MATH 113 EXAM 1 Review PDF

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MATH 113 EXAM 1 Review...

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Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. 3 1) y = , [4, 7] x- 2 A) 7

1 3

B)

C) - 3 10

1) D) 2

Use the graph to evaluate the limit. 2) lim f(x) x→0

2)

y 4 3 2 1

-4

-3

-2

-1

1

2

4 x

3

-1 -2 -3 -4

A) 0

B) -2

C) 2

D) does not exist

3) lim f(x) x→0

3) y 4 3 2 1

-4

-3

-2

-1

1

2

3

4

x

-1 -2 -3 -4

A) 0

B) -2

C) 1

1

D) does not exist

Find the limit. 4)

lim (3x5 - 2x4 - 4x3 + x2 - 5 ) x→-2 A) -97

4)

B) -33

C) 47

D) -161

x2 + 2x + 1

5) lim x→2

A) does not exist

5) B) 9

C) ±3

D) 3

Find the limit if it exists. 6)

lim (x + 248 ) 3 /5 x→-5 A) 81

6) B) -27

C) 27

D) 9

Find the limit, if it exists. x+6 7) lim x→6 (x - 6) 2 A) 0

B) Does not exist

A) 1/4

C) Does not exist

D) 0

x3 + 12x2 - 5x 5x

9) B) Does not exist

C) 0

D) 5

x2 - 25 lim x → 5 x2 - 9x + 20 A) Does not exist

11)

D) 6

8) B) 1/2

A) -1

10)

C) -6

1+x- 1 x

8) lim x→0

9) lim x→0

7)

lim x → 11

10) B) 0

C) 5

D) 10

11 - x 11 - x

A) -1

11) B) 0

C) Does not exist

D) 1

Find the limit. 12)

lim x→-! A)

x + 7 cos(x + !) 7- !

12) B) 0

C) 1

2

D) -

7- !

Provide an appropriate response. 13) It can be shown that the inequalities -x ≤ x cos

1 ≤ x hold for all values of x ≥ 0. x

13)

1 Find lim x cos if it exists. x x→0 A) does not exist

B) 0.0007

C) 0

D) 1

Use the table of values of f to estimate the limit. x2 + 7x + 12 14) Let f(x) = , find lim f(x). x→-4 x2 + 5 x + 4 x f(x)

-4.1

-4.01

-4.001

14)

-3.999

-3.99

-3.9

A) x -4.1 -4.01 -4.001 -3.999 -3.99 -3.9 ; limit = 0.4333 f(x) 0.4548 0.4355 0.4336 0.4331 0.4311 0.4103 B) x -4.1 -4.01 -4.001 -3.999 -3.99 -3.9 ; limit = 0.2333 f(x) 0.2548 0.2355 0.2336 0.2331 0.2311 0.2103 C) x -4.1 -4.01 -4.001 -3.999 -3.99 -3.9 ; limit = 1.4 f(x) 1.4082 1.4008 1.4001 1.3999 1.3992 1.3922 D) x -4.1 -4.01 -4.001 -3.999 -3.99 -3.9 ; limit = 0.3333 f(x) 0.3548 0.3355 0.3336 0.3331 0.3311 0.3103 Find the limit L for the given function f, the point c, and the positive number ε. Then find a number δ > 0 such that, for all x, 0 < |x - c|< δ ⇒ |f(x) - L| < ε. 15) f(x) = -3x + 8 , c = -2, ε = 0.03 15) A) L = 14; δ = 0.01 B) L = -2; δ = 0.01 C) L = 14; δ = 0.02 D) L = 2; δ = 0.02 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Prove the limit statement 16) lim (2x - 5) = -3 x→1

16)

3

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to estimate the specified limit. f(x) f(x) and lim 17) Find lim + x→(!/2) x→(!/2)

17)

y 6 5 4 3 2 1 -π

-π 2 -1

π 2

π

3π 2

2π

x

-2 -3 -4 -5 -6

A) 6 ; 1

B) !; !

C)

! ! ; 2 2

D) 1; 6

Determine the limit by sketching an appropriate graph. 3x -3 ≤ x < 0, or 0 < x ≤ 2 lim f(x), where f(x) = 3 x=0 18) x → -3 + 0 x < -3 or x > 2

A) Does not exist

B) 2

C) -0

4

18)

D) -9

19)

lim x → 4-

f(x), where f(x) =

A) Does not exist

2 4 16 - x 6

04 ≤ ≤ xx < < 46 x=6

B) 0

19)

C) 6

D) 4

Find the limit. 20)

7x2 10 + x

lim x→-2+

20)

A) Does not exist

21)

A)

22)

13 -

lim h→0 -

lim + x→3 A) -

-7 4

C)

7 2

D)

7 2

8h2 + 9 h + 13 h

9 2

B)

13

21)

B) Does not exist

C)

-9 2

13

D)

-9 26

5x x - 3 x-3 15

22) B) Does not exist

C) 0

D)

15

Find all points where the function is discontinuous. 23)

A) x = 0, x = 3

23)

B) x = 0

C) x = 3

5

D) None

Answer the question. 24) Is f continuous at x = 4?

24) d

f(x) =

x3 , -2x, 2, 0,

10

-2 < x ≤ 0 0≤x < 2 2< x ≤4 x= 2

8 6 4 2 (2, 0) -4

-3

-2

-1

1

2

3

4

t

-2 -4 -6 -8 -10

A) No

B) Yes

Find the intervals on which the function is continuous. 3 25) y = - 3x x +7

25)

A) discontinuous only when x = 7 C) continuous everywhere

B) discontinuous only when x = -10 D) discontinuous only when x = -7

4 26) y = 2x - 4 A) continuous on the interval 2, ∞ C) continuous on the interval 2, ∞

26) B) continuous on the interval -∞, 2 D) continuous on the interval - 2, ∞

Find the limit and determine if the function is continuous at the point being approached. 2! 27) lim cos ln (e x) 3 x→1 A) does not exist; no

B) 1; yes

1 C) - ; yes 2

1 D) - ; no 2

27)

Find numbers a and b, or k, so that f is continuous at every point. 28) x 2, x < -4 ax + b, -4 ≤ x ≤ 2 x + 2, x > 2 A) a = -2, b = 8

28)

f(x) =

B) a = 2, b = 8

C) a = -2, b = -8

D) Impossible

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 29) Use the Intermediate Value Theorem to prove that x(x - 2) 2 = 2 has a solution between 1 and 3.

6

29)

30) Use the Intermediate Value Theorem to prove that 3x3 + 9 x2 - 3 x + 5 = 0 has a solution

30)

between -4 and -3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the function f whose graph is given, determine the limit. 31) Find lim f(x). x→∞

31)

f(x) 4 3 2 1 -4 -3 -2 -1 -1

1

2

3

x

4

-2 -3 -4

A) -2

B) does not exist

C) ∞

D) 0

32) Find lim f(x). x→-1

32) y 4

2 A -4

-2

2

x

4

-2

-4

A) -1

B) -

4 5

C)

4 5

D) does not exist

Find the limit. 33)

-1 + (6 /x) lim x→-∞ 5 - (1/x2) A) ∞

33) B) -∞

C)

7

1 5

D) -

1 5

34)

x2 - 6 x + 11 lim x→∞ x3 + 8 x2 + 18 A) 1

35)

B) ∞

C)

11 18

D) 0

-7x2 - 9 x + 2 lim x→-∞ -13x2 - 8 x + 3 A) ∞

36)

34)

35) B)

7 13

C)

2 3

D) 1

lim cos 5 x x x→-∞ A) -∞

36) B) 0

C) 1

D) 5

Divide numerator and denominator by the highest power of x in the denominator to find the limit. 37)

25x2 + x - 3 (x - 11)(x + 1)

lim x→∞ A) 5

38)

B) 25

C) 0

D) ∞

5x + 3 3x2 + 1

lim x→∞ A)

37)

5 3

38)

B)

5

C) 0

3

D) ∞

Find the limit. 39)

1 lim x +7 + x → -7 A) -∞

40)

C) ∞

D) 0 40)

B) -∞

C) 0

D) ∞

tan x lim + x→(!/2) A) ∞

42)

B) -1

1 lim x -9 x →9 A) -1

41)

39)

41) B) 0

C) 1

D) -∞

1 +8 lim + 1/5 x →0 x A) ∞

42) B) Does not exist

C) 8

Graph the rational function. Include the graphs and equations of the asymptotes. 8

D) -∞

43) y =

x 2 x +x +4

43) y 4

2

-10 -8

-6

-4

-2

2

4

6

8

10 x

-2

-4

A) asymptote: y = 1

B) asymptote: y = 0 y

-10 -8

-6

-4

y

4

4

2

2

-2

2

4

6

8

10 x

-10 -8

-6

-4

-2

2

-2

-2

-4

-4

C) asymptote: y = 0

-6

-4

6

8

10 x

4

6

8

10 x

D) asymptotes: x = 4, x = -4 y

-10 -8

4

y

4

4

2

2

-2

2

4

6

8

10 x

-10 -8

-6

-4

-2

2

-2

-2

-4

-4

9

2 44) y = 2x 4 - x2

44) y 8 6 4 2

-8

-6

-4

-2

2

4

6

x

8

-2 -4 -6 -8

A) asymptotes: x = -2, x = 2, y = 0

B) asymptotes: x = -2, x = 2, y = 2

y

-8

-6

-4

y

8

8

6

6

4

4

2

2

-2

2

4

6

8

x

-8

-6

-4

-2

2

-2

-2

-4

-4

-6

-6

-8

-8

C) asymptotes: x = -2, x = 2, y = -2

-6

-4

8

x

8

x

y

8

8

6

6

4

4

2

2

-2

6

D) asymptotes: x = -2, x = 2, y = 0

y

-8

4

2

4

6

8

x

-8

-6

-4

-2

2

-2

-2

-4

-4

-6

-6

-8

-8

10

4

6

45) y =

2 - x2 2x + 4

45) y 8 6 4 2

-8

-6

-4

-2

2

4

6

x

8

-2 -4 -6 -8

A) asymptotes: x = -2, y = -x + 2

B) asymptotes: x = -2, y = -

y

1 1 x+ 4 2

y 8 6

8

4

6

2

4 2

-8

-6

-4

-2

2

4

6

8

x

-2

-8

-6

-4

-2

2

-4

-2

-6

-4

-8

-6

4

6

8

x

8

x

-8

C) asymptotes: x = -2, y = -

1 x+1 2

D) asymptotes: x = -4, y = -

y

-8

-6

-4

y

8

8

6

6

4

4

2

2

-2

1 1 x+ 8 2

2

4

6

8

x

-8

-6

-4

-2

2

-2

-2

-4

-4

-6

-6

-8

-8

11

4

6

46) y =

2 - 2x - x 2 x

46) y 8 6 4 2

-8

-6

-4

-2

2

4

6

x

8

-2 -4 -6 -8

A) asymptotes: x = 0, y = -x - 4

B) asymptotes: x = 0, y = -x

y

-8

-6

-4

y

8

8

6

6

4

4

2

2

-2

2

4

6

8

x

-8

-6

-4

-2

2

-2

-2

-4

-4

-6

-6

-8

-8

C) asymptotes: x = 0, y = -x + 2

-6

-4

8

x

6

8

x

y

8

8

6

6

4

4

2

2

-2

6

D) asymptotes: x = 0, y = -x - 2

y

-8

4

2

4

6

8

x

-8

-6

-4

-2

2

-2

-2

-4

-4

-6

-6

-8

-8

12

4

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find a function that satisfies the given conditions and sketch its graph. 47) lim g(x) = -5, lim g(x) = 5, lim g(x) = 5 , lim g(x) = -5. + x→-∞ x→∞ x→0 x→0

47)

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

2

4

6

8

x

-4 -6 -8

48)

lim f(x) = 0, lim f(x) = ∞, lim f(x) = ∞. + x→±∞ x→-5 x→-5

48)

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

2

4

6

8

x

-4 -6 -8

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. 49) y = x2 - x, (-4, 20) A) y = -9x + 12

B) y = -9x - 16

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

49)

C) y = -9x - 12

D) y = -9x + 16

C) y = 3

1 D) y = - x + 53 2

50) f(x) = 10

50)

Find the slope of the curve at the indicated point. 51) y = 8 , x = 7 4+x A) m = -

8 11

B) m = -

51)

8 121

C) m =

13

8 121

D) m =

8 11

Calculate the derivative of the function. Then find the value of the derivative as specified. ′ 52) g(x) = x 3 + 5x; g (1) A) g (x) = x2 + 5; g (1) = 6 ′ ′ C) g (x) = 3x 2 + 5; g (1) = 8 ′

53) g(x) = -

′

′

2 ′ ; g (-2) x ′

53) B) g (x) = - 2x2; g (- 2) = - 8 ′

′

A) g (x) = - 2; g (- 2) = - 2 ′

C) g (x) = -

52)

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

′

′ 1 2 ; g (- 2) = 2 x2

′

D) g (x) =

14

′

′ 1 2 ; g (-2) = 2 x2

Answer Key Testname: UNTITLED1

1) C 2) D 3) B 4) A 5) D 6) C 7) B 8) B 9) A 10) D 11) C 12) A 13) C 14) D 15) A 16) Let ε > 0 be given. Choose δ = ε/2. Then 0 < x - 1 < δ implies that (2x - 5) + 3 = 2x - 2 = 2(x - 1) = 2 x - 1 < 2δ = ε Thus, 0 < x - 1 < δ implies that (2x - 5) + 3 < ε 17) A 18) D 19) B 20) D 21) C 22) D 23) C 24) B 25) D 26) C 27) C 28) A 29) Let f(x) = x(x - 2) 2 and let y 0 = 2. f(1) = 1 and f(3) = 3. Since f is continuous on [1, 3] and since y 0 = 2 is between f(1) and f(3), by the Intermediate Value Theorem, there exists a c in the interval (1, 3) with the property that f(c) = 2. Such a c is a solution to the equation x(x - 2) 2 = 2. 30) Let f(x) = 3x3 + 9 x2 - 3 x + 5 and let y 0 = 0. f(-4) = -31 and f(-3) = 14. Since f is continuous on [-4, -3] and since y 0 = 0 is between f(-4) and f(-3), by the Intermediate Value Theorem, there exists a c in the interval (-4 , -3) with the property that f(c) = 0. Such a c is a solution to the equation 3x3 + 9 x2 - 3 x + 5 = 0. 31) 32) 33) 34) 35) 36) 37) 38) 39)

C C D D B B A B C 15

Answer Key Testname: UNTITLED1

40) 41) 42) 43) 44) 45) 46)

B D A C C C D

47) (Answers may vary.) Possible answer: f(x) =

5, x > 0 -5, x < 0

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

2

4

6

8

x

-4 -6 -8

48) (Answers may vary.) Possible answer: f(x) =

1 . x +5

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

2

4

6

8

x

-4 -6 -8

49) 50) 51) 52) 53)

B D B C D

16...