Name: ________________________ Class: ___________________ Date: __________ ID: A

2

2413final review

Short Answer

1. Find the limit.

limx 4

x 5x 1

2. Find the limit (if it exists).

limx 5

x 4 3x 5

3. Find the lmit.

limx

tanx3

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃

4. Find the x-values (if any) at which f x( ) x 3 x 3

is not continuous.

5. Suppose that limx c

f x( ) 11 and limx c

g x( ) 3. Find the following limit.

limx c

f x( ) g x( )ÈÎÍÍÍ

˘˚˙̇̇

Name: ________________________ ID: A

2

6. Find the vertical asymptotes (if any) of the function f(x) x2 4

x2 3x 2.

7. Determine the following limit. (Hint: Use the graph to calculate the limit.)

limx 1

x2 4Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜

8. Find the derivative of the function.

y cos 2x4 6ÊËÁÁÁ

ˆ¯˜̃̃

Name: ________________________ ID: A

3

9. Let f x( ) x2 5, x 1

1, x 1

Ï

ÌÓ

ÔÔÔÔÔÔÔÔÔÔ

.

Determine the following limit. (Hint: Use the graph to calculate the limit.)

limx 1

f x( )

10. Discuss the continuity of the function f x( ) x2 4x 2

.

Name: ________________________ ID: A

4

11. Find the x-values (if any) at which the function f x( ) x 2

x2 6x 8 is not continuous. Which of the discontinuities

are removable?

12. Use the graph as shown to determine the following limits, and discuss the continuity of the function at x 3.

(i) limx 3

f(x) (ii) limx 3

f(x) (iii) limx 3

f(x)

13. Find the derivative of the function sinx

2x cos x. Simplify your answer.

14. Find the derivative of the function f x( ) x5 9

x4 .

15. Find the derivative of the function f x( ) 4x2 4cos x( ) .

Name: ________________________ ID: A

5

16. Find the derivative of the function.

f t( ) 4 7t5ÊËÁÁÁ

ˆ¯˜̃̃

8

17. Find the derivative of the function y 8cos 4x .

18. Find the second derivative of the function f x( ) sin5x6 .

19. Find dydx

by implicit differentiation.

x2 5x 9xy y2 4

20. The radius r of a sphere is increasing at a rate of 6 inches per minute. Find the rate of change of the volume when r = 11 inches.

21. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent.

y x( ) 6x

x 9( ) 2

22. Find the second derivative of the function.

f x( ) 3x3 7ÊËÁÁÁ

ˆ¯˜̃̃

7

Name: ________________________ ID: A

6

23. Use the Product Rule to differentiate f s( ) s5 cos s.

24. A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure). When he is 13 feet from the base of the light, at what rate is the tip of his shadow moving?

25. Use the Quotient Rule to differentiate the function f x( ) 8x

x5 3.

26. Use the quotient rule to differentiate the following function f s( ) 2s

s5 7 and evaluate f 2( ).

Name: ________________________ ID: A

7

27. Find the indefinite integral of the following function.

cosx

sin8xdx

28. Locate the absolute extrema of the given function on the closed interval [60, 60].

f(x) 60x

x2 36

29. Find an equation of the tangent line to the graph of the function f x( ) x 2 at the point 18,4ÊËÁÁ

ˆ¯̃̃ .

30. Find all critical numbers of the function g x( ) x4 4x2 .

31. For the function f x( ) 4x3 48x2 6:

(a) Find the critical numbers of f (if any);(b) Find the open intervals where the function is increasing or decreasing; and(c) Apply the First Derivative Test to identify all relative extrema.

Then use a graphing utility to confirm your results.

32. Determine the open intervals on which the graph of y 6x3 8x2 6x 5 is concave downward or concave upward.

33. A rectangular page is to contain 144 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Name: ________________________ ID: A

8

34. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 38 feet.

35. Find the indefinite integral 12s3 12s 3ÊËÁÁÁ

ˆ¯˜̃̃ds .

36. Write the limit limx 0

4ci 7 ciÊËÁÁ

ˆ¯̃̃

2xi

i 1

n

, as a definite integral on the interval 0,8ÈÎÍÍÍ

˘˚˙̇̇ where ci is any point in the

ith subinterval.

37. Find the indefinite integral and check the result by differentiation.

3z2 12z 9

z4 dz

38. Find the indefinite integral 2secy tany secyÊËÁÁ

ˆ¯̃̃dy .

Name: ________________________ ID: A

9

39. Find the slope m of the line tangent to the graph of the function g x( ) 9 x2 at the point 4, 7ÊËÁÁ

ˆ¯̃̃ .

40. Solve the differential equation.

dfds

12s3 , f 1( ) 2

41. The rate of growth dPdt

of a population of bacteria is proportional to the square root of t, where P is the population

size and t is the time in days 0 t 10( ). That is, dPdt

k t . The initial size of the population is 500. After one

day the population has grown to 600. Estimate the population after 5 days. Round your answer to the nearest integer.

42. Find the limit (if it exists).

limx 11

11 x

x2 121

43. Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

limx 4

8x2 40x 32x 4

Name: ________________________ ID: A

10

44. The graph of f consists of line segments, as shown in the figure. Evaluate the definite integral f x( )dx

0

11

using

geometric formulas.

45. A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when its base is 13 feet from the wall? Round your answer to two decimal places.

Name: ________________________ ID: A

11

46. Find the indefinite integral of the following function.

cos4x dx

47. The volume of a cube with sides of length s is given by V s3 . Find the rate of change of volume with respect to s when s 6 centimeters.

48. Find the indefinite integral x2 3 x3 dx .

49. Find the average value of the function f x( ) 48 12x2 over the interval 5 s 5.

50. Find an equation of the line that is tangent to the graph of the function f(x) 8x2 and parallel to the line 16x y 6 0.

Name: ________________________ ID: A

12

51. Determine the area of the given region.

y x sinx, 0 x

52. Determine all values of x , (if any), at which the graph of the function has a horizontal tangent.

y x( ) x 3 12x 2 8

53. The graph of the function f is given below. Select the graph of f .

Name: ________________________ ID: A

13

54. Find the slope of the graph of the function at the given value.

f x( ) 2x2 5x 7

x2 when x 4

55. Suppose the position function for a free-falling object on a certain planet is given by s t( ) 16t2 v 0 t s0 . A

silver coin is dropped from the top of a building that is 1372 feet tall. Determine the average velocity of the coin over the time interval 3, 4È

ÎÍÍÍ

˘˚˙̇̇ .

56. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.

81 t2 d t

0

9

57. Evaluate the following definite integral

9s2 4ÊËÁÁÁ

ˆ¯˜̃̃ds

2

4

58. Evaluate 1 lnx( ) 5

xdx

1

e

.

59. A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. If water is flowing into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep.

Name: ________________________ ID: A

14

60. Use implicit differentiation to find dydx

.

7x2 8lnxy 14

61. Find the indefinite integral x2

4x3 9 dx.

62. Find x2 14x 10

x 12dx .

63. Find the indefinite integral.

2xe 4x2

dx

64. Find f x( ) if f x( ) 7 4x( )e4x .

65. Find the derivative of the function y ln x x2 7Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃ .

66. Find the derivative of the function f x( ) ln11x

x2 8

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃ .

ID: A

1

2413final reviewAnswer Section

SHORT ANSWER

1. 1

2. 16

3. 3 4. f x( ) is not continuous at x 3 and the discontinuity is nonremovable. 5. –8 6. x 1 7. 5

8. y 8x3 sin 2x4 6ÊËÁÁÁ

ˆ¯˜̃̃

9. 6 10. f x( ) is discontinuous at x 2. 11. x 2 ( removable), x 4 (not removable) 12. 1 ,1 ,1 , not continuous

13. 1 2x cos x 2sinx

2x cos x( ) 2

14. f x( ) 1 36

x5

15. f x( ) 8x 4sin x( )

16. f t( ) 280t4 4 7t5ÊËÁÁÁ

ˆ¯˜̃̃

7

17. y 32sin4x

18. f x( ) 150x4 cos 5x6 900x10 sin5x6

19. dydx

2x 5 9y

2y 9x

20. dV

dt 2904 in3 / min

21. x 9

22. f 63x 7 3x3ÊËÁÁÁ

ˆ¯˜̃̃

514 60x3Ê

ËÁÁÁ

ˆ¯˜̃̃

23. f s( ) s5 sins 5s4 cos s

24. 50

3 ft/sec

ID: A

2

25. f x( ) 8 3 4x5ÊËÁÁÁ

ˆ¯˜̃̃

x5 3ÊËÁÁÁ

ˆ¯˜̃̃ 2

26. f 2( ) 54

125

27. sin x( ) 7

7C

28. absolute max: 6, 5ÊËÁÁ

ˆ¯̃̃; absolute min 6, 5Ê

ËÁÁˆ¯̃̃

29. y x8 7

4

30. critical numbers: x 0, x 2, x 2 31. (a) x = 0 , 8

(b) increasing: ,0ÊËÁÁ

ˆ¯̃̃ 8,Ê

ËÁÁˆ¯̃̃ ; decreasing: 0,8Ê

ËÁÁˆ¯̃̃

(c) relative max: f 0( ) 6 ; relative min: f 8( ) 1018

32. concave upward on ,49

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃; concave downward on

49

,Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃

33. 14,14

34. x 764 feet; y 38

4 feet

35. 3s4 6s2 3s C

36. 4x 7 x( ) 2dx

0

8

37. 3z 6

z2 3

z3 C

38. 2secy 2tany C 39. m 8

40. f s( ) 3s4 5

41. P 5( ) 1618 bacteria

42. 122

43. –24 44. 2 45. 1.71 ft/sec

46. sin4x

4C

47. 108 cm2

ID: A

3

48. 2 3 x3ÊËÁÁÁ

ˆ¯˜̃̃

32

9C

49. –52 50. 16x y 8 0

51. 0.5 2 2 52. x 0 and x 8

53.

54. f 4( ) 35932

55. –112 ft/sec

56. 814

57. 240

58. 1 lnx( ) 5

xdx

1

e

212

59. 81

50ft/min

60. y 7x2 4ÊËÁÁÁ

ˆ¯˜̃̃

4x

61. 112

ln 4x3 9

C

62. x2 14x 10

x 12dx 1

2x2 26x 322ln x 12 C

63. 14

e4x2

C

64. f x( ) 80 64x( )e4x

ID: A

4

65. 1x x

x2 7

66. f x( ) 1x 2x

x2 8