MATH 113 EXAM 2 Review PDF

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MATH 113 EXAM 2 Review...

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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an equation for the tangent to the curve at the given point. 3 1) y = x , (6 , 108) 2 A) y = 18 x + 216

B) y = 216 x + 54

1)

C) y = 18 x - 216

x - x + 3 , (100, 3 ) 1 1 A) y = - x + 3 B) y = - x + 53 2 2

D) y = 54x - 216

2) f(x) = 10

2) 1 C) y = x - 53 2

D) y = 3

Graph the equation and its tangent. 3) Graph y = x 2 - 3 x - 9 and the tangent to the curve at the point whose x-coordinate is -2. y 10

5

-10

-5

10 x

5 -5

-10

A)

B) y

-10

y

10

10

5

5

-5

5

10 x

-10

-5

5

-5

-5

-10

-10

1

10 x

3)

C)

D) y

y

-10

10

10

5

5

-5

-10

10 x

5

-5

10 x

5

-5

-5

-10

-10

4) Graph y = x3 + 2 and the tangent to the curve at the point whose x-coordinate is 0.

4)

y 8 6 4 2 -8 -6 -4 -2 -2

2

4

6

8

x

-4 -6 -8

A)

B) y

y

8

8

6

6

4

4

2

2

-8 -6 -4 -2 -2

2

4

6

8

x

-8 -6 -4 -2 -2

-4

-4

-6

-6

-8

-8

2

2

4

6

8

x

C)

D) y

y

8

8

6

6

4

4

2

2

-8 -6 -4 -2 -2

2

4

6

x

8

-8 -6 -4 -2 -2

-4

-4

-6

-6

-8

-8

2

4

6

8

x

Find the slope of the curve at the indicated point. 8 5) y = , x=7 4+x A) m = -

8 121

B) m = -

5)

8 11

C) m =

8 121

8 11

D) m =

Solve the problem. 6) Find the points where the graph of the function have horizontal tangents. f(x) = 6 x2 + 3 x - 2 A) (-15 , 3193 )

B) -

1 19 ,4 8

C) (0, 2)

5 1 x+ 4 4

B) y =

D)

1 31 ,4 4

x + 1 that has slope 1 . 4

7) Find an equation of the tangent to the curve f(x) = A) y = -

6)

1 x 4

C) y =

5 1 x+ 4 4

7) D) y =

5 1 x4 4

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 8) Does the graph of f(x) =

3, 0, -3,

x 0

8)

0, 5,

x> 0 have a vertical tangent at the point (0, 5)? Give x≤0

9)

reasons for your answer.

9) Does the graph of f(x) = reasons for your answer.

3

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. ′ 10) f(x) = 5x + 9; f (2) ′ ′ ′ ′ A) f (x) = 9; f (2) = 9 B) f (x) = 5; f (2) = 5 ′ ′ ′ ′ C) f (x) = 5x; f (2) = 10 D) f (x) = 0; f (2) = 0

11)

dr if r = dθ θ =3

10)

4

11)

28 - θ

A)

dr 4 dr = 4 ; θ =3 = - 125 dθ 3/2 dθ (28 - θ)

B)

dr 2 dr = 2 ; θ =3 = 125 dθ (28 - θ) 3/2 dθ

C)

dr 2 dr 2 ; =θ =3 = - 125 dθ 3/2 dθ (28 - θ)

D)

dr 4 dr 4 ; = θ =3 = 125 dθ (28 - θ) 3/2 dθ

Find the indicated derivative. dp 1 if p = 12) dq q +2 A) C)

13)

12)

1 2(q + 2)

B) -

q +2

1 2(q + 2)

D) -

q +2

1 2

q +2 1

(q + 2)

q +2

dt x if t = dx 2x - 2 A) -

2x (2x - 2) 2

13) B) -

2

C) -

(2x - 2) 2

2 2x - 2

D)

4x - 2 (2x - 2) 2

Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 2 14) f(x) = 3 x + , x = 5 14) x A)

73 25

B)

77 5

C)

77 25

D)

73 5

Find an equation of the tangent line at the indicated point on the graph of the function. 15) w = g(z) = 4 +

6 - z, (z, w) = (5, 5 ) 15 1 A) w = z + 2 2 C) w = -

15) B) w = -

15 1 z+ 2 2

D) w =

4

15 1 z2 2

15 1 z2 2

Use the formula f'(x) = lim f(z) - f(x) to find the derivative of the function. z- x z→ →x 16) g(x) = A)

x x +5

16)

x

B) -

(x + 5 ) 2

17) g(x) = 4x + A) 4 +

5

C)

(x + 5 ) 2

5

D)

(x + 5 ) 2

x2 x +5

x

17)

1 2

B)

x

1 2

C) 4 + 1 x

x

D) 4 -

1 2

The graph of a function is given. Choose the answer that represents the graph of its derivative. 18) y 15 10 5 -15 -10

-5

5

15 x

10

-5 -10 -15

A)

B) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-15 -10

-5

-5

-5

-10

-10

-15

-15

C)

5

10

15 x

5

10

15 x

D) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-15 -10

-5

-5

-5

-10

-10

-15

-15

5

x

18)

19)

19) y 15 10 5 -15 -10

-5

5

15 x

10

-5 -10 -15

A)

B) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-15 -10

-5

-5

-5

-10

-10

-15

-15

C)

5

10

15 x

5

10

15 x

D) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-15 -10

-5

-5

-5

-10

-10

-15

-15

6

20)

20) y 15 10 5 -15 -10

-5

5

15 x

10

-5 -10 -15

A)

B) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-15 -10

-5

-5

-5

-10

-10

-15

-15

C)

5

10

15 x

5

10

15 x

D) y

-15 -10

y

15

15

10

10

5

5

-5

5

10

15 x

-5 -5

-10

-10

-15

-15

Given the graph of f, find any values of x at which f 21)

A) x = -3, 0, 3

-15 -10

-5

′

is not defined. 21)

B) x = -2, 0, 2

C) x = -3, 3

7

D) x = -2, 2

22)

22)

A) x = 1

B) x = 0

C) x = 2

D) x = 0, 1, 2

23)

23)

A) x = 2, 5 C) x = 5

B) x = 2 D) Defined for all values of x

Determine if the piecewise defined function is differentiable at the origin. 6x - 7 if x < 0 24) f(x) = 2 x + 5 x + 7 if x ≥ 0 A) Differentiable

24)

B) Not differentiable

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? 25) x = 0 25) y

4

2

-4

-2

2

4

x

-2

-4

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable

8

26) x = -1

26) y

4

2

-4

-2

2

4

x

-2

-4

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable 27) x = 1

27) y

4

2

-4

-2

2

4

x

-2

-4

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. 5 28) Over what intervals of x-values, if any, does the function y = x decrease as x increases? 5 For what values of x, if any, is y 29) Does the curve y = answer.

′

28)

negative? How are your answers related?

x ever have a negative slope? If so, where? Give reasons for your

9

29)

30) Does the curve y = (x + 3) 3 have any horizontal tangents? If so, where? Give reasons for your answer.

30)

31) Does the curve y = x3 + 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?

31)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 2 7 32) r = s3 s A)

32)

7 6 4 s2 s

33) y = 3 x2 + 12x + 4e x A) 3x + 4e x

B) - 6 + 7 s4 s2

C)

7 2 4 s2 s

D) - 6 + 7 s2 s2

B) 6x + 12 + 4e x

C) 6x + 12 + e x

D) 6x + 4e x

B) 6 + 60 x-5

C) 6x + 9 - 15 x-4

D) 6 - 60 x-5

33)

Find the second derivative. 34) y = 3 x2 + 9 x + 5 x-3 A) 6 + 60 x-1

34)

′

Find y . 35) y = 1 + 3 x2 A) -

x2 - 1 + 3 x2

4 - 6x x5

35) B) -

1 +6x x5

C)

4 +6x x5

D)

4 +6x x3

Find the derivative of the function. 2 36) y = x - 3x + 2 x7 - 2

36)

8 7 6 ′ A) y = -5x + 18x - 13x - 4x + 6 7 2 (x - 2)

8 7 6 ′ B) y = -5x + 18x - 14x - 4x + 6 7 2 (x - 2)

8 7 6 ′ C) y = -5x + 19x - 14x - 4x + 6 (x7 - 2) 2

8 7 6 ′ D) y = -5x + 18x - 14x - 3x + 6 (x7 - 2) 2

-1 37) f(t) = (2 - t)(2 + t3 ) 3 2 ′ A) f (t) = 2t - 6t - 2 2 (2 + t3 )

3 2 ′ B) f (t) = - 2t + 6 t - 2 2 (2 + t3 )

2t3 - 6t2 - 2 2 + t3

3 2 ′ D) f (t) = - 4t + 6 t - 2 (2 + t3 ) 2

′

C) f (t) =

37)

10

θ -3

38) r =

38)

θ +3 3

′

A) r = -

′

B) r =

θ(θ + 3 ) 2 6

′

C) r = (θ + 3 )

′

D) r =

θ2 - 9

3 θ +3 3 θ(θ + 3 ) 2

Find the derivative. 39) y = 6 x2e -x A) 6xe -x(x + 2)

39) B) 12xe -x(1 - x)

C) 6xe -x(2 - x)

7

x6 + x6e 6 A) x1/7 + 6 ex6 e -1 7

40) y =

D) 6xe x(2 - x)

C)

40) 6 B) x-1/7 + 6 ex6 e -1 7

6 1/7 x + 6 x6e -1 7

D)

6 -1/7 x + 6 x6e -1 7

Find the derivative of the function. q6 + 4 q8 + 6 41) p = 2q q

41)

A)

dp = 6 q11 + 12q 5 + 12q 3 - 24 dq q3

B)

dp = 1 q11 + 2q5 + 3 q 3 + 24 dq 2 q3

C)

dp 24 = 6 q11 dq q3

D)

dp 24 = 8 q15 + 20 q9 + 24q 7 dq q3

Find the second derivative of the function. x4 + 3 42) y = x2

42)

A)

d2y 18 =2x4 dx2

B)

d2y 18 =1+ x4 dx2

C)

d2y = 2 + 18 x4 dx2

D)

d2y = 2x - 6 x3 dx2

2 43) y = (x - 9 )(x + 4x) x3

43)

A)

d2y 10 216 =2 x4 x3 dx

B)

d2y 10 216 =2 x x2 dx

C)

d2y 10 216 + = x4 x3 dx2

D)

d2y 72 5 + = 2 2 x3 x dx

11

Provide an appropriate response. 44) Find an equation for the tangent to the curve y = 8 x at the point (1, 4). x2 + 1 A) y = 4

B) y = 4x

C) y = x + 4

44) D) y = 0

45) Find all points (x, y) on the graph of f(x) = 2x2 -3x with tangent lines parallel to the line y = 5x +6. A) (4, 2) B) (0, 0), (2, 2) C) (2, 2) D) (2, 8 ) 46) Find all points (x, y) on the graph of y = y = 3 x - 2. A) (6 , 2)

x with tangent lines perpendicular to the line (x - 3 )

B) (0, 0), (3, 2)

C) (0, 0)

45)

46)

D) (0, 0), (6, 2)

The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 47) s = - t3 + 8 t2 - 8 t, 0 ≤ t ≤ 8 47) Find the body's speed and acceleration at the end of the time interval. A) -72 m/sec, -32 m/sec2 B) 8 m/sec, 0 m/sec2 C) 72 m/sec, -8 m/sec2

D) 72 m/sec, -32 m/sec2

Solve the problem. 48) The area A = "r2 of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 4 ft? A) 8 " ft2/ft B) 8 ft2/ft C) 16" ft2/ft D) 4" ft2/ft Find the derivative. 49) y = 5 + 4 sec x x

49)

′ A) y = - 5 - 4 csc x x2

′ B) y = - 5 + 4 tan2x x2

′ C) y = 5 - 4 sec x tan x x2

′ D) y = - 5 + 4 sec x tan x x2

50) s = t3 tan t - t ds = - t3 sec2 t + 3 t2 tan t + 1 A) dt 2 t C)

48)

50) ds = t3 sec2 t + 3 t2 tan t - 1 B) dt 2 t

ds = 3 t2 sec2 t - 1 dt 2 t

D)

ds = t3 sec t tan t + 3 t2 tan t - 1 dt 2 t

51) y = (csc x + cot x)(csc x - cot x)

51) B) y = - csc2 x

′

′

A) y = 1 ′

′

C) y = - csc x cot x

D) y = 0

12

52) y = sin x + 9x 9x sin x

52)

A)

dy cos x 9 = + dx 9 cos x

B)

dy sin x - x cos x 9 x cos x - 9 sin x + = dx sin2x 81x2

C)

dy x cos x - sin x 9 sin x - 9 x cos x + = dx sin2x 9x2

D)

dy = x cos x + sin x + 9 sin x + 9 x cos x dx sin2x 9x2

53) s = sin t - e -t ds = cost - e -t A) dt C)

54) p =

53) ds = -cos t + e -t B) dt

ds = -cost - e -t dt

D)

ds = cos t + e -t dt

sec q + csc q csc q

54)

A)

dp = sec2 q + 1 dq

B)

dp = sec q tan q dq

C)

dp = sec2 q dq

D)

dp = - csc q cot q dq

Find the indicated derivative. ′ ′ 55) Find y if y = 6x sin x. ′ ′ A) y = 6 cos x - 12x sin x ′ ′ C) y = - 12 cos x + 6 x sin x

55) ′ ′

B) y = 12 cos x - 6 x sin x ′ ′ D) y = - 6 x sin x

56) Find y (4) if y = -6 cos x. A) y (4) = 6 cos x

56) B) y (4) = -6 sin x D) y (4) = -6 cos x

C) y (4) = 6 sin x Find the limit. 57)

lim x→"/3 A) 1

5 2 + sin(" sec x)

57) B) 0

Solve the problem. 58) Find the tangent to y = 2 - sin x at x = ". A) y = x - " + 2 B) y = - x + " - 2

52 + 1

C) 5

D)

C) y = - x + 2

D) y = x - 2

58)

13

59) Does the graph of the function y = 10x + 5 sin x have any horizontal tangents in the interval 0 ≤ x ≤ 2"? If so, where? " 2" 2" A) Yes, at x = , x = B) Yes, at x = 3 3 3 D) Yes, at x = 2" , x = 4" 3 3

C) No

′

′

Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 5 , u = 2x - 7 60) y = u2 A) -

59)

10 2x - 7

B)

60)

20x 2x - 7

C) -

61) y = sin u, u = cos x A) sin(cos x) sin x C) - cos(cos x) sin x

20 2x - 7

D) -

20 (2x - 7) 3 61)

B) - cos x sin x D) cos x sin x

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 62) y = (-2x + 11) 5

62)

dy A) y = u5 ; u = -2x + 11; = -2(-2x + 11) 5 dx

dy B) y = u5 ; u = -2x + 11; = 5 (-2x + 11) 4 dx

C) y = 5 u + 11; u = x5 ; dy = -10 x4 dx

D) y = u5 ; u = -2x + 11; dy = -10 (-2x + 11) 4 dx

63) y = cos 4 x

63)

A) y = cos u; u = x4; dy = - 4x3 sin(x4) dx

B) y = u4; u = cos x; dy = - 4 cos 3 x sin x dx

dy C) y = u4; u = cos x; = 4 cos 3 x sin x dx

D) y = cos u; u = x4;

dy = - sin(x4) dx

64) y = e 3 x/2

64)

dy 2 3x/2 A) y = e u; u = 3 x/2; = e dx 3 C) y = e u; u = 3 x/2;

dy 3 3x/2 B) y = e u; u = 3 x/2; = e dx 2

dy 3 = xe 3x/2 dx 2

D) y = e u; u = 3 x/2;

dy 2 3x/2 = xe dx 3

Find the derivative of the function. 65) q =

17r - r5 17 - 5 r4

A) 2

17r - r5

65) 1

B) 2

C)

17r - r5

14

-5r 4 17r - r5

1

D) 2
...