Unit 5 Limits and Derivatives Test Review

[Pages:5]Unit 5 Limits and Derivatives Test Review

(x + h)2 x2

1. What is lim

h!0

h

?

A. 2xh B. 2x

C. h

D. 0

5. At which of the ve points shown on the graph

dy

d2y

are dx and dx2 both negative?

2. The functions f and g are di erentiable and have the values shown in the table.

If A = f g, then A0(4) =

A. 0 B. 114 C. 83 D. 29

x f f0 05 1 28 3 4 14 9 6 26 27

g g0

7

1 4

51

34

1 16

3. The functions f and g are di erentiable and have the values shown in the table.

If A = f (g(x)), then A0( 8) =

A. 72 B. 18 C. 54 D. 9

x f f 0 g g0 84 3 2 6 6 10 12 0 9 2 20 9 6 18 2 30 15 12 24

4. The table shows the position of an object moving along a line at 10 second intervals.

Estimate the velocity, in units/sec, at t = 35.

t (sec)

0

position 4

10 20 30 40 12 26 44 68

A. 2.400 B. 2.400

C. 11.200 D. 3.842

A. A

B. C

C. D

D. E

6.

Find all open intervals on which f (x) = x2

x 5x + 4

is increasing.

A. ( 1; 1) B. ( 2; 2)

C. ( 1; 2) D. (2; 1)

7.

Let

f

(x)

=

x

3 2

on

[0; 4]

Which of the hypotheses of Rolle's Theorem are not satis ed on the given interval?

I. f is di erentiable in (a; b)

II. f is continuous at x = a and x = b

III. f (a) = f (b) = 0 IV. there is no c in (a; b) for which f 0(c) = 0

A. III and IV B. IV only

C. I and IV D. III only

page 1

8. Given the function f (x) = x(x2 8) 5 satis es the hypothesis of the Mean Value Theorem on the interval [1; 4], nd a number C in the interval (1; 4) which satis es this theorem.

p A. 7

B. 12

C. 7

D. 5

14. For what x coordinate(s) does the function de ned by f (x) = 3x5 5x3 8 have a relative maximum?

A. 1 only B. 0 and 1

C. 1 only D. 1 and 1

9. If y = 12px, what is the tangent line used to nd approximate values of y for x near 9?

A. y = 36 2(x 9) B. y = 36 + 2(x 9)

C. y = 36 2(x 9) D. y = 2 36(x 9)

15. Find the derivative of x2f (x).

A. x xf 0(x) + 2f (x) B. 2xf 0(x)

C. x xf (x) + 2f 0(x)

D.

1 3

x3

+

f 0(x) 2

10. If f (x) = x3 3x2 x + 7, determine its point of in ection.

A. (1; 4) B. (2; 1)

C. (3; 4) D. ( 1; 4)

11. Given that f 0(x) = x sin x for 0 < x < 4 , then f has a local maximum for x

A. 2.029 B. 3.141 C. 1.820 D. 1.571

1 cos 2x

12. lim

x!0

x2

is

A. 2

B.

3 2

C. 4

D. 0

16. If f (x) p= (2x + 3)5, then the fth derivative of f (x) at x = 2 is equal to

A. 3840

p

p

p

B. 120 2 C. 240 2 D. 25 2

2

17. Suppose f (x) = x5 and let h(x) be the inverse of f . Find h0(32).

A.

1 80

B. 80

C.

1 32

D.

1 32

18. Di erentiate: f (x) = arcsin(7x)

7 A.

p 1 7x2 7

B. p 1 49x2

7 C.

p 1 49x2 7

D. p 1 x2

13.

A

curve

is

de

ned by

y = esin 2x.

Find

dy dx .

A. esin 2x cos 2x B. 2esin 2x cos 2x

C. sin 2xecos 2x D. 4 sin 4x

19. Find f 0(x) given f (x) = cos4(3x).

A. 4 cos3(3x) B. 12 sin 3x cos3(3x)

C. 4 cos3(3x) D. 12 cos2(3x)

page 2

Unit 5 Limits and Derivatives Test Review

20.

Find

dy dx

given

y2

3xy + x2 = 7.

A.

3y 2y

2x 3x

2x B. 3 2y

C.

2x y

2y 3x D. 3y 2x

24. In a right triangle the hypotenuse is of xed length of 15 units, one side is increasing in length by 4 units per second while the third side is decreasing in size. At a certain instant the increasing side is of length 9 units. Find the rate of change of the third side at the same instant.

A. 3 units/sec

B.

3 2

units/sec

C.

5 4

units/sec

D.

3 4

units/sec

21. What is the slope of the tangent line to

xy + ln 2x =

1 2

at

the

point

(

1 2

;

1)?

A. 6

B.

2 3

C.

1 6

D.

1 6

22. An open box can be made from a square piece of material, by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box of maximum volume if the material has dimension 6 cm by 6 cm.

A. 4 cm 4 cm 1 cm

B. 3 cm 3 cm 3 cm

C. 3:5 cm 3:5 cm 1:25 cm

D. 5 cm 5 cm 0:5 cm

23. A balloon rises vertically at the rate of 10 ft/s. A person on the ground 100 ft away from the spot below the rising balloon watches the balloon ascend; at what rate is the distance between balloon and observer changing when the balloon is 100 ft above ground?

p A. 5 2 ft=s

p B. 2 ft=s

2

p C. 11 2 ft=s

2 p D. 2 2 ft=s

25. Let f be de ned as follows:

( x2 f (x) = x

9 3

for x 6= 3,

1

for x = 3

Which of the following are true about f ?

I. lim f (x) exists

x!3

II. f (3) exists

III. f (x) is continuous at x = 3

A. None B. II only

C. I and II only D. I, II, and III

8x3 1

26.

lim

x!

1 2

10x2

= 7x + 1

A. 2

B.

8 3

C.

1 2

D. 0

sin 9x

27.

lim

x!0

sin 5x

=

A.

9 5

B. 1

C. 0

D.

5 9

page 3

Unit 5 Limits and Derivatives Test Review

jxj

28.

lim

x!0

x

is

A. 1 B. 1

C. 0

D. 1

e5x

30.

lim

x!1

ln

2x

is

A. 1

B. 0

C. 5e

D. 1

5 29. lim is

x!0+ x2

A. 1

B. 1 C. e

D. 1

page 4

Unit 5 Limits and Derivatives Test Review

1.

Answer:

B

2.

Answer:

D

3.

Answer:

C

4.

Answer:

A

5.

Answer:

A

6.

Answer:

B

7.

Answer:

D

8.

Answer:

A

9.

Answer:

B

10.

Answer:

A

11.

Answer:

B

12.

Answer:

A

13.

Answer:

B

14.

Answer:

A

15.

Answer:

A

16.

Answer:

A

17.

Answer:

A

18.

Answer:

B

19.

Answer:

B

20.

Answer:

A

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Unit 5 Limits and Derivatives Test Review

21. Answer:

22. Answer:

23. Answer:

24. Answer:

25. Answer:

26. Answer:

27. Answer:

28. Answer:

29. Answer:

30. Answer:

4/18/2018 A A A A C A A B B D

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