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MATH 1910 Exam 2 Review Problems Be sure to review the notes and homework as well when preparing for the exam.

Find the slope of the line tangent tangent to the graph a the given point 1) y  x2 + 5x + 3, x  1 A) m  2

B) m  7

C) m  10

D) m  6

Find an equation for the tangent to the curve at the given point 2) f(x)  6 x  x  1, (36, 1) A) y  1

B) y  

1 x  19 2

C) y 

1 x  19 2

D) y  

1 x1 2

5

x

Graph the equation and its tangent. 3) Graph y  3 x2 and the tangent to the curve at the point whose xcoordinate is 1. y 10

-5

5

x

-10

A)

B) y

y

10

10

-5

5

x

-5

-10

-10

1

C)

D) y

y 10

10

-5

5

x

-5

5

x

-10

-10

Calculate the derivative of the function. Then find the value of the derivative as specified 2 4) g(x)   ; g  ( 2) x A) g  (x) 

2 x2

C) g  (x)  

; g  ( 2) 

1 2

B) g  (x)   2; g ( 2)   2

2 1 ; g  ( 2)   2 2 x

D) g  (x)   2x2 ; g  ( 2)   8

Find the derivative. 1 5) y  x5 6 A)

5 4 x 6

B)

1 5 x 30

5 6 x 6

C) 30x4

D)

C) 2t + 6

D) 4t2 + 6

6) s  2t2 + 6t + 8 A) 4t + 6

B) 2t2 + 6

Find the derivative of the function by first expanding or simplifying the expression 5x 2 + 30x 7) f(x)  5x A) f'(x)  5 8) f(x)  (6x + 5)2 A) f'(x)  12x + 10

B) f'(x)  10x

C) f'(x)  1

D) f'(x)  x  6

B) f'(x)  72x + 60

C) f'(x)  36x + 30

D) f'(x)  36x + 25

C) y  50x - 252

D) y  -2

Find an equation of the tangent line at x  a. 9) y  x3  25x - 2; a  5 A) y  48x - 252 B) y  50x - 2

Provide an appropriate response. 10) The curve y  ax2  bx  c passes through the point (2, 28) and is tangent to the line y = 2x at the origin. Find a, b, and c. A) a  2, b  0, c  6 B) a  7 , b  0, c  0 C) a  6, b  2, c  0 D) a  0, b  6, c  2

2

Find the second derivative. 11) y  3x 3 - 8x2 + 2ex A) 16x - 12 + 2ex

B) 16x - 18 + 2ex

C) 12x - 16 + 2ex

D) 18x - 16 + 2ex

Solve the problem. 12) 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) 65 ft/sec; 130 ft/sec B) 64 ft/sec; 128 ft/sec C) 128 ft/sec; 65 ft/sec D) 32 ft/sec; 64 ft/sec Find y  . 13) y  (4x - 2)(3x3  x2  1) A) 48x 3 - 10x2 + 30x + 4

B) 36x3 + 30x 2 - 10x + 4 D) 12x3 + 10x 2 - 30x + 4

C) 48x 3 - 30x2 + 4x + 4 Find the derivative of the function. x2+ 5 14) g(x)  x2 + 6x A) g  (x) 

2x3 - 5x2 - 30x x2 (x + 6)2

B) g  (x) 

x4 + 6x3 + 5x 2 + 30x x2 (x + 6)2

C) g  (x) 

4x3 + 18x2 + 10x + 30 x2 (x + 6)2

D) g  (x) 

6x2 - 10x - 30 x2 (x + 6)2

15) y 

x 2 + 8x + 3 x

A) y  

2x + 8 2x3/2

B) y  

3x2 + 8x - 3 x

C) y  

3x2 + 8x - 3 2x3/2

D) y  

2x + 8 x

Provide an appropriate response. 16) Find an equation for the tangent to the curve y  A) y  5x

10x x2 + 1

at the point (1, 5).

B) y  5

C) y  0

D) y  x  5

B) 14xe-x(1  x)

C) 7xex(2  x)

D) 7xe -x(2  x)

Find the derivative. 17) y  7x2 e-x A) 7xe -x(x  2) Solve the problem. 18) The size of a population of mice after t months is P  100(1  0.2t  0.02t2). Find the growth rate at t = 16 months. A) 184 mice/month B) 168 mice/month C) 84 mice/month D) 42 mice/month

3

Find the second derivative of the function x4 + 6 19) y  x2 A)

36 d2 y 1 dx2 x4

B)

36 d 2y 2 dx2 x4

C)

12 d2 y  2x  dx2 x3

D)

36 d2 y 2 dx2 x4

Find the derivative. 5 20) y   10 sec x x A) y  

5 x2

C) y   

21) y 

 10 sec x tan x

5 x2

B) y   

 10 tan2 x

D) y   

5 x2

 10 csc x  10 sec x tan x

8 1  sin x cot x

A) y   8 csc x cot x  sec2 x C) y   8 cos x  csc2 x

22) p 

5 x2

B) y   8 csc x cot x  csc2 x D) y    8 csc x cot x  sec2 x

6 + sec q 6 - sec q

A)

dp 12 sin q  dq (6 cos q - 1)2

B)

dp 12 sin q  dq (6 cos q - 1)2

C)

dp 12 tan2 q  dq (6 - sec q)2

D)

dp 2 sec2 q tan q  dq (6 - sec q)2

Solve the problem. 23) Does the graph of the function y  tan x  x have any horizontal tangents in the interval0 ≤ x ≤ 2π? If so, where? A) No

B) Yes, at x  π

C) Yes, at x  0, x  π, x  2π

D) Yes, at x 

π 3π ,x 2 2

24) Does the graph of the function y  5x  10 sin x have any horizontal tangents in the interval0 ≤ x ≤ 2π? If so, where? 2π 4π ,x A) No B) Yes, at x  3 3 C) Yes, at x 

π 2π ,x 3 3

D) Yes, at x 

4

2π 3

The equation gives the position s f(t) of a body moving on a coordinate line (s in meters, t in seconds) 25) s (m) 5 4 3 2 1

-1

t (sec) 1

2

3

4

5

6

7

8

9 10

-2 -3 -4 -5

When is the body moving forward? A) 0  t  8 C) 0  t  3, 3  t  5, 5  t  8

B) 0  t  1, 3  t  4, 5  t  7, 9  t  10 D) 0  t  1, 3  t  4, 5  t  7

The function s  f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds 26) s  6t2  4t  5, 0  t  2 Find the body's speed and acceleration at the end of the time interval. A) 28 m/sec, 12 m/sec2 B) 28 m/sec, 24 m/sec2 C) 33 m/sec, 12 m/sec2

D) 16 m/sec, 2 m/sec2

Solve the problem. 27) The position of a body moving on a coordinate line is given by s = t2 - 5t + 5, with s in meters and t in seconds. When, if ever, during the interval 0 ≤ t ≤ 5 does the body change direction? A) t  5 sec B) t  10 sec C) no change in direction D) t  2.5 sec 28) At time t  0, the velocity of a body moving along the saxis is v = t2 - 7t + 6. When is the body moving backward? A) t  6 B) 0  t  1 C) 0  t  6 D) 1  t  6 29) A ball dropped from the top of a building has a height of s  256  16t2 meters after t seconds. How long does it take the ball to reach the ground? What is the ball's velocity at the moment of impact? A) 4 sec, 128 m/sec B) 8 sec,  64 m/sec C) 4 sec, 128 m/sec D) 16 sec,  512 m/sec 30) A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of s = 120t - 6t2 meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point? A) 1200 m, 20 sec B) 600 m, 10 sec C) 2280 m, 20 sec D) 1190 m, 10 sec

5

Write the function in the form y  f(u) and u  g(x). Then find dy/dx as a function of x 31) y  (-3x + 8)5 A) y  5u  8; u  x5 ;

dy  -15x4 dx

C) y  u5 ; u  -3x  8;

dy  -15(-3x + 8)4 dx

B) y  u5 ; u  -3x  8;

dy  5(-3x + 8)4 dx

D) y  u5 ; u  -3x  8;

dy  -3(-3x + 8)5 dx

32) y  cos4 x A) y  cos u; u  x4 ;

dy   4x3 sin(x4 ) dx

B) y  u4 ; u  cos x;

dy  4 cos3 x sin x dx

C) y  u4 ; u  cos x;

dy   4 cos3 x sin x dx

D) y  cos u; u  x4 ;

dy   sin(x4 ) dx

33) y  csc(cot x) A) y  csc u; u  cot x;

dy  csc3 x cot x dx

B) y  cot u; u  csc x;

dy  csc2 (csc x) csc x cot x dx

C) y  csc u; u  cot x;

dy   csc(cot x) cot(cot x) dx

D) y  csc u; u  cot x;

dy  csc(cot x) cot(cot x) csc2 x dx

34) y  e9 x/2 A) y  eu; u  9x/2;

dy 9 9x/2  e dx 2

B) y  eu; u  9x/2;

dy 2 9x/2  xe C) y  eu; u  9x/2; dx 9

dy 9 9x/2  xe dx 2

dy 2 9x/2  e D) y  eu; u  9x/2; dx 9

Find the derivative of the function. 35) q  12r - r7 A)

12 - 7r6 2 12r - r7

B)

-7r 6

C)

12r - r7

36) y  x4 cos x  9x sin x  9 cos x A) x4 sin x  4x3 cos x  9x cos x

2 12r - r7

cos x 4 1 + sin x

37) h(x) 

-4 cos3 x

B) 4

(1 + sin x)4

C) -

D)

1 2 12 - 7r6

B) 4x3 sin x  9 cos x  9 sin x D) x4 sin x  4x3 cos x  9x cos x  18 sin x

C) x4 sin x  4x 3 cos x  9x cos x

A)

1

4 sin x cos x 3 cos x 1 + sin x

D) 4

6

sin x 3 cos x

cos x 3 1 + sin x

Find dy/dt. 38) y  cos( 4t + 12) 1 A)  sin( 4t + 12) 2 4t + 12

2

B)

C) sin( 4t + 12)

4t + 12

D) sin

sin( 4t + 12)

2 4t + 12

Solve the problem. 39) The position of a particle moving along a coordinate line is s  5 + 4t with s in meters and t in seconds. Find the particle's acceleration at t = 1 sec. 1 2 4 4 A)  B) m/sec2 C)  D) m/sec2 m/sec2 m/sec2 27 3 27 27 Use implicit differentiation to find dy/dx. 40) 2xy  y2  1 A)

y x-y

B)

41) x 3  3x2y  y3  8 x2 + 3xy A) x 2 + y2

x y-x

B) 

C)

x2 + 3xy x 2 + y2

y y-x

C) 

x2 + 2xy x 2 + y2

D)

x x-y

D)

x2 + 2xy x 2 + y2

42) x  sec(2 y) A) 2 sec(2 y) tan(2 y)

B)

1 sec(2y) tan(2y) 2

C)

1 cos(2y) cot(2y) 2

D) cos(2y) cot(2y)

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested 43) y 4  x 3  y2  11x, slope at (0, 1) 7 11 11 11 A) B) C) D) 2 4 2 6 Solve the problem. 44) The graph of y  f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f. y

y (3, 5)

(6, 5)

(-3, 2)

(-5, 0)

(0, -1)

x

x

7

A)

B) y

-6

-4

y

6

6

4

4

2

2

-2

2

4

6

x

-6

-4

-2

-2

-2

-4

-4

-6

-6

C)

2

4

6

x

2

4

6

x

D) y

y

-6

-4

6

6

4

4

2

2

-2

2

4

6

x

-6

-4

-2

-2

-2

-4

-4

-6

-6

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? 45) x  1 y 4

2

-4

-2

2

4

x

-2

-4

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable 8

46) x  1 y 4

2

-4

-2

2

4

x

-2

-4

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

9

Answer Key Testname: EXAM 2 REVIEW

1) B 2) B 3) D 4) A 5) A 6) A 7) C 8) B 9) C 10) C 11) D 12) B 13) C 14) D 15) C 16) B 17) D 18) C 19) B 20) D 21) D 22) A 23) C 24) B 25) B 26) A 27) D 28) D 29) A 30) B 31) C 32) C 33) D 34) A 35) A 36) A 37) A 38) B 39) C 40) C 41) C 42) C 43) C 44) B 45) B 46) C

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