Test bank for calculus 2nd edition by briggs

Choose the one alternative that best completes the statement or answers the question... A fa exists, the limit of fx as x→a from the left exists, and the limit of fx as x→a from theright

MULTIPLE CHOICE Choose the one alternative that best completes the statement or answers the question Find the average velocity of the function over the given interval.

6)

7) h(t) = sin (3t), 0, π

6A) 6

8)

Use the table to find the instantaneous velocity of y at the specified value of x.

9) x = 1

0 0.20.40.60.81.01.21.4

00.020.080.180.320.50.720.98

00.010.040.090.160.250.360.49

00.120.481.081.9234.325.88

11)

12) x = 2.

x y00.51.01.52.02.53.03.54.0

103858707470583810

-0.05263-0.00503-0.00050.00000.00050.004980.04762

t 1.9 1.99 1.999 2.001 2.01 2.1s(t) 16.810 17.880 17.988 18.012 18.120 19.210 ; instantaneous velocity is 18.0B)

t 1.9 1.99 1.999 2.001 2.01 2.1s(t) 16.692 17.592 17.689 17.710 17.808 18.789 ; instantaneous velocity is 17.70C)

t 1.9 1.99 1.999 2.001 2.01 2.1s(t) 5.043 5.364 5.396 5.404 5.436 5.763 ; instantaneous velocity is ∞D)

t 1.9 1.99 1.999 2.001 2.01 2.1s(t) 5.043 5.364 5.396 5.404 5.436 5.763 ; instantaneous velocity is 5.40

14)

t -0.1 -0.01 -0.001 0.001 0.01 0.1s(t) -4.9900 -4.9999 -5.0000 -5.0000 -4.9999 -4.9900 ; instantaneous velocity is-5.0

C)

t -0.1 -0.01 -0.001 0.001 0.01 0.1s(t) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; instantaneous velocity is ∞D)

t -0.1 -0.01 -0.001 0.001 0.01 0.1s(t) -2.9910 -2.9999 -3.0000 -3.0000 -2.9999 -2.9910 ; instantaneous velocity is-3.0

II limx→0f(x) = LrIII lim f(x) does not exist

21)

22) Given lim

x→0-f(x) = Ll, lim

x→0+f(x) = Lr , and Ll = Lr, which of the following statements is false?

I limx→0f(x) = Ll

II limx→0f(x) = LrIII lim

x→0f(x) does not exist.

II limx→0+f(x) does not exist.

III limx→0-f(x) = L

IV limx→0+f(x) = LA) I and II only B) III and IV only C) II and III only D) I and IV only

23)

24) What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x

approaches some value of a?

A) f(a) exists, the limit of f(x) as x→a from the left exists, and the limit of f(x) as x→a from theright exists

B) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and

at least one of these limits is the same as f(a)

C) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, andthese two limits are the same

D) Either the limit of f(x) as x→a from the left exists or the limit of f(x) as x→a from the rightexists

24)

Use the graph to evaluate the limit.

25) lim

x→-1f(x)

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

1

-1

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

26)

27) lim

x→0f(x)

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

-2 -4

x

y 12 10 8 6 4 2

-2 -4

28)

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

30)

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

32)

-1 -2 -3 -4

x

y 4 3 2 1

-1 -2 -3 -4

Use the table of values of f to estimate the limit.

35) Let f(x) = x2 + 8x - 2, find lim

x→2f(x).

f(x)A)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.692 17.592 17.689 17.710 17.808 18.789 ; limit = 17.70B)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = ∞C)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.810 17.880 17.988 18.012 18.120 19.210 ; limit = 18.0D)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = 5.40

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = 1.20B)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = ∞C)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485 ; limit = 4.0D)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236 ; limit = 5.10

36)

37) Let f(x) = x2 - 5, find lim

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -4.9900 -4.9999 -5.0000 -5.0000 -4.9999 -4.9900 ; limit = -5.0C)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = ∞D)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = -15.0

f(x) 0.5263 0.5025 0.5003 0.4998 0.4975 0.4762 ; limit = 0.5B)

f(x) -0.5263 -0.5025 -0.5003 -0.4998 -0.4975 -0.4762 ; limit = -0.5C)

f(x) 0.4263 0.4025 0.4003 0.3998 0.3975 0.3762 ; limit = 0.4D)

f(x) 0.6263 0.6025 0.6003 0.5998 0.5975 0.5762 ; limit = 0.6

38)

39) Let f(x) = x2 - 3x + 2

x2 + 3x - 10, find limx→2f(x).

f(x)A)

f(x) 0.0304 0.0416 0.0427 0.0430 0.0441 0.0549 ; limit = 0.0429B)

f(x) -1.0690 -1.0067 -1.0007 -0.9993 -0.9934 -0.9355 ; limit = -1C)

f(x) 0.2304 0.2416 0.2427 0.2430 0.2441 0.2549 ; limit = 0.2429D)

2 - 2 cos(x) < 1 hold for all values of x close

to zero What, if anything, does this tell you about x sin(x)

2 - 2 cos(x) ? Explain.

42)

MULTIPLE CHOICE Choose the one alternative that best completes the statement or answers the question.

43) Write the formal notation for the principle "the limit of a quotient is the quotient of the limits" andinclude a statement of any restrictions on the principle

A) If limx→a g(x) = M and limx→a f(x) = L, then limx→a

g(x)f(x) =

limx→a g(x)limx→a f(x)

= M

L, provided that

L ≠ 0

B) limx→a

g(x)f(x) = g(a)f(a), provided that f(a) ≠ 0

C) limx→a

g(x)f(x) = g(a)f(a).

D) If limx→a g(x) = M and limx→a f(x) = L, then limx→a

g(x)f(x) =

limx→a g(x)limx→a f(x)

= M

L, provided thatf(a) ≠ 0

B) The limit of a sum or a difference is the sum or the difference of the functions

C) The sum or the difference of two functions is continuous

D) The limit of a sum or a difference is the sum or the difference of the limits

44)

45) The statement "the limit of a constant times a function is the constant times the limit" follows from

a combination of two fundamental limit principles What are they?

A) The limit of a product is the product of the limits, and a constant is continuous

B) The limit of a product is the product of the limits, and the limit of a quotient is the quotient ofthe limits

C) The limit of a function is a constant times a limit, and the limit of a constant is the constant

D) The limit of a constant is the constant, and the limit of a product is the product of the limits

Give an appropriate answer.

58)

65) lim

x→0

1 + x - 1x

78)

79) lim

h → 0

(x + h)3 - x3h

Provide an appropriate response.

81) It can be shown that the inequalities -x ≤ x cos 1

x ≤ x hold for all values of x ≥ 0

Find limx→0x cos

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.810 17.880 17.988 18.012 18.120 19.210 ; limit = 18.0B)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.692 17.592 17.689 17.710 17.808 18.789 ; limit = 17.70C)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = 5.40D)

x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763 ; limit = ∞

84)

85) If f(x) = x4 - 1

x - 1 , find limx → 1 f(x)

f(x)A)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 3.439 3.940 3.994 4.006 4.060 4.641 ; limit = 4.0B)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit = ∞C)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 4.595 5.046 5.095 5.105 5.154 5.677 ; limit = 5.10D)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit = 1.210

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858 ; limit = -1.20C)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -4.09476 -4.00995 -4.00100 -3.99900 -3.98995 -3.89526 ; limit = -4.0D)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574 ; limit = -2.10

86)

87) If f(x) = x - 4

x - 2, find limx → 4 f(x).

f(x)A)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = 1.20B)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485 ; limit = 4.0C)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745 ; limit = ∞D)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236 ; limit = 5.10

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = -15.0C)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -1.4970 -1.4999 -1.5000 -1.5000 -1.4999 -1.4970 ; limit = ∞D)

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) -4.9900 -4.9999 -5.0000 -5.0000 -4.9999 -4.9900 ; limit = -5.0

88)

89) If f(x) = x + 1

x + 1 , find limx → 1 f(x).

f(x)A)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 2.15293 2.13799 2.13656 2.13624 2.13481 2.12106 ; limit = 2.13640B)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 0.72548 0.70888 0.70728 0.70693 0.70535 0.69007 ; limit = 0.7071C)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 0.21764 0.21266 0.21219 0.21208 0.21160 0.20702 ; limit = ∞D)

x 0.9 0.99 0.999 1.001 1.01 1.1f(x) 0.21764 0.21266 0.21219 0.21208 0.21160 0.20702 ; limit = 0.21213

f(x) -0.02516 -0.00250 -0.00025 0.00025 0.00250 0.02485 ; limit = 0B)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit = 1.95C)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 3.9000 2.9000 1.9000 2.0000 3.0000 4.0000 ; limit = ∞D)

x 3.9 3.99 3.999 4.001 4.01 4.1f(x) 1.47736 1.49775 1.49977 1.50022 1.50225 1.52236 ; limit = 1.50

90)

For the function f whose graph is given, determine the limit.

91) Find lim

x→5-f(x) and limx→5+f(x).

x -2 -1 1 2 3 4 5 6 7 8 9 10 11

y 8 6 4 2

-2 -4 -6 -8

x -2 -1 1 2 3 4 5 6 7 8 9 10 11

y 8 6 4 2

-2 -4 -6 -8

y 5 4 3 2 1

-1 -2 -3 -4 -5

x -5 -4 -3 -2 -1 1 2 3 4 5

y 5 4 3 2 1

-1 -2 -3 -4 -5

92)

93) Find lim

x→3f(x).

x -5 -4 -3 -2 -1 1 2 3 4 5

y 5 4 3 2 1

-1 -2 -3 -4 -5

x -5 -4 -3 -2 -1 1 2 3 4 5

y 5 4 3 2 1

-1 -2 -3 -4 -5

93)

94) Find lim

x→-3f(x).

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

Find the limit.

Find all vertical asymptotes of the given function.

113)

114) f(x) = -2x(x + 2)

2x2 - 5x - 7A) x = - 2

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

117)

118) f(x) = x

x2 + x + 2A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

118)

119) f(x) = x2 - 3

x3A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

y 4

2

-2

-4

x -10 -8 -6 -4 -2 2 4 6 8 10

y 4

2

-2

-4

119)

120) f(x) = 1

x + 1A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

B)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

D)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

120)

121) f(x) = x - 1

x + 1A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

B)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

D)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

121)

122) f(x) = 1

(x + 2)2A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

B)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

D)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

122)

123) f(x) = 2x2

4 - x2A)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

B)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10C)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

D)

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

x -10 -8 -6 -4 -2 2 4 6 8 10

y 10 8 6 4 2

-2 -4 -6 -8 -10

135)

Divide numerator and denominator by the highest power of x in the denominator to find the limit.

148) h(x) = 9x4 - 5x2 - 6

5x5 - 8x + 4A) y = 5

x y

156)

157) f(0) = 5, f(1) = -5, f(-1) = -5, lim

x→±∞ f(x) = 0.

x y

d 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 (1, -5)

t -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

d 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 (1, -5)

175)

176) Is f continuous at f(0)?

f(x) =

-x2 + 1,2x,-5,-2x + 4 1,

d 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 (1, -5)

t -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

d 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 (1, -5)

d 10 8 6 4 2

-2 -4 -6 -8 -10

(2, 0)

t -5 -4 -3 -2 -1 1 2 3 4 5

d 10 8 6 4 2

-2 -4 -6 -8 -10 (2, 0)

d 10 8 6 4 2

-2 -4 -6 -8 -10

(2, 0)

t -5 -4 -3 -2 -1 1 2 3 4 5

d 10 8 6 4 2

-2 -4 -6 -8 -10 (2, 0)

178)

179) Is the function given by f(x) = x + 1

x2 - 8x + 12 continuous at x = 2? Why or why not?

A) Yes, lim

x→ 2f(x) = f(2)B) No, f(2) does not exist and lim

x→ 2 f(x) does not exist

179)

180) Is the function given by f(x) = 10x + 9 continuous at x = - 109 ? Why or why not?

A) No, lim

x→ - 910

f(x) does not exist B) Yes, lim

x→- 910

f(x) = f - 9

10

180)

181) Is the function given by f(x) = x2 - 4, for x < 0

-2, for x ≥ 0 continuous at x = -2? Why or why not?

continuous at x = 3? Why or why not?

183)

184) y = 3

(x + 4)2 + 8A) discontinuous only when x = 24 B) continuous everywhereC) discontinuous only when x = -4 D) discontinuous only when x = -32

184)

185) y = x + 3

x2 - 7x + 12A) discontinuous only when x = 3 B) discontinuous only when x = 3 or x = 4C) discontinuous only when x = -3 or x = 4 D) discontinuous only when x = -4 or x = 3

185)

186) y = 2

x2 - 16A) discontinuous only when x = -4B) discontinuous only when x = 16C) discontinuous only when x = -4 or x = 4

186)

187) y = 2

x + 1 -

x22A) discontinuous only when x = -1 B) continuous everywhereC) discontinuous only when x = -3 D) discontinuous only when x = -2 or x = -1

187)

188) y = sin (3θ)

2θ

C) discontinuous only when θ = π

188)

189) y = 2 cos θ

θ + 9A) discontinuous only when θ = π

193)

194) lim

x→∞

11x - 1x3

194)