MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Estimate the slope of the tangent line to the curve at the given point.

1)

1)

_______ A)

1

B)

2

C)

-1

D)

1/2

2)

2)

_______ A)

1/4

B)

-1/4

C)

-4

D)

-1/2

3)

3)

_______ A)

-

B)

-1

C)

1

D)

-

4)

4)

_______ A)

B)

C)

3

D)

Find the equation for the tangent to the curve at the given point. 5)

f(x) = x2 + 5x ; (4, 36)

5)

_______ A)

y = +

B)

y = -39x - 80

C)

y = - +

D)

y = 13x - 16

6)

f(x) = 5x2 + x; (-4, 76)

6)

_______ A)

y = 39x + 232

B)

y = -39x - 80

C)

y = 13x - 16

D)

y = 39x + 80

7)

f(x) = ;

7)

_______ A)

y = x +

B)

y = -39x - 80

C)

y = - x +

D)

y = 13x - 16

8)

f(x) = ;

8)

_______ A)

y = -39x - 80

B)

y = x +

C)

y = - x +

D)

y = 13x - 16

9)

f(x) = x2 + 11x - 15 ; (1, -3)

9)

_______ A)

y = -39x - 80

B)

y = - x +

C)

y = 13x - 16

D)

y = x +

10)

f(x) = - 2;

10)

______ A)

y = -7x - 9

B)

y = x +

C)

y = - x +

D)

y = x + 1

11)

f(x) = x - ; (1, 0)

11)

______ A)

y = -7x + 28

B)

y = x + 1

C)

y = x -

D)

y = -7x - 9

12)

f(x) = 3x2 + 5x - 7; (-2, -5)

12)

______ A)

y = -7x - 19

B)

y = x -

C)

y = -7x + 28

D)

y = x + 1

13)

f(x) = - 7x, (1, -6)

13)

______ A)

y = -4x - 2

B)

y = -4x

C)

y = 3x - 9

D)

y = 3x - 5

14)

f(x) = , ( 2, 3)

14)

______ A)

y = x +

B)

y = x -

C)

y = x +

D)

y = x -

Find the slope of the curve at the point indicated. 15)

y = , x = 6

15)

______ A)

36

B)

12

C)

-12

D)

1

16)

y = 2 - , x = 4

16)

______ A)

-6

B)

-8

C)

2

D)

-14

17)

y = 10x - , x = 6

17)

______ A)

-2

B)

10

C)

12

D)

-26

18)

y = 4 - 9x, x = 3

18)

______ A)

-33

B)

15

C)

27

D)

33

19)

y = 3/x, x = -4

19)

______ A)

B)

-

C)

-

D)

-

20)

y = -8/x, x = -9

20)

______ A)

B)

C)

-

D)

21)

y = 2 , x = 49

21)

______ A)

B)

C)

-

D)

22)

y = 5 , x = -2

22)

______ A)

-160

B)

-40

C)

-32

D)

320

Solve the problem. 23)

Find the points where the graph of the function has horizontal tangents.f(x) = 3 + 3x - 2

23)

______ A)

( -9, 214)

B)

(0, 2)

C)

D)

24)

Find the points where the graph of the function has horizontal tangents.f(x) = - 21x

24)

______ A)

( , -42 )

B)

(- , 14 ), ( , -14 ) C)

(-7, -196), ( , 196)

D)

(- , 42 ), (0, 0), ( , -42 )

25)

Find equations of all tangents to the curve f(x) = that have slope -1.

25)

______ A)

y = -x + 8

B)

y = x + 8, y = x - 8 C)

y = -x + 8, y = -x - 8

D)

y = x - 8

26)

Find equations of all tangents to the curve f(x) = that have slope -1.

26)

______ A)

y = -x - 23

B)

y = -x + 27, y = -x - 23 C)

y = -x - 23, y = -x - 27

D)

y = -x + 27

27)

Find an equation of the tangent to the curve f(x) = that has slope .

27)

______ A)

y = - x +

B)

y = x -

C)

y = x

D)

y = x +

28)

Find an equation of the tangent to the curve f(x) = - 2x + 1 that has slope 2.

28)

______ A)

y = 2x

B)

y = 2x - 1

C)

y = 2x + 1

D)

y = 2x + 2

29)

A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant . Find the rate of change of surface area with respect to radius when the radius is (Answer can be left in terms of π).

29)

______ A)

120.0π /mm

B)

60.1π /mm C)

120.2π /mm

D)

60.2π /mm

30)

A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to temperature T when the width is and the length is if and

30)

______ A)

1.9 x /°C

B)

4 x /°C C)

0.8 x /°C

D)

3.6 x /°C

31)

A cubic salt crystal expands by accumulation on all sides. As it expands outward find the rate of change of its volume with respect to the length of an edge when the edge is

31)

______ A)

8.67 /mm

B)

0.01 /mm C)

0.0867 /mm

D)

0.87 /mm

32)

The equation for free fall at the surface of Planet X is with t in seconds. Assume a rock is dropped from the top of a cliff. Find the speed of the rock at

32)

______ A)

38.26 m/sec

B)

37.26 m/sec

C)

17.63 m/sec

D)

18.63 m/sec

33)

Assume that a watermelon dropped from a tall building falls in t sec. Find the watermelon's average speed during the first 5 sec of fall and the speed at the instant

33)

______ A)

160 ft/sec; 81 ft/sec

B)

40 ft/sec; 80 ft/sec C)

81 ft/sec; 162 ft/sec

D)

80 ft/sec; 160 ft/sec

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.

34)

Does the graph of f(x) = have a tangent at the origin? Give reasons for your answer.

34)

_____________

35)

Does the graph of f(x) = have a tangent at the origin? Give reasons for your answer.

35)

_____________

36)

Does the graph of f(x) = have a tangent at the origin? Give reasons for your answer.

36)

_____________

37)

Does the graph of f(x) = have a vertical tangent at the origin? Give reasons for your answer.

37)

_____________

38)

Does the graph of f(x) = have a vertical tangent at the point (0, 5)? Give reasons for your answer.

38)

_____________

39)

The graph of the function y = f(x) = is shown below. The "V"-shaped graph comes to a sharp point at Without doing any calculations, decide whether the function has a tangent at Give reasons for

your answer. [Hint: consider the signs of the tangent lines on either side of and what implication this has in terms of the limit definition of the slope of a tangent line.].

39)

_____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified.

40)

f(x) = 5x + 9; (2)

40)

______ A)

(x) = 5x; (2) = 10

B)

(x) = 9; (2) = 9 C)

(x) = 5; (2) = 5

D)

(x) = 0; (2) = 0

41)

g(x) = 3x2 - 4x; (3)

41)

______ A)

(x) = 2x- 4; (3) = 2

B)

(x) = 6x - 4; (3) = 14 C)

(x) = 3x - 4; (3) = 5

D)

(x) = 6x; (3) = 18

42)

f(x) = x2 + 7x - 2; (0)

42)

______ A)

(x) = 2x - 2; (0) = - 2

B)

(x) = 2x; (0) = 0 C)

(x) = x + 7; (0) = 7

D)

(x) = 2x + 7; (0) = 7

43)

g(x) = x3 + 5x; (1)

43)

______ A)

(x) = 3 + 5x; (1) = 8

B)

(x) = + 5; (1) = 6 C)

(x) = 3 ; (1) = 3

D)

(x) = 3x2 + 5; (1) = 8

44)

f(x) = ; (-1)

44)

______ A)

(x) = - ; (-1) = -8

B)

(x) = ; (-1) = 8 C)

(x) = 8; (-1) = 8

D)

(x) = - 8 ; (-1) = - 8

45)

g(x) = - ; (-2)

45)

______ A)

(x) = - 2; (- 2) = - 2

B)

(x) = - ; (- 2) = - C)

(x) = ; (-2) =

D)

(x) = - 2 ; (- 2) = - 8

46)

f(x) = ; (0)

46)

______ A)

(x) = 8; (0) = 8

B)

(x) = - ; (0) = -2 C)

(x) = - 8 ; (0) = - 32

D)

(x) = ; (0) = 2

47)

if s = - t

47)

______ A)

= 2t + 1; = 9

B)

= 2t - 1; = 7 C)

= t - 1; = 3

D)

= 2 - t; = -2

48)

if r =

48)

______ A)

= ; =

B)

= - ; = - C)

= - ; = -

D)

= ; =

Find the slope of the tangent line at the given value of the independent variable. 49)

f(x) = 9x + , x = 2

49)

______ A)

B)

C)

D)

50)

g(x) = , x = 9

50)

______ A)

-

B)

C)

D)

-

51)

s = -4 + -2 , t = -1

51)

______ A)

10

B)

-10

C)

-22

D)

22

Find an equation of the tangent line at the indicated point on the graph of the function. 52)

y = f(x) = , (x, y) = ( 5, 31.25)

52)

______ A)

y = x -

B)

y = x -

C)

y = x +

D)

y = x +

53)

w = g(z) = - 3, (z, w) = ( -4, 13)

53)

______ A)

w = -8z - 38

B)

w = -4z - 19

C)

w = -8z - 19

D)

w = -8z - 35

54)

y = f(x) = + 1, (x, y) = ( -2, 5)

54)

______ A)

y = -4x - 7

B)

y = -2x - 3

C)

y = -4x - 3

D)

y = -4x - 6

55)

y = f(x) = - x, (x, y) = ( -3, 12)

55)

______ A)

y = -7x - 6

B)

y = -7x + 6

C)

y = -7x + 9

D)

y = -7x - 9

56)

y = f(x) = x - , (x, y) = ( -1, -2)

56)

______ A)

y = 3x + 1

B)

y = -3x + 1

C)

y = -x - 1

D)

y = -x + 1

57)

s = h(t) = - 16t + 4, (t, s) = ( 4, 4)

57)

______ A)

s = 32t - 124

B)

s = 4

C)

s = 32t + 4

D)

s = 36t - 124

58)

y = f(x) = 6 - x + 1, (x, y) = ( 36, 1)

58)

______ A)

y = - x + 19

B)

y = 1

C)

y = x - 19

D)

y = - x + 1

Find the value of the derivative. 59)

if s = 1 - 4

59)

______ A)

-32

B)

1

C)

32

D)

-31

60)

if y = 4 -

60)

______ A)

3

B)

1

C)

- 1

D)

5

The graph of a function is given. Choose the answer that represents the graph of its derivative. 61)

61)

______ A)

B)

C)

D)

62)

62)

______ A)

B)

C)

D)

63)

63)

______ A)

B)

C)

D)

64)

64)

______ A)

B)

C)

D)

65)

65)

______ A)

B)

C)

D)

66)

66)

______ A)

B)

C)

D)

Given the graph of f, find any values of x at which is not defined. 67)

67)

______ A)

x = 2

B)

x = -1

C)

x = 0

D)

x = 1

68)

68)

______ A)

x = -3, 0, 3

B)

x = -2, 0, 2

C)

x = -3, 3

D)

x = -2, 2

69)

69)

______ A)

x = 2

B)

x = -2, 0, 2

C)

x = 0

D)

x = -2, 2

70)

70)

______ A)

x = 1

B)

x = 2

C)

x = 0, 1, 2

D)

x = 0

71)

71)

______ A)

x = 1, 2, 3

B)

x = 2 C)

x = 1, 3

D)

Defined for all values of x

72)

72)

______ A)

x = 5

B)

x = 2, 5 C)

x = 2

D)

Defined for all values of x

73)

73)

______ A)

x = -1, 0, 1

B)

x = -1, 1 C)

x = 0

D)

Defined for all values of x

74)

74)

______ A)

x = 0

B)

x = -2, 2 C)

x = -2, 0, 2

D)

Defined for all values of x

75)

75)

______ A)

x = 0, 3

B)

x = 3 C)

x = 0

D)

Defined for all values of x

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.

76)

y = x y = 2x

76)

______ A)

Since (x) = 1 while (x) = 2, f(x) is not differentiable at x = 0. B)

Since (x) = -2 while (x) = -1, f(x) is not differentiable at x = 0. C)

Since (x) = 2 while (x) = 1, f(x) is not differentiable at x = 0. D)

Since (x) = 1 while (x) = 1, f(x) is differentiable at x = 0.

77)

y = y = -1

77)

______ A)

Since (x) = 0 while (x) = 1, f(x) is not differentiable at B)

Since (x) = -1 while (x) = 0, f(x) is not differentiable at C)

Since (x) = 0 while (x) = -1, f(x) is not differentiable at D)

Since (x) = 0 while (x) = 0, f(x) is differentiable at

78)

y = y =

78)

______ A)

Since (x) = while (x) = 1, f(x) is not differentiable at B)

Since (x) = while (x) = 2, f(x) is not differentiable at C)

Since (x) = 2 while (x) = , f(x) is not differentiable at D)

Since (x) = 2 while (x) = 2, f(x) is differentiable at

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable?

79)

x = 0

79)

______ A)

Differentiable B)

Continuous but not differentiable C)

Neither continuous nor differentiable

80)

x = 0

80)

______ A)

Differentiable B)

Continuous but not differentiable C)

Neither continuous nor differentiable

81)

x = -1

81)

______ A)

Differentiable B)

Continuous but not differentiable C)

Neither continuous nor differentiable

82)

x = 1

82)

______ A)

Differentiable B)

Continuous but not differentiable C)

Neither continuous nor differentiable

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.

83)

Does the curve y = + 4x - 10 have a tangent whose slope is -2? If so, find an equation for the line and the point of tangency. If not, why not?

83)

_____________

84)

Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.

84)

_____________

85)

Is there any difference between finding the derivative of f(x) at and finding the slope of the line tangent to f(x) at Explain.

85)

_____________

1)

D 2)

B 3)

B 4)

A 5)

D 6)

B 7)

C 8)

B 9)

C 10)

C 11)

C 12)

A 13)

A 14)

A 15)

B 16)

B 17)

A 18)

B 19)

D 20)

A 21)

A 22)

A 23)

C 24)

B 25)

C 26)

C 27)

D 28)

B 29)

C 30)

D 31)

C 32)

B 33)

D 34)

The function f is differentiable at x = o and hence has a tangent at the origin. 35)

The function f is not differentiable at x = 0 and hence does not have a tangent at the origin. 36)

The graph does not have a tangent at the origin. The slope of the tangent lines as x approaches 0 from the left is -2, whereas the slope of the tangent lines as x approaches 0 from the right is 1. Since the one-sided limits are not equal, then, according to the limit definition of the slope of the tangent line, no tangent line exists.

37)

The function is not differentiable at the x = 0 and hence does not have a tangent at the origin. 38)

The function is not differentiable at x = 0 because it is discontinuous at x = 0. The graph does not have a vertical tangent at (0, 5) since f remains bounded as x approaches zero from either side.

39)

The function does not have a tangent at x = 0. The tangents to the left of x = 0 all have negative slopes, whereas those to the right of x = 0 all have positive slopes. Thus, the limit of the slope as x approaches 0 from the left cannot equal the limit of the slope as x approaches 0 from the right and therefore, according to the definition of the slope of the tangent line, no tangent line exists.

40)

C 41)

B 42)

D 43)

D 44)

A 45)

C 46)

B 47)

B 48)

A 49)

D 50)

D 51)

A 52)

B 53)

C 54)

C 55)

D 56)

A 57)

A 58)

A 59)

A 60)

B 61)

C 62)

C 63)

C 64)

C 65)

C 66)

B 67)

C 68)

D 69)

C 70)

D 71)

D 72)

C 73)

C 74)

C 75)

A 76)

C 77)

C 78)

B 79)

A 80)

C 81)

B 82)

C 83)

The curve has no tangent whose slope is -2. The derivative of the curve, = 3 + 4, is always positive and thus never equals -2.

84)

Yes, a tangent line to a graph can intersect the graph at more than one point. For example, the graph has a horizontal tangent at x = 0. It intersects the graph at both (0, 0) and (2, 0).

85)

There is no difference at all.