Mid-Term 2013 Review

Name___________________________________

Find the slope of the line tangent tangent to the graph a the given point. 1) y= x2 + 10x, x = 8 A) m = 144

B) m = 18

C) m = 16

D) m = 26

Answer: D 2) y =

6 ,x=7 3+x

A) m =

3 5

B) m =

3 50

3 5

C) m = -

D) m = -

3 50

Answer: D Find an equation for the tangent to the curve at the given point. 3) h(x) = t3 - 36t - 5, (6, -5) A) y = 72t - 437

B) y = 72t - 5

C) y = -5

D) y = 67t - 437

Answer: A 4) f(x) = 10 x - x + 3, (100, 3) A) y = 3

B) y =

1 x - 53 2

C) y = -

1 x + 53 2

Answer: C Calculate the derivative of the function. Then find the value of the derivative as specified. 5) g(x) = x3 + 5x; g (1) A) g (x) = 3x2 + 5x; g (1) = 8

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

C) g (x) = 3x2; g (1) = 3

D) g (x) = x2 + 5; g (1) = 6

Answer: B

1

D) y = -

1 x+3 2

6)

dr if r = dt t =3

4 28 - t

A)

dr 4 dr 4 ; = = dt (28 - t)3/2 dt t =3 125

B)

dr 2 dr 2 ; = = dt (28 - t)3/2 dt t =3 125

C)

dr 2 dr 2 ; ===3 dt 3/2 dt t 125 (28 - t)

D)

dr 4 dr 4 ; ===3 dt 3/2 dt t 125 (28 - t)

Answer: B Find y . 7) y = (5x - 2)(6x + 1) A) 60x - 3.5

B) 30x - 7

C) 60x - 7

Answer: C 8) y = (3x - 5)(4x3 - x2 + 1) A) 12x3 + 23x2 - 69x + 3

B) 36x3 + 69x2 - 23x + 3

C) 48x3 - 23x2 + 69x + 3

D) 48x3 - 69x2 + 10x + 3

Answer: D

2

D) 60x - 17

Solve the problem. 9) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f.

Answer:

3

10) Use the following information to graph the function f over the closed interval [-5, 6]. i) The graph of f is made of closed line segments joined end to end. ii) The graph starts at the point (-5, 1). iii) The derivative of f is the step function in the figure shown here.

Answer:

Provide an appropriate response. 11) Find all points (x, y) on the graph of y =

A) (0, 0), (14, 2)

x with tangent lines perpendicular to the line y = 7x - 2. (x - 7)

B) (0, 0), (7, 2)

C) (0, 0)

Answer: A

4

D) (14, 2)

Find the second derivative. 23x3 -8 12) y = 6

A) 23x - 8

B) 23x

C)

23 2 x 2

Answer: B The graph of a function is given. Choose the answer that represents the graph of its derivative. 13)

A)

B)

C)

D)

Answer: D

5

D)

23 x 6

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative.

14) u(1) = 4, u (1) = -6, v(1) = 7, v (1) = -3. d (2u - 4v) at x = 1 dx

A) -20

B) 36

C) 0

D) -24

Answer: C Solve the problem. 15) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the watermelon's average speed during the first 4 sec of fall and the speed at the instant t = 4 sec. A) 32 ft/sec; 64 ft/sec

B) 65 ft/sec; 130 ft/sec

C) 64 ft/sec; 128 ft/sec

D) 128 ft/sec; 65 ft/sec

Answer: C Find y . 16) y =

2 2 +x -x x x

A)

8 + 2x x3

B) -

4 - 2x x3

C) -

8 - 2x x3

D) -

8 + 2x x

Answer: C Find the derivative of the function. x2 + 8x + 3 17) y = x

A) y =

2x + 8 2x 3/2

B) y =

3x2 + 8x - 3 x

C) y =

Answer: C

6

3x2 + 8x - 3 2x3/2

D) y =

2x + 8 x

Solve the problem. 18) Under standard conditions, molecules of a gas collide billions of times per second. If each molecule has diameter t, the average distance between collisions is given by 1 L= , 2 t2 n where n, the volume density of the gas, is a constant. Find

A)

dL =dt

2

2 t3 n

B)

dL 1 = dt 2 2 t3 n

dL . dt

C)

dL =dt

1 2 tn

D)

dL = dt

Answer: A Find the derivative. 19) s = t7 tan t -

t

A)

ds 1 = 7t6 sec2 t dt 2 t

B)

ds 1 = - t7 sec2 t + 7t6 tan t + dt 2 t

C)

ds 1 = t7 sec t tan t + 7t6 tan t dt 2 t

D)

ds 1 = t7 sec2 t + 7t6 tan t dt 2 t

Answer: D 20) y = (csc x + cot x)(csc x - cot x) A) y = - csc x cot x

B) y = - csc2 x

C) y = 1

Answer: D 21) y =

2 1 + sin x cot x

A) y = - 2 csc x cot x + sec2 x

B) y = 2 csc x cot x - sec2 x

C) y = 2 cos x - csc2 x

D) y = 2 csc x cot x - csc2 x

Answer: A Find the indicated derivative. 22) Find y

if y = 6x sin x.

A) y

= 6 cos x - 12x sin x

B) y

= - 6x sin x

C) y

= - 12 cos x + 6x sin x

D) y

= 12 cos x - 6x sin x

Answer: D

7

D) y = 0

2

2 t3n

Solve the problem. 23) Find the tangent to y = 2 - sin x at x = . A) y = - x +

-2

B) y = x - 2

C) y = x -

+2

D) y = - x + 2

Answer: C 24) Does the graph of the function y = tan x - x have any horizontal tangents in the interval 0 A) No

B) Yes, at x =

C) Yes, at x = 0, x = , x = 2

D) Yes, at x =

2

,x=

x

2 ? If so, where?

3 2

Answer: C The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 25) s = 5t2 + 3t + 7, 0 t 2 Find the body's speed and acceleration at the end of the time interval. A) 23 m/sec, 20 m/sec2

B) 23 m/sec, 10 m/sec2

C) 30 m/sec, 10 m/sec2

D) 13 m/sec, 2 m/sec2

Answer: B Solve the problem. 26) Suppose that the dollar cost of producing x radios is c(x) = 400 + 20x - 0.2x2 . Find the marginal cost when 40 radios are produced. A) $4

B) -$880

C) $36

D) $880

Answer: A Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 27) 3 1 9 6 7 4 3 3 2 -6 f2(x) · g(x), x = 3

A) 84

B) 25

C) 61

Answer: D

8

D) 115

Find the derivative of the function. 28) q =

20r - r7

A)

1 2 20r - r7

B)

20 - 7r6

-7r6

C)

2 20r - r7

20r - r7

D)

Answer: B 29) s = sin

A) C)

7 t 7 t - cos 2 2 7 7 t 7 7 t cos sin 2 2 2 2

B)

7 7 t 7 7 t cos + sin 2 2 2 2

7 7 t 7 7 t cos sin 2 2 2 2

D) cos

7 t 7 t + sin 2 2

Answer: C cos x 5 1 + sin x

30) h(x) =

A) 5 C)

cos x 4 1 + sin x

-5 cos4 x (1 + sin x)5

B) -5

sin x 4 cos x

D) -

4 sin x cos x 4 cos x 1 + sin x

Answer: C Find dy/dt. 31) y = cos7 ( t - 16) A) -7 cos6( t - 16) sin( t - 16)

B) -7 cos6 ( t - 16) sin( t - 16)

C) 7 cos6( t - 16)

D) -7 sin6 ( t - 16)

Answer: A Solve the problem. 32) Find dy/dx given y = (a) x2 sin x3 (b) sin7 x cos x (c) cos4 x 4 (d) sin (x 3 + x 2 + x + 1) Answer: (a) 3x 4 cos x3 + 2x sin x3 (b) 7 cos2 x sin6 x - sin8 x (c) -16x 3 cos3 x 4 sin x 4 (d) (3x 2 + 2x + 1) cos (x 3 + x 2 + x + 1)

9

1 2 20 - 7r6

33) Find the exact coordinates of the inflection points and critical points marked on the graph of 2 f(x) = x3 - 2x2 -6x. 3

Answer: maximum at (-1,

10 22 ); minimum at (3, -18); inflection point at (1, ) 3 3

34) Find the exact coordinates of the inflection points and critical points marked on the graph of f(x) = 4x5 - 100x3 -5.

Answer: maximum at (- 15, 2318.79); minimum at ( 15, 30 30 ( , -1442.77), (, 1432.77) 2 2

10

-2328.79); inflection points at (0, -5),

35) Match the graph of the function (1 - 5) with the graph of its second derivative (A-E). (1) (A)

(2)

(B)

(3)

(C)

(4)

(D)

(5)

(E)

Answer: 1 C, 2 A, 3 B, 4 E, 5 D

11