state which conditions required by the Continuity Test are not met. Classify each

discontinuity as removable or nonremovable.

a. 2

4)(

2

2

xx

xxf

b. 2

5)(

x

xxf

c. 2

2)(

x

xxf

6. For what value of b is

2,

2,)(

2 xbx

xxxg continuous at every x?

7. Use the Intermediate Value Theorem to show that there is a root of the equation 4 3 0x x

between 1 and 2.

8. For the function defined by

2

1 cos , 1

1 , 1 0

, 0 3( )

9 , 3 4

7, 4

5

x

x x

x x

e xf x

x x

xx

a. Graph ( )f x . Clearly indicate all important points.

b. Discuss limits of ( )f x at all important points.

c. Discuss the continuity of ( )f x at all important points.

d. Discuss the differentiability of ( )f x at all important points.

9. Find the derivative using the limit definition of the derivative.

a. 73)( xxf

b. xxxf 2)( 2

c. 1

1)(

xxf

d. 12)( xxf

5

10. Find the derivative of the function.

a. 5 35

3 3

25)( x

x

xxf

b. )ln(tan)( xxf

c. 26ln)( xxf d. 5

22)( xexxf

e. 62)53()( xxxf

f. 53

27)(

x

xxf

g. 4

2sin)(

x

xxf

h. xxf 4cos4)( 3

i. 21cos)( xxf j. xxxf 3sin)( 12

k. )(tan)( 1 xexf

l. xxf 9)(

m. xxxf sec)( 4

n. 1)( 2 xxf

o.

xxf

1sin)( 1

p. xxxf 2)(

11. Find dx

dyfor each of the following.

a. xxy ln

b. 433 yx

c. 132 yxxy

d. xyeyxy 3423

e. xyy

y

1

1sin

12. Find the equation of the line tangent to 24)( 53

xxf at the point (1, 2).

13. Find the equation of the tangent line to the graph of 3)( xxf and parallel to the line 3 1 0x y .

14. For 1( ) tan 3f x x

a. Find the equation of the line tangent to ( )f x at the point 13 4, .

b. Find the equation of the line tangent to 1( )f x at the point 14 3,

.

15. Given that 4

( ) 6f xx

a. Show that f satisfies the hypotheses of the Mean Value Theorem on the interval [1, 4] b. Find all numbers c that satisfy the conclusion of the Mean Value Theorem.

16. Find the coordinates of all relative/local extrema and points of inflection, and any asymptotes. Indicate any relative/local extrema that are also absolute extrema. Give the intervals on which the

function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave

down. Then sketch the graph.

a. 21

)(x

xxf

b. xxxf cossin)( , on the interval [0, 2π]

c. ( ) sin2

xf x

, on the interval (0, 4π)

d. xxxxf 126)( 23

e. xxxf ln)(

f. x

xxf

ln)(

g. 9

)(2

x

xxf

h. ( )x

xf x

e

i. 1 2( ) tanf x x

6

17. For each of the following functions, find the intervals where f is concave up and where f is concave down. State any points of inflection.

a. 1234)( 23 xxxxf

b. 22,cos1

sin)(

x

x

xxf

18. For each of the following functions, find the intervals where the function is increasing and where the function is decreasing.

a. 353)( 24 xxxf

b. 20,sin2cos)( 2 xxxxf

19. Given 2( ) xf x xe , 2( ) 1 2xf x e x , 2( ) 4 1xf x e x ,

a. Find all critical numbers of f. b. Find any local/relative extrema using the Second Derivative Test. c. Give the intervals on which f is increasing, and those on which f is decreasing. d. Give the intervals on which f is concave up, and those on which f is concave down. e. Find all points of inflection of f.

20. Sketch the graph of a continuous function f that satisfies all of the stated conditions.

f (0) = 1; f (2) = 3; 0)2()0( ff

)(xf < 0 if x > 2 or x < 0

)(xf > 0 if 0 < x < 2

)(xf > 0 if x < 1; )(xf < 0 if x > 1

21. Sketch the graph of a continuous function f that satisfies all of the stated conditions. f (0) = 4; f (2) = 2; f (5) = 6;

0)2()0( ff

)(xf > 0 if x > 2 or x < 0

)(xf < 0 if 0 < x < 2

)(xf < 0 if x < 1 or if 3 < x < 5

)(xf > 0 if 1 < x < 3 or if x > 5

22. The number of individuals (in thousands) in a population of animals is projected to be given by

2

6000( ) 250

25P t

t

, where t is measured in years since January 1, 2010.

a. Find dPdt

and 2

2

d P

dt.

b. Evaluate ( ), ,dPdt

P t and 2

2

d P

dt at 0, 5t t and 10t , and explain their meanings in terms

of the population and its rate of growth.

c. When is the population projected to be growing most rapidly?

7

23. A projectile is fired directly upward with a velocity of 144 ft/sec. Its height in feet above the ground

after t seconds is given by 216144100)( ttts . Find the following.

a. How long does it take for the projectile to hit the ground?

b. What is the speed of the projectile at t = 7 seconds?

c. What is the velocity of the projectile when it hits the ground?

d. What is the maximum height of the projectile?

24. The graph below shows the position ( )s f t of an object moving up and down, with displacement

measured from the origin. Use the graph to answer the following questions about the object’s motion.

a. During approximately what time intervals is the object moving toward the origin?

b. During approximately what time intervals is the object moving away from the origin?

c. At approximately what times does the object change directions?

d. During approximately what time intervals is the object stationary (not moving)?

e. At approximately what times (and, if applicable, during what time intervals) is

the velocity of the object equal to zero?

f. At approximately what times (and, if applicable, during what time intervals) is

the acceleration of the object equal to

zero?

g. During approximately what time intervals is the acceleration of the object positive?

h. During approximately what time intervals is the acceleration of the object negative?

25. An open-top rectangular box is constructed from a 10- by 16- inch piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find the dimensions of the box

of largest volume and find the maximum volume.

26. A rectangle is bounded by the x-axis and the semicircle 225 xy . What length and width should

the rectangle have so that its area is a maximum?

27. The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.

28. Petroleum is leaking from an offshore well and forms a circular oil slick on the surface of the water. If the area of the slick is increasing at the rate of 8 km

2 per day, at what rate is the radius increasing at the

instant when the radius is 3 km?

29. A spherical balloon is being inflated. As the radius of the balloon increases at the rate of 1 ft./minute, find the rate of change of the volume when the radius is 3 ft.

30. A baseball diamond is a square with each side 90 ft long. A batter hits the ball and runs from home plate toward first base at a speed of 24 ft/sec.

a. At what rate is his distance from second base changing when he is one third of the way to first base?

b. At what rate is his distance from third base changing at the same instant?

1st base

home plate

90 ft

3rd base

2nd base

8

31. The height of a rocket is given by 250)( tts feet after t seconds. A ground level camera is located

2400 feet from the launch pad. As seen by the camera, what is the rate of change of the angle of

elevation of the rocket 10 seconds after liftoff?

32. Find y and use the approximation x dx to find dy for the function 1045 23 xxxy at

1x with 1.0x .

33. The radius of a circle increased from 2.00 to 2.02 meters. Use differentials to (a) estimate the resulting change in area and (b) express the estimate as a percentage change in area.

34. If 2

cos)(x

xf and F(x) is an antiderivative of f (x) with F(0) = 3, find F(x).

35. If 4( ) 2x

f x e

and F(x) is an antiderivative of f (x) with F(0) = 1, find F(x).

36. Find the indefinite integrals and evaluate the definite integrals.

a. 23

2

cos 3

dxx

b. dxx 49

16

2

5

c. 3

0

2 )12( dxxx

d. e

dxx

2

3

e. 75 dxx

f.

3)sin1()(cos dxxx

g.

4sec 2 tan 2 x x dx

h.

2

tan sec x x dx

i. 2

1

1 5 dxx

j.

2

9

5dx

x

k.

0 22

3

41

1dx

x

l.

2

14

1dx

xx

m. sin x x dx

n.

5 dxe x

o. 9

4 dxx

p. dxx ln

q. 2 )32( dxx

r.

2

sin

5dx

x

s. dtett42

t. 2cot x dx

u.

dx

x

x

241

v.

dxx

253

1

w.

dxx3cos

1

2

x. 1tan x dx

y. tan x dx

37. Find )(xF .

a. F(x) = x

dtt

1

2 sin

b. F(x) = 2

5

3 1x

dtt

c. F(x) = x t dte

sin

0

2

9

38. Use the trapezoidal rule to approximate.

a. 2

3 0

4

1dx

x , n = 4

b.

2

0sin x dx

, n = 4

39. Use rectangles and (a) an upper sum, and (b) a lower sum to approximate the area of the region between 2xy e and the x-axis, on the interval [0, 1]. Use 4 subintervals of equal width.

40. Use Simpson’s Rule with n = 4 to approximate 2

2

1ln x dx .

41. Use the error bound formula for Simpson’s Rule to find an upper bound SE for the error in estimating

2

2

1ln x dx using Simpson’s Rule with n = 4.

42. How many subdivisions should be used in the Trapezoid Rule to approximate 2

1

1ln(2) dx

x with an error

whose absolute value is less than410 ?

43. Set up a definite integral to find the area of the region bounded by the given curves. Be sure to use proper notation.

a. 2xy and 22 xxy

b. xy sin , xy cos , 0x , and 2x

c. yyx 42 and 22 yyx

44. Set up a definite integral to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Be sure to use proper notation.

a. 1,0,0, xxyey x ; rotated about the x-axis.

b. 0,8,3 xyxy ; rotated about the y-axis.

c. 0,2 32 yxxy ; rotated about the y-axis.

d. 2, xyxy ; rotated about the line 1x .

45. Find the average value of 2( ) 3 2f x x x on the interval [1,4] .

46. Find the arc length of the graph of y = ln(cos x) from x = 0 to4

x

.

47. Find the area of the surface generated by revolving the curve 2y x , 1 ≤ x ≤ 2, about the x-axis.

48. The line segment x = 1 – y, 0 ≤ y ≤ 1, is revolved about the y-axis to generate a cone. Find its lateral surface area.

49. The following data give the velocity of an attack submarine taken at 6-minute intervals during a submerged trial run.

Time t (hr) 0 0.1 0.2 0.3 0.4 0.5 0.6

Velocity v (mph) 6 24 40 45 38 27 9

a. Use Simpson’s Rule to estimate the distance traveled by the submarine during the 36-minute submerged trial run.

b. Use your answer to estimate the average velocity of the submarine over the 36-minute trial run.

10

1

50. The graph of ( )y f x is shown in the figure at right. The shaded

region labeled A has area 8 square units, and 6

2( ) 16f x dx . Use

this information to find the find the following values:

a. 2

0( )f x dx

b. 6

0( )f x dx

c. 6

0( )f x dx

d. 2

02 ( )f x dx

e. 2

0( ) 2f x dx

f. the average value of f on [0,6]

51. For each of the following statements, determine whether the statement is true or false, and justify your claim.

a. If a function f is continuous at x c , then f must be differentiable at x c .

b. If a function f is differentiable at x c , then f must be continuous at x c .

c. If a function f is not differentiable at x c , then f must be discontinuous at x c .

d. If x c is a critical number of a function f , then f must attain a local maximum or a local minimum

at x c .

e. The function 2 , 1( )

2 , 1

x xf xx x

is continuous at 1x .

f. The function 2 , 1( )

2 , 1

x xf xx x

is differentiable at 1x .

g. If ( ) ( )f x g x for all x in ,a b and both f and g are continuous on ,a b , then

( ) ( )b b

a af x dx g x dx .

h. If f is an even function that is continuous for all ,x then ( ) 0a

af x dx

.

i. If f is an odd function that is continuous for all ,x then ( ) 0a

af x dx

.

52. State each of the following:

a. The definition of lim ( )x a

f x b

where a and b are real numbers.

b. The definition of continuity for a function ( )f x .

c. The Intermediate Value Theorem.

d. The definition of the derivative for a function ( )f x .

e. The Extreme Value Theorem.

f. The Mean Value Theorem for derivatives.

g. The definition of ( )b

af x dx .

h. The Mean Value Theorem for integrals.

i. The Fundamental Theorem of Calculus (both parts).

11

Answers to the Math 241 final review

1a. 1 2a. 45

1b. –1 2b. 0

1c. DNE (Does Not Exist) 2c. 45

1d. –1 2d. 51

1e. DNE 2e.

1f. –3 2f. DNE

1g. –3 2g.

1h. –3 2h. 1

1i. –3 2i. –

1j. DNE 2j. –2

1k. –1 2k. 0

1l. –1 2l. 16

1m. –1 2m. 34

1n. DNE 2n. DNE

1o. DNE 2o. 2

1p. –2 2p. 0 (See Thomas’ Calculus ET, 12th ed, p. 107)

1q. –1 2q. DNE

2r. 0

2s. 0

3. 20

1 cos 1lim

2x

x

x

by Sandwich Theorem since

2

0 0

1 1 1 1lim and lim

2 24 2 2 2x x

x

4a. Let 0 be given.

Let 2

.

Then 0 2x

22

x

2 2x

2( 2)x

2 4x

2 1 3x .

Since this is true for any 0 , 2

lim (2 1) 3x

x

.

12

4b. Let 0 be given. Without loss of generality, we may assume 1 .

Let 24 . Note that 24 0 since 0 1 .

So 0 4x

24 4x

2 24 4 4x

Now note that for any 0 , 2 24 4 so we have

2 2 24 4 4 4x 2 24 4 4 4x

2 2

2 2x

We assumed 1 , so we will have 2 0 , so we can write

2 2x

2x

2x

Since this is true for any 0 , 4

lim 2x

x

.

5a. (For the Continuity Test, see Thomas, p. 94.) Continuous on ( , 1) ( 1,2) (2, ) ; discontinuity at

x = 2 since (2)f is undefined, and discontinuity at x = ‒1 since ( 1)f is undefined and 1

lim ( ) DNEx

f x

;

x = 2 is a removable discontinuity and x = ‒1 is a nonremovable discontinuity.

5b. Continuous on ( ,2) (2, ) ; discontinuity at x = 2 since (2)f is undefined and 2

lim ( ) DNEx

f x

;

x = 2 is a nonremovable discontinuity

5c. Continuous on ( , 2) ( 2, ) ; discontinuity at x = ‒2 since ( 2)f is undefined and

2lim ( ) DNE

xf x

; x = ‒2 is a nonremovable discontinuity

6. 12

b

7. 4( ) 3f x x x is a polynomial, which is continuous for all values of x, and (1) 1 0f , and

(2) 15 0f . Applying Intermediate Value Theorem with 1a , 2b , and 0 0y , there must be a

value x c with c between 1 and 2 for which ( ) 0f c .

13

8a.

8b. 1

lim ( ) 0x

f x

and 1

lim ( ) 0x

f x

, so 1

lim ( ) 0x

f x

;

0

lim ( ) 1x

f x

and 0

lim ( ) 1x

f x

, so 0

lim ( ) 1x

f x

;

3

3

lim ( )x

f x e

and 3

lim ( ) 0x

f x

, so 3

lim ( ) DNEx

f x

;

4

lim ( ) 7x

f x

and 4

lim ( ) 7x

f x

, so 4

lim ( ) 7x

f x

;

5

lim ( )x

f x

and 5

lim ( )x

f x

, so 5

lim ( ) DNEx

f x

;

lim ( ) DNEx

f x

, and lim ( ) 1x

f x

8c. f is discontinuous at 1x since ( 1)f is undefined;

f is continuous at 0x since 0

(0) 1 lim ( )x

f f x

;

f is discontinuous at 3x since 3

lim ( ) DNEx

f x

;

f is continuous at 4x since 4

(4) 7 lim ( )x

f f x

;

f is discontinuous at 5x since (5)f is undefined and 5

lim ( ) DNEx

f x

f is continuous for all other values of x.

8d. f cannot be differentiable at 1,x 3x , or 5x since f is not continuous at those values of x.

To check differentiability at 0x and 4x , examine the behavior of the slopes of the tangent lines on either side of these values.

0 0 0

lim ( ) lim 1 lim 1 1ddx

x x x

f x x

, and 0 0 0

lim ( ) lim lim 1x xddx

x x x

f x e e

so f is

differentiable at 0x .

24 4 4

lim ( ) lim 9 lim 2 8ddx

x x x

f x x x

, and

24 4 4

7 7lim ( ) lim lim 7

5 ( 5)

ddx

x x x

f xx x

. The slopes approach different values on either

side of 4x so f is not differentiable at 4x

Therefore f is not differentiable at 1,3,4, and 5x , but is differentiable for all other values of x.

(4, 7)

33,e

(3,0)(0,1)

( 1,0)

14

9a. Start: h

xhxxf

h

)73(7)(3lim)(

0

Answer: 3)( xf

9b. Start: h

xxhxhxxf

h

)2()(2)(lim)(

22

0

Answer: 22)( xxf

9c. Start: h

xf xhx

h

11

11

0lim)(

Answer:

21

1)(

xxf

9d. Start:

h

xhxxf

h

1212lim)(

0

Answer:

12

1)(

xxf

10a. 5 2

4

45

3

3

1015)(

x

x

xxf

10b. x

xxf

tan

sec)(

2

10c. x

xf2

)(

10d. 5222522 4)( xx exexf

10e. )103()53(6)( 52 xxxxf

10f. 2)53(

31)(

xxf

10g. 5

2sin4cossin2)(

x

xxxxxf

10h. )4 )(sin4 (cos48)( 2 xxxf

10i. 41

2)(

x

xxf

10j. 2

21

91

33sin2)(

x

xxxxf

10k. x

x

e

exf

21)(

10l. )9 (ln9)( xxf

10m. xxxxxxf sec4tansec)( 34

10n. 1

)(2

x

xxf

10o. 12

1)(

xxxf

10p. 2ln2)( 2 xxxf x

11a.

ln2 ln

xdy x x

dx x

11b. 2

2

y

x

dx

dy

11c. 3

2

x

y

dx

dy

11d. x

x

eyxy

yexy

dx

dy

322

4

43

2

11e. 2

1 1sin cos y y

dy y

dx y xy

12. y = 5

2

5

12x

13. y = 3x – 2 and y = 3x + 2

14a. 3 14 2 3y x or 3 2

2 4y x 14b. 1 23 3 4y x

or 223 6

y x

15

15a. f is continuous for all x except 0, so f is continuous on [1,4] ; 4

2( )

xf x exists for all x except 0, so f

is differentiable on (1,4) .

15b. 2c

16a. Relative/local and absolute min: 21,1 ;

relative/local and absolute max: 21,1 ;

points of inflection:

4

3

4

3 ,3 ,0,0 ,,3 ;

increasing on ( 1,1) ;

decreasing on ( , 1) (1, ) ;

concave up on 3,0 3, ; concave down on , 3 0, 3 ; horizontal asymptote 0y

16b. Relative/local and absolute min: 2,4

5 ;

relative/local and absolute max: 2,4 ;

points of inflection: 0, and 0,4

74

3 ;

increasing on 54 40, ,2

;

decreasing on 54 4, ;

concave up on 3 74 4, ;

concave down on 3 74 40, ,2

16c. Relative/local and absolute min: (3π,-1);

relative/local and absolute max: (π,1);

point of inflection: (2π,0);

increasing on (0, ) (3 ,4 ) ;

decreasing on ( ,3 ) ;

concave up on (2 ,4 ) ;

concave up on ( ,3 )

16

16d. No extrema;

point of inflection: (2,8);

increasing on ( , ) ;

concave up on (2, ) ;

concave down on ( ,2)

16e. Relative/local and absolute min: (1,1);

no maxima;

increasing on (1, ) ;

decreasing on (0,1) ;

no point of inflection;

concave up on (0, ) ;

vertical asymptote 0x

16f. No minima;

relative/local and absolute max: e

e 22 , ;

point of inflection:

34

38

3

8,

e

e ;

increasing on 20,e ;

decreasing on 2 ,e ;

concave up on 8

3 ,e

;

concave down on 8

30,e

;

horizontal asymptote 0y ;

vertical asymptote 0x

16g. No extrema;

point of inflection: 0,0 ; decreasing on ( , 3) ( 3,3) (3, ) ;

concave up on ( 3,0) (3, ) ;

concave down on ( , 3) (0,3) ;

horizontal asymptote 0y ;

vertical asymptotes 3,3 xx

6543210-1-2-3-4-5-6

5

4

3

2

1

0

-1

-2

-3

-4

-5

17

16h. No minima;

relative/local and absolute max: 11, e ;

point of inflection: 2

22,

e

;

increasing on ( ,1) ;

decreasing on (1, ) ;

concave up on (2, ) ;

concave down on ( ,2) ;

horizontal asymptote 0y

16i. Relative/local absolute min: (0,0) ;

no maxima;

points of inflection: 4

1,

63

and 4

1,

63

;

increasing on (0, ) ;

decreasing on ( ,0) ;

concave up on 4 4

1 1,

3 3

;

concave down on 4 4

1 1, ,

3 3

;

horizontal asymptote 2

y

17a. Concave down on 41, ; concave up on ,

41 ; inflection point at

85

41 ,

17b. Concave up on ,2 ,0 ; concave down on 0, 2, ; inflection point at 0,0

18a. Decreasing on

65

65 0, , ; increasing on

, 0,65

65

18b. Decreasing on 2,,02

32 ; increasing on

23

2,

19a. 1

2x

19b. 12 0f , so local/relative min is 12e

at 12

x

19c. increasing on 12, ; decreasing on 12

,

19d. concave up on (1, ) ; concave down on ( ,1)

19e. point of inflection at 211, e

18

20. 21.

Answers may vary. Answers may vary.

22a.

2

2

12,000

25

dP t

dtt

,

22

2 32

12,000 25 3

25

td P

dt t

22b.

22c. 5

32.89t years after January 1, 2010. (November 19, 2012)

t ( )P t dP

dt

2

2

d P

dt

0 10 On January 1, 2010, the

population was 10,000 animals

0

On January 1, 2010, the

population was not growing or

declining

19.2

On January 1, 2010, the rate of

growth of the population was

increasing at a rate of 19,200

animals per year each year

5 130 On January 1, 2015, the

population is projected to be

130,000 animals

24

On January 1, 2015, the

population is projected to be

growing at a rate of 24,000

animals per year

‒4.8

On January 1, 2015, the rate of

growth of the population is

projected to be decreasing at a rate

of 4,800 animals per year each

year

10 202 On January 1, 2020, the

population is projected to be

202,000 animals

7.68

On January 1, 2020, the

population is projected to be

growing at a rate of 7,680 animals

per year

‒1.6896

On January 1, 2020, the rate of

growth of the population is

projected to be decreasing at a rate

of 1689.6 animals per year each

year

19

23a. sec2

1069 9.65 sec

23b. 80 ft/sec

23c. ft/sec10616 –164.73 ft/sec

23d. 424 ft

24a. Estimates may vary slightly: 0,0.3 , 1.5,3 , (4,5) , (6,7) seconds

24b. Estimates may vary slightly: 0.3,1.5 , (5,6) , (7,9) seconds 24c. Estimates may vary slightly: 1.5, 6t t seconds

24d. Estimates may vary slightly: (3,4) seconds

24e. Estimates may vary slightly: 1.5, 3,4 , 6t t seconds

24f. Estimates may vary slightly: 0.5, 2, 3,4 , 5, 7t t t t seconds 24g. Estimates may vary slightly: [0,0.5), (2,3), (5,7) seconds

24h. Estimates may vary slightly: (0.5,2), (4,5), (7,9) seconds

25. Length = 12'', width = 6'', height = 2''; Volume = 144 in3

26. Length = 25 , width = 2

25

27. Side of the square = 349

310

, side of the triangle=

349

30

28. km/day 3

4

dt

dr

29. /minft 36 3dt

dV

30a. 48

13.313 ft/sec13

dS

dt

30b.

247.589 ft/sec

10

dT

dt

31. sec/47.4sec/rad769

60

dt

d

32. 8.0,915.0 dyy

20

33a. 0.08π m2 33b. 2%

34. 3 2

sin 2)( x

xF

35. 9 8)( 4

x

exF

36a. –6

36b. 1,614,3187 72

7 7(7 4 )

36c. 332

36d. 2ln33

36e.

3

2215

(5 7)x C

36f. Cx 44

1)sin1(

36g. 2 sec 2x + C

36h. Cx )(tan23

3

2

36i. 5 ln 2

36j. Cx

3

1 sin 5

36k. 6

36l. Cx 2sec 1

36m. Cxxx sincos

36n. xe 55

1 + C

36o. 3

38

36p. Cxxx ln

36q. xxxCx 96or 32 2334

61 3 + C

36r. Cx cot5

36s. 2 4 4 41 1 1

4 8 32

t t tt e te e C

36t. Cxx cot

36u. 21

41 4x C

36v.

Cx

535

1

36w. 13

tan 3x C

36x. 1 212tan ln 1x x x C

36y. ln cos x C or ln sec x C

37a. 2sin x

37b. 1 2 6 xx

37c. xe x cos2sin

21

38a. 32 13691 44 4 35 9 315

324 2 2 2 2 4.346

9T

38b. 2 2 2 294 8 16 4 160 2sin 2sin 2sin sin 0.251T

39a. upper sum = 1/16 1/4 9/1614 1.705e e e e

39b. lower sum = 1/16 1/4 9/1614 1 1.276e e e

40. 25 9 4914 12 16 4 160 4 ln 2 ln 4 ln ln 4 0.773S

41.

5

4

12 1 10.00026042 (note: round up for upper bound)

180 4 3840SE

42. 41n

43a. 1 1

2 2 2

0 0(2 ) OR 2 2 A x x x dx x x dx

43b. /4 /2 /4

0 /4 0cos sin + sin cos OR 2 cos sin A x x dx x x dx x x dx

43c. 3 3

2 2 2

0 0(2 ) ( 4y) OR 6 2 A y y y dy y y dy

44a. 1 1

2 2

0 0( ) OR x xV e dx e dx

44b. 8 8 2 2

2 2/3 3 43

0 0 0 0( ) = OR 2 (8 ) = 2 8V y dy y dy V x x dx x x dx

44c. 2 2

2 3 3 4

0 0 = 2 (2 ) 2 2V x x x dx x x dx

22

44d. 1 1

2 2 2

0 0(1 ) (1 ) OR 2 ( 1)( ) V y y dy V x x x dx

45. 16

46. ln(1 2)

47. 8

(3 3 2 2)3

48. 2

49a. 0.16 3 6 4 24 2 40 4 45 2 38 4 27 9 18.5S miles

49b. 1856

30.83 mph

50a. ‒8

50b. 8

50c. 24

50d. 16

50e. ‒12

50f. 43

51a. F

51b. T

51c. F

51d. F

51e. F

51f. F

51g. T

51h. F

51i. T

All page references are for Thomas’ Calculus, Early Transcendentals, 12th edition

52a. p. 77

52b. p. 93

52c. p. 99

52d. p. 126

52e. p. 223

52f. p. 231

52g. p. 314

52h. p. 325

52i. p. 327, 328