Chapter 2 - MDC Faculty Home Pages

MAC 2233 

Group Activity/Review #2-Chapter 2: Application of Differentiation 

Prof Nguyen 

---------------------- 

Name______________________________ 

Class time__________________________ 

SECTION 2.1 

Find the relative extrema of the function, if they exist. 

"Note: to find relative extrema, we need only consider those inputs for which the derivative is 0 

or for which it does not exits. If a relative maximum or minimum occurs, then the first 

cooridinate of that extremum will be a critical value. Note that the function f has a relative 

minimum at a critical value c, if the f is decreasing (f'(x) < 0) to the left of c and increasing ( f'(x) 

> 0) to the right of c. The function f has a relative maximum a critical value c , if f is increasing ( 

f'(x) >0 ) to the left of c and decreasing ( f'(x) < 0) to the right of c. We use derivatives of function 

to find critical values (locations of max and mins). Next, determine where f'(x) does not exist or 

where f'(x) = 0. " 

1) f(x) = x2 - 4x + 7 1) 

2) f(x) = 2 

3 x3 - 3 

2 x2 - 9x + 2 2) 

3) f(x) = 3x4 + 16x3 + 24x2 + 32 3) 

1 

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4) f(x) = (x + 5)2/3 + 3 4) 

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

Find the location and value of all relative extrema for the function. 

5) 

6) 

A) Relative minimum of 0 at 0. 

B) Relative minimum of -2 at -3 ; Relative maximum of 2 at 3. 

C) None 

D) Relative minimum of -2 at -3 ; Relative minimum of 0 at 0 ; Relative maximum of 2 at 3. 

A) Relative minimum of -1 at -3. 

B) Relative minimum of 3 at 3 ; Relative minimum of 0 at 0 ; Relative maximum of -3 at -3. 

C) Relative minimum of 3 at 3 ; Relative maximum of -3 at -3. 

D) None 

2 

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5) 

6)

7) 

8) 

A) None 

B) Relative minimum of 0 at 0. 

C) Relative maximum of 5 at -2 ; Relative maximum of 1 at 2. 

D) Relative maximum of 5 at -2 ; Relative minimum of 0 at 0 ; Relative maximum of 1 at 2. 

A) Relative maximum of 2 at 1. 

B) Relative minimum of -1 at -2. 

C) Relative minimum of 1 at 2 ; Relative maximum of -1 at -2. 

D) None 

Graph the function by first finding the relative extrema. 

9) f(x) = x3 + 2x2 - x - 2 

12 

8 

4 

y 

-6 -4 -2 2 4 6 

-4 

-8 

-12 

x 

3 


9) 

7) 

8)

10) h(x) = x4 - 2x2 + 1 

8 

4 

y 

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

11) f(x) = x2/3 + 3 

6 

4 

2 

-4 

-8 

y 

-5 5 

-2 

-4 

-6 

x 

x 


Section 2.2 

Find f"(x) for the function. 

12) f(x) = 8x2 + 3x - 3 

A) 16x + 3 B) 16 C) 0 D) 8 

13) f(x) = 3x4 - 8x2 + 4 

A) 12x2 - 16 B) 36x2 - 16x C) 36x2 - 16 D) 12x2 - 16x 

4 


10) 

11) 

12) 

13)

14) f(x) = x2 + x 

A) 2x3/2 - 1 

x3/2 

1 

15) f(x) = 

x2 - 1 

A) 6x2 - 2 

(x2 - 1)3 

B) 8x3/2 + 1 

4x3/2 

B) 6x2 - 2 

(x2 - 1)4 

Find the requested value of the second derivative of the function. 

16) f(x) = x4 + 4x3 - 2x + 5; Find f′′ (1). 

C) 8x3/2 - 1 

4x3/2 

C) 6x2 + 2 

(x2 - 1)3 

D) 2x3/2 + 1 

x3/2 

D) 6x2 + 2 

(x2 - 1)4 

A) 36 B) 31 C) 40 D) -35 

17) f(x) = 7x2 + 6x - 3; Find f′′ (0). 

A) 14 B) 7 C) 0 D) -14 

5 


14) 

15) 

16) 

17)

NOTE: 

* Test for Concavity 

-If f"(x) >0 on an interval I, then the graph of f is concave up. (f' is increasing, so f is turning up 

on I). 

-If f"(x)

20) f(x) = 2x3 + 3x2 - 12x - 2 

A) Relative minimum: - 1 11 

, 

2 4 

B) Relative minimum: -2, 3 , relative maximum: 1, - 3 

2 

C) Relative maximum: -2, 3 , relative minimum: 1, - 3 

2 

D) Relative maximum: 1 21 

, - 

2 4 

21) f(x) = (x - 5)2/3 

A) Relative minimum: (5, 0) B) Relative minimum: (-5, 0) 

C) Relative maximum: (-5, 0) D) There are no relative extrema. 

22) f(x) = 20x3 - 3x5 

A) Relative minimum: (-2, -64), relative maximum at (0,0) 

B) Relative maximum at (0,0), relative minimum: ( 2, 64) 

C) Relative minimum: (-2, -64), relative minimum at (0,0), relative maximum: ( 2, 64) 

D) Relative minimum: (-2, -64), relative maximum: (2, 64) 

Identify the intervals where the function is changing as requested. 

23) Increasing 

A) (-3, 3) B) (-2, 2) C) (-2, ∞) D) (-3, ∞) 

7 


20) 

21) 

22) 

23)

24) Decreasing 

A) (2, -1) B) (-1, 2) C) (1, 2) D) (2, 1) 

25) Increasing 

A) (1, 5) B) (1, 6) C) (0, 6) D) (0, 5) 

26) Decreasing 

A) (3, ∞) B) (0, 3) C) (-1, 0) D) (-∞, 0) 

Determine where the given function is increasing and where it is decreasing by using the Second -Derivative Test 

27) f(x) = 2x3 - 9x2 + 12x 

A) Increasing on (-∞, 1], decreasing on [1, ∞) 

B) Decreasing on (-∞, 0] and [2, ∞), increasing on [0, 2] 

C) Increasing on (-∞, 1] and [2, ∞), decreasing on [1, 2] 

D) Decreasing on (-∞, 1] and [2, ∞), increasing on [1, 2] 

8 


24) 

25) 

26) 

27)

Determine where the given function is increasing and where it is decreasing. 

28) f(x) = -4x2 - 2x - 12 

A) Increasing on -∞, 1 

4 

B) Increasing on -∞, - 1 

4 

C) Increasing on -∞, - 1 

4 

D) Decreasing on -∞, - 1 

4 

1 

, decreasing on , ∞ 

4 

1 

, decreasing on - , ∞ 

4 

1 

and (0, ∞), decreasing on - , 0 

4 

1 

, increasing on - , ∞ 

4 

Determine where the given function is increasing and where it is decreasing by using the Second -Derivative Test 

29) f(x) = (x + 3)2/3 - 6 

A) Increasing on (-∞, -3], decreasing on [-3, ∞) 

B) Decreasing on (-∞, ∞) 

C) Decreasing on (-∞, -3], increasing on [-3, ∞) 

D) Decreasing on (-∞, -3] and [0, ∞), increasing on [-3, 0) 

Determine where the given function is increasing and where it is decreasing. 

30) f(x) = x4 - 18x2 + 4 

A) Increasing on (-∞, -3] and [0, 3], decreasing on [-3, 0] and [3, ∞) 

B) Increasing on (-∞, -3] and [3, ∞), decreasing on [-3, 3] 

C) Decreasing on (-∞, -3] and [3, ∞), increasing on [-3, 3] 

D) Decreasing on (-∞, -3] and [0, 3], increasing on [-3, 0] and [3, ∞) 

Find the points of inflection. 

31) f(x) = x3 - 3x2 + 2x + 15 

A) (-1, -3) B) (1, -1) C) (1, 15) D) (1, -3) 

9 


28) 

29) 

30) 

31)

32) f(x) = 4 

3 x3 - 12x2 + 10x + 46 

A) (3, 0) B) (3, -26) C) (3, 4) D) (0, 4) 

33) f(x) = (x + 2)2/3 - 7 

A) (-2, -7), (0, 22/3 - 7) B) (-2, -7) 

C) (0, 22/3 - 7) D) No points of inflection exist 

Find the largest open intervals where the function has the indicated concavity. 

34) Concave upward 

A) (0, ∞) B) (0, 3) C) (-3, ∞) D) (-3, 3) 

35) Concave upward 

A) (-∞, 0) B) (0, ∞) C) None D) (-∞, 0), (0, ∞) 

36) Concave downward 

A) (-1, 0) , (1, ∞) B) (-∞, -1) C) (-1, 0) D) (0, 1) 

10 


32) 

33) 

34) 

35) 

36)

Suppose that the function with the given graph is not f(x), but f′(x). Find the open intervals where the function is 

concave upward or concave downward, and find the location of any inflection points. 

37) 

38) 

120 

80 

40 

y 

-5 -4 -3 -2 -1 

-40 

1 2 3 4 5 

-80 

-120 

x 

A) Concave upward on (-∞, 0); concave downward on (0, ∞); inflection point at 0 

B) Concave upward on (-∞, -2) and (2, ∞); concave downward on (-2, 2); inflection points at -2 

and 2 

C) Concave upward on (-2, 2); concave downward on ( -∞, -2) and (2, ∞); inflection points at -2 

and 2 

D) Concave upward on (-∞, -2) and (2, ∞); concave downward on (-2, 2); inflection points at 

-120 and 120 

8 

6 

4 

2 

y 

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

-2 

-4 

-6 

-8 

x 

A) Concave upward on (-∞, -1) and (0, 1); concave downward on ( -1, 0) and (1, ∞; ); inflection 

points 

at -1, 0, and 1 

B) Concave upward on (-1, 0) and (1, ∞); concave downward on (-∞, -1) and (0, 1); inflection 

points 

at -4, 0, and 4 

C) Concave upward on (-∞, 0); concave downward on (0, ∞); inflection point at 0 

D) Concave upward on (-1, 0) and (1, ∞); concave downward on (-∞, -1) and (0, 1); inflection 

points 

at -1, 0, and 1 

11 


37) 

38)

Determine where the given function is concave up and where it is concave down. 

39) f(x) = x3 + 3x2 - x - 24 39) 

40) G(x) = 1 

4 x4 - x3 + 5 40) 

41) f(x) = (x + 3)2/3 - 6 41) 

Graph the function. 

42) f(x) = x3 + x2 - 5x - 5 

10 

y 

-5 5 

-10 

x 

12 


42)


Solve the problem. 

43) The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately 

R(x) = 500x - 0.01x2 and C(x) = 160x + 100,000, where x denotes the number of clocks made. What 

is the maximum annual profit? 

A) $2,890,000 B) $2,990,000 C) $3,090,000 D) $2,790,000 

44) The percent of concentration of a certain drug in the bloodstream x hr after the drug is 

administered is given by K(x) = 5x 

. How long after the drug has been administered is the 

x2 + 4 

SECTION 2.3 

concentration a maximum? Round answer to the nearest tenth, if necessary. 

A) 2 hr B) 0.2 hr C) 5 hr D) 0.4 hr 

Determine the vertical asymptote(s) of the given function. If none exists, state that fact. 

x + 4 

45) g(x) = 

x2 - 1 

A) x = -1, x = 1 B) x = -1, x = 1, x = -4 

C) x = 1, x = -4 D) x = 0, x = 1 

x + 9 

46) f(x) = 

x2 + 1 

A) x = -1, x = -9 B) x = -1, x = 1, x = -9 

C) x = -1, x = 1 D) none 

-3x2 

47) R(x) = 

x2 + 6x - 55 

A) x = - 55 B) x = -11, x = 5, x = -3 

C) x = -11, x = 5 D) x = 11, x = -5 

x - 1 

48) R(x) = 

x3 + 3x2 - 40x 

A) x = -8, x = 5 B) x = -8, x = 0, x = 5 

C) x = -5, x = -30, x = 8 D) x = -5, x = 0, x = 8 

13 


43) 

44) 

45) 

46) 

47) 

48)

Determine the horizontal asymptote of the given function. If none exists, state that fact. 

49) g(x) = x2 + 4x - 8 

x - 8 

50) h(x) = 

A) y = 8 B) y = 0 

C) y = 1 D) no horizontal asymptotes 

8x - 3 

x - 5 

A) y = 8 B) y = 5 

C) y = 0 D) no horizontal asymptotes 

51) h(x) = 2x3 - 6x 

3x3 - 7x + 5 

A) y = 0 B) y = 2 

3 

C) y = 6 

7 

52) h(x) = 7x4 - 8x2 - 4 

6x5 - 2x + 2 

A) y = 0 B) y = 4 

C) y = 7 

6 

Find the indicated intercept(s) of the graph of the function. 

53) y-intercept of f(x) = x2 - 11x + 4 

x2 + 8x + 2 

D) no horizontal asymptotes 

D) no horizontal asymptotes 

A) (0, 2) B) (0, 8) C) (0, 4) D) none 

54) y-intercept of f(x) = x3 + 2 

x2 + 2 

A) (0, 3) B) (0, 1) C) (0, 2) D) none 

55) x-intercepts of f(x) = 

2x + 3 

x - 6 

A) - 3 

3 

, 0 B) , 0 C) (6, 0) D) (-6, 0) 

2 2 

x - 4 


x2 + 6x - 2 

A) (4, 0) B) (6, 0) C) (-4, 0) D) none 

14 


49) 

50) 

51) 

52) 

53) 

54) 

55) 

56)

(x - 3)(2x + 5) 


x2 + 6x - 4 

A) (-3, 0), 5 

5 

, 0 B) (3, 0), - , 0 C) (3, 0), (-5, 0) D) none 

2 2 

58) x-intercepts of f(x) = x2 - x - 20 

x2 + 5 

. 

A) (-5, 0), (4, 0) B) (-4, 0), (5, 0) C) (- 20, 0) D) (-5, 0), (0, 0) 

Graph the rational function. 

x - 1 

59) f(x) = 

x + 2 

x 

60) f(x) = 

x2 - 25 

y 

x 

15 


59) 

60) 

57) 

58)

x - 3 

61) f(x) = 

x2 - 9 


SECTION 2.4 

Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval, and 

indicate the x-values at which they occur. 

62) f(x) = 9 + 2x - x2; [0, 3] 

10 

8 

6 

4 

2 

y 

1 2 3 

x 

A) Absolute maximum = 8 at x = 2; absolute minimum = 6 at x = 0 

B) Absolute maximum = 11 at x = 1; absolute minimum = 6 at x = 3 

C) Absolute maximum = 10 at x = 1; absolute minimum = 6 at x = 3 

D) Absolute maximum = 10 at x = 1; absolute minimum = 9 at x = 0 

16 


61) 

62)

Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval, and 

indicate the x-values at which they occur. 

63) f(x) = 6x 

; [-3, 3] 

63) 

x2 + 1 

3 

y 

-3 3 x 

-3 

A) Absolute maximum = 1.8 at x = 1; absolute minimum = -1.8 at x = -1 

B) Absolute maximum = 3 at x = 1; absolute minimum = -3 at x = -1 

C) Absolute maximum = 1.8 at x = -1; absolute minimum = 0 at x = 0 

D) Absolute maximum = 3 at x = 1; absolute minimum = 0 at x = 0 

Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. 

When no interval is specified, use the real line ( -∞, ∞). 

64) f(x) = 1 

3 x3 - 4x; [-8, 8] 

64) 

A) Absolute maximum: 416 

416 

, absolute minimum: - 

3 3 

B) Absolute maximum: 5.33, absolute minimum: -5.33 

C) Absolute maximum: 5.33, absolute minimum: - 416 

3 

D) Absolute maximum: 416 

, absolute minimum: -5.33 

3 

17 


Find the absolute maximum and absolute minimum values of the function, if they exist, on the indicated interval. 

65) f(x) = x4 - 32x2 + 2; [-5, 5] 

A) Absolute maximum: 0, absolute minimum: -254 

B) Absolute maximum: -254 

C) Absolute minimum: 0 

D) Absolute maximum: 2, absolute minimum: -254 

66) f(x) = -x2 + 6x - 9 : [3, 3] 

A) Absolute maximum: 0; absolute minimum: 0 

B) Absolute maximum: 18; absolute minimum: 0 

C) Absolute maximum: 0; absolute minimum: 1 

4 

D) Absolute maximum: 1; absolute minimum: 0 

67) F(x) = 3 x ; [0, 8] 

A) Absolute maximum: 2, absolute minimum: -2 

B) Absolute maximum: 8, absolute minimum: 0 

C) Absolute maximum: 0, absolute minimum: -2 

D) Absolute maximum: 2, absolute minimum: 0 

18 


65) 

66) 

67)

Find the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. 

When no interval is specified, use the real line ( -∞, ∞). 

68) f(x) = x2 + 16 

; (0, ∞) 

68) 

x 

200 

y 

14 x 

A) No absolute maximum; absolute minimum: 2 

B) No absolute maximum; absolute minimum: 12 

C) Absolute maximum: 196; absolute minimum: 2 

D) No absolute extrema 

69) f(x) = x4 - 4x2 

12 

8 

4 

y 

-4 4 x 

-4 

-8 

-12 

-16 

-20 

A) Absolute maximum: 0; absolute minimum: -4 

B) Absolute maximum: 0; no absolute minimum 

C) No absolute maximum; absolute minimum: -4 


70) f(x) = - 1 

3 x3 + 16x - 1; (-∞, 0) 

A) Absolute maximum: 4; absolute minimum: -4 

B) No absolute maximum; absolute minimum: -43.7 

C) Absolute minimum: -41.7; absolute maximum: 43.7 


19 


69) 

70)

71) f(x) = 4x + 8 


A) Absolute maximum: 8; no absolute minimum 

B) Absolute maximum: 4; no absolute minimum 

C) No absolute maximum; absolute minimum: 4 


72) Assume that the temperature T of a person during a certain illness is given by 

T(t) = -0.1t2 + 1.3t + 98.6, 0 ≤ t ≤ 12 where T = the temperature (°F) at time t, in days. Find the 

maximum value of the temperature and when it occurs. Round your answer to the nearest tenth, 

if necessary. 

A) 102.8°F at 6.5 days B) 101.3°F at 5.2 days 

C) 101.8°F at 6.5 days D) 102.8°F at 3.9 days 

73) The total-revenue and total-cost functions for producing x clocks are R(x) = 520x - 0.01x2 and 

C(x) = 120x + 100,000, where 0 ≤ x ≤ 25,000. What is the maximum annual profit? 

A) $4,100,000 B) $4,200,000 C) $4,000,000 D) $3,900,000 

20 


71) 

72) 

73)

74) P(x) = -x3 + 27 

2 x2 - 60x + 100, x ≥ 5 is an approximation to the total profit (in thousands of dollars) 

from the sale of x hundred thousand tires. Find the number of tires that must be sold to maximize 

profit. 

A) 550,000 B) 500,000 C) 500,000 D) 450,000 

75) In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year 

during the years 1990 - 2000 are given in the figure. Consider the closed interval [1990, 2000]. 

Rate per 500K People 

300 

250 

200 

150 

100 

50 

A 

B 

C 

D 

E 

F 

1990 1992 1994 1996 1998 2000 

Year 

A (1990, 171) D (1993, 282) G (1996, 188) L (1999, 238) 

B (1991, 204) E (1994, 211) H (1997, 258) M (2000, 271) 

C (1992, 255) F (1995, 141) K (1998, 247) 

Give all relative maxima and minima on the interval and the years when they occur. 

A) Relative maxima of 282 in 1993, 258 in 1997, 271 in 2000 

Relative minima of 171 in 1990, 141 in 1995, 238 in 1999 

B) Relative maxima of 282 in 1993 and 258 in 1997 

Relative minima of 141 in 1995 and 238 in 1999 

C) Relative maxima of 282 in 1993, 258 in 1997, 271 in 2000 

Relative minima of 141 in 1995 and 238 in 1999 

D) Relative maxima of 282 in 1993 and 258 in 1997 

Relative minima of 171 in 1990, 141 in 1995, 238 in 1999 

G 

H 

21 


K 

L 

M 

74) 

75)

SECTION 2.5 


76) A carpenter is building a rectangular room with a fixed perimeter of 420 ft. What are the 

dimensions of the largest room that can be built? What is its area? 

A) 210 ft by 210 ft; 44,100 ft2 B) 42 ft by 378ft; 15,876 ft2 


C) 105 ft by 105 ft; 11,025 ft2 D) 105 ft by 315 ft; 33,075 ft2 

77) A company wishes to manufacture a box with a volume of 40 cubic feet that is open on top and is 

twice as long as it is wide. Find the width of the box that can be produced using the minimum 

amount of material. Round to the nearest tenth, if necessary. 

A) 3.2 ft B) 3.6 ft C) 6.4 ft D) 7.2 ft 

78) Find the number of units that must be produced and sold in order to yield the maximum profit, 

given the following equations for revenue and cost: 

R(x) = 60x - 0.5x2 

C(x) = 4x + 7. 

A) 63 units B) 57 units C) 56 units D) 64 units 

22 


76) 

77) 

78)

79) Find the maximum profit given the following revenue and cost functions: 

R(x) = 108x - x2 

C(x) = 1 

3 x3 - 3x2 + 96x + 36 

where x is in thousands of units and R(x) and C(x) are in thousands of dollars. 

A) 108 thousand dollars B) 72 thousand dollars 

C) 18 thousand dollars D) 36 thousand dollars 

80) An appliance company determines that in order to sell x dishwashers, the price per dishwasher 

must be 

p = 600 - 0.4x. 

It also determines that the total cost of producing x dishwashers is given by 

C(x) = 3000 + 0.2x2. 

What is the maximum profit? 

A) $147,000 B) $153,000 C) $150,000 D) $297,000 

81) A hotel has 230 units. All rooms are occupied when the hotel charges $110 per day for a room. For 

every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room 

costs $34 per day to service and maintain. What should the hotel charge per day in order to 

maximize daily profit? 

A) $177 B) $187 C) $170 D) $77 

23 


79) 

80) 

81)

82) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a 

certain city, where p(x) = 49 - x 

. How many candy bars must be sold to maximize revenue? 

22 

A) 539 thousand candy bars B) 1078 candy bars 

C) 539 candy bars D) 1078 thousand candy bars 

83) A baseball team is trying to determine what price to charge for tickets. At a price of $10 per ticket, 

it averages 40,000 people per game. For every increase of $1, it loses 5,000 people. Every person at 

the game spends an average of $5 on concessions. What price per ticket should be charged in 

order to maximize revenue? 

A) $13.50 B) $3.00 C) $3.50 D) $6.50 

84) SECTION 2.6 

A company estimates that the daily revenue (in dollars) from the sale of x cookies is given by 

R(x) = 885 + 0.02x + 0.0003x2. 

Currently, the company sells 900 cookies per day. Use marginal revenue to estimate the increase in 

revenue if the company increases sales by one cookie per day. 

A) $92.00 B) $0.56 C) $0.92 D) $56.00 

24 


82) 

83) 

84)

85) A company finds that when it spends x million dollars on advertising, its profit P, in thousands of 

dollars, is given by 

P(x) = 970 + 15x - 4x2 

Currently the company spends 17 million dollars on advertising. Use the marginal profit to 

estimate the change in profit if the company increases its advertising expenditure by one million 

dollars. 

A) -121 thousand dollars B) 970 thousand dollars 

C) 119 thousand dollars D) 255 thousand dollars 

86) Suppose that the daily cost, in dollars, of producing x televisions is 

C(x) = 0.003x3 + 0.1x2 + 62x + 620, 

and currently 60 televisions are produced daily. Use C(60) and the marginal cost to estimate the 

daily cost of increasing production to 63 televisions daily. Round to the nearest dollar. 

A) $5673 B) $5635 C) $5481 D) $5667 

Section 2.7 

Differentiate implicitly to find the slope of the curve at the given point. 

87) y3 + yx2 + x2 - 3y2 = 0; (-1, 1) 

A) 3 

2 

88) xy3 - x5y2 = -4; (-1, 2) 

A) - 6 

5 

B) -1 C) -2 D) - 1 

2 

B) - 3 

2 

25 

C) - 3 

4 

D) 2 

3 


85) 

86) 

87) 

88)

89) xy + x = 2; (1, 1) 

A) 2 B) - 2 C) 1 

2 

Find dy/dx by implicit differentiation. 

90) xy2 = 4 

A) 2x 

y 

91) y2 - xy + x2 = 7 

2x - y 

A) 

x + 2y 

92) x3 + y3 = 8 

A) x2 

y2 

93) y3 + 5xy + 4x3 - 4x = 0 

4 + 5y - 12x2 

A) 

3y2 - 5x 

B) - y 

2x 

B) 

2x + y 

x - 2y 

B) - y2 

x2 

4 - 5y - 12x2 

B) 

3y2 - 5x 

26 

C) x 

2y 

C) 

2x + y 

x + 2y 

C) y2 

x2 

C) 

4 - 5y - 12x2 

3y2 + 5x 

D) - 1 

2 

D) - 2y 

x 

D) 

2x - y 

x - 2y 

D) - x2 

y2 

D) 

4 + 5y - 12x2 

3y2 + 5x 


89) 

90) 

91) 

92) 

93)

94) 9x3 - x2y3 = 7 

A) 27x2 - 2xy2 

xy2 

Calculate dy/dt using the given information. 

95) x3 + y3 = 9; dx/dt = -3, x = 1, y = 2 

A) - 3 

4 

B) 27x2 - 2xy2 

3x2y2 

B) 4 

3 

96) xy + x = 12; dx/dt = -3, x = 2, y = 5 


C) 27x2 - 2xy3 

3x2y2 

C) 3 

4 

D) 27x2 - 2xy3 

3xy2 

D) - 4 

3 

A) 3 B) -9 C) 9 D) -3 

97) Given the revenue and cost functions R = 34x - 0.4x2 and C = 4x + 9, where x is the daily 

production, find the rate of change of revenue with respect to time when x = 15 units and 

dx 

= 7 units per day. 

dt 

A) $204.4 per day B) $198 per day C) $154 per day D) $126 per day 

27 


94) 

95) 

96) 

97)

98) Given the revenue and cost functions R = 30x - 0.3x2 and C = 3x + 13, where x is the daily 

production, find the rate of change of profit with respect to time when x = 10 units and 

dx 

= 8 units per day. 

dt 

A) $168 per day B) $192 per day C) $211.2 per day D) $210 per day 

99) A heart attack victim is given a blood vessel dilator to increase the radii of the blood vessels. After 

receiving the dilator, the radii of the affected blood vessels increase at about 2% per minute. 

According to Poiseulle's law, the volume of blood flowing through a vessel and the radius of the 

vessel are related by the formula V = kr4 where k is a constant. What will be the percentage rate of 

increase in the blood flow after the dilator is given? 

A) 8%/min B) 6%/min C) 12%/min D) 10%/min 

100) A zoom lens in a camera makes a rectangular image on the film that is 8 cm length x 5 cm width. 

As the lens zooms in and out, the size of the image changes. Find the rate at which the area of the 

image begins to change (dA/df) if the length of the frame changes at 0.5 cm/s and the width of the 

frame changes at 0.1 cm/s. 

A) 3.3 cm2/s B) 8 

5 cm2/2 C) 0.66 cm2/s D) 4.5 cm2/s 

28 


98) 

99) 

100)

Answer Key 

Testname: MAC2233-GA-CHP2 

1) Relative minimum at (2, 3) 

2) Relative maximum at - 3 79 

41 

, ; relative minimum at 3, - 

2 8 2 

3) Relative minimum at (0, 32) 

4) Relative minimum at (-5, 3) 

5) B 

6) D 

7) B 

8) C 

9) 

10) 

11) 

12) B 

13) C 

14) C 

12 

8 

4 

y 

-6 -4 -2 2 4 6 

-4 

-8 

-12 

8 

4 

y 

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

6 

4 

2 

-4 

-8 

y 

-5 5 

-2 

-4 

-6 

x 

x 

x 

29 


Answer Key 


15) C 

16) A 

17) A 

18) A 

19) D 

20) C 

21) A 

22) D 

23) B 

24) C 

25) D 

26) C 

27) C 

28) B 

29) C 

30) D 

31) C 

32) C 

33) D 

34) A 

35) B 

36) A 

37) B 

38) D 

39) Concave up on (-1, ∞), concave down on (-∞, -1) 

40) Concave up on (-∞, 0) and (2, ∞), concave down on (0, 2) 

41) Concave down on (-∞, -3) and (-3, ∞) 

42) 

43) D 

44) A 

45) A 

46) D 

47) C 

48) B 

49) D 

50) A 

51) B 

52) A 

10 

y 

-5 5 

-10 

x 

30 


Answer Key 


53) A 

54) B 

55) A 

56) A 

57) B 

58) B 

59) 

60) 

61) 

62) C 

63) B 

64) A 

65) D 

66) A 

12 

8 

4 

y 

-12 -8 -4 4 8 12 x 

-4 

-8 

-12 

5 

y 

-10 10 

-5 

4 

2 

-6 -4 -2 2 4 6 

-2 

-4 

x 

31 


Answer Key 


67) D 

68) B 

69) C 

70) B 

71) D 

72) A 

73) D 

74) C 

75) A 

76) C 

77) A 

78) C 

79) D 

80) A 

81) B 

82) A 

83) D 

84) B 

85) A 

86) D 

87) C 

88) B 

89) B 

90) B 

91) D 

92) D 

93) C 

94) C 

95) C 

96) C 

97) C 

98) A 

99) A 

100) A 

32

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