MTH 135 Mid-Term Exam for week 5
Name___________________________________ Date: ___________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) The critical values of f(x) = 4 – 48x + 24 are x = -2 and x = 2. Use the first derivative test to determine which of the
critical values correspond to a local minimum. 1) _______
A) x = -2 B) x = 2
C) neither x = 2 nor x = -2 correspond to a local minimum D) x = 2 and x = -2
2) Determine the interval(s) where is decreasing. 2) _______
A) (-∞, 0) and (6, ∞) B) (0, 3) and (6, ∞) C) (0, 6) D) (0, 3) and (3, 6)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the que stion.
3) Identify the intervals where (x) > 0.
3) _____________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
4) Find the critical values and determine the intervals where f(x) is decreasing and the intervals where f(x) is increasing
for 4) _______
A) decreasing on (-1, 0) and (1, ∞); increasing on (-∞, -1) and (0, 1)
B) increasing on (-1, 0); decreasing on (-∞, -1) and (0, 1)
C) decreasing on (-1, 0) and (1, ∞); increasing on (-∞, -1)
D) increasing on (-1, 0) and (1, ∞); decreasing on (-∞, -1) and (0, 1)
Sketch a graph of the function.
5) f(x) = 12x –
5) _______
A)
B)
C)
D)
Solve the problem.
6) The Olympic flame at the 1992 Summer Olympics was lit by a flaming arrow. As the arrow moved d feet horizontally
from the archer, assume that its height h(d), in feet, was approximated by the function Find
the relative maximum of the function. 6) _______
A) (350, 129.4) B) (175, 61.25) C) (0, 6.9) D) (175, 68.15)
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
7) Concave upward
7) _______
A) (-2, ∞); no inflection points B) (-∞, ∞); x = -2
C) (-∞, ∞); no inflection points D) (-∞, -2); no inflection points
Provide an appropriate response.
8) Find f”(x) for f(x) = . 8) _______
A) f”(x) = 4x + 5 B) f”(x) = 12x + 15 C) f”(x) = 24x + 30 D) f”(x) = 384x + 480
9) Find f”(x) for f(x) = 5 – 6 + 7. 9) _______
A) f”(x) = 20 – 12x B) f”(x) = 20 – 12 C) f”(x) = 60 – 12 D) f”(x) = 60 – 12x
10) Determine the interval(s) over which is concave upward. 10) ______
A) (-3, ∞) B) (-∞, 3) C) (-∞, ∞) D) (-∞, -3)
11) Find the inflection point(s) for f(x) = – + 6. 11) ______
A) (0, 0) B) (0, 6) and (2, 2) C) (0, 0) and (2, 2) D) (0, 6) and (2, -4)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the que stion.
Solve the problem.
12) A backpack manufacturer is planning to expand its work force. They estimate that the number of backpacks
produced by hiring new workers is given by Determine when the rate of backpacks is
increasing and when it is decreasing. Determine the point of diminishing returns and the maximum rate of change of
backpack production. 12) _____________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the limit, if it exists.
13) Find . 13) ______
A) 0 B) C) ∞ D) 2
14) Find: 14) ______
A) – B) C) D) –
Find the domain and intercepts.
15) f(x) = 15) ______
A) Domain: [-64, ∞); y intercept: 8; x intercept: -64
B) Domain: (-∞, ∞); y intercept: 8; x intercept: -64
C) Domain: (-64, ∞); y intercept: 8; x intercept: -64
D) Domain: [-64, ∞); y intercept: -64; x intercept: 8
Solve the problem.
16) Suppose that the total-cost function for a certain company to produce x units of a product is given by
Graph the average cost function A(x) = C(x)/x.
16) ______
A)
B)
C)
D)
Provide an appropriate response.
17) Find the absolute minimum value of f(x) = 5 + 4x + for x > 0. 17) ______
A) min f(x) = f(2) = 21 B) min f(x) = f(1) = 25 C) min f(x) = f(0) = 5 D) min f(x) = f(2) = 13
18) Find the absolute maximum and minimum values of f(x) = 9 – 54 + 81x + 13 on the interval [-6, 2].
18) ______
A) max f(x) = f(1) = 4361 min f(x) = f(-6) = 49
B) max f(x) = f(1) = 49 min f(x) = f(-6) = -4361
C) max f(x) = f(1) = 4361 min f(x) = f(-6) = -49
D) max f(x) = f(-6) = -4361 min f(x) = f(1) = 49
19) Find the absolute minimum value of for Round your answer to three decimal places. 19) ______
A) 1 at x = 2.718 B) 3 at x = 0.7439 C) D) 2.718 at x = 1
20) Find two numbers whose sum is 360 and whose product is a maximum. 20) ______
A) 1 and 359 B) 179 and 181 C) 10 and 350 D) 180 and 180
Solve the problem.
21) A carpenter is building a rectangular room with a fixed perimeter of What are the dimensions of the largest
room that can be built? What is its area?
21) ______
A) 45 ft by 135 ft; 6,075 B) 45 ft by 45 ft; 2,025
C) 90 ft by 90 ft; 8,100 D) 18 ft by 162ft; 2,916
22) A computer software company sells 20,000 copies of a certain computer game each year. It costs the company $1.00 to
store each copy of the game for one year. Each time it must produce additional copies, it costs the company $625 to set up
production. How many copies of the game should the company produce during each production run in order to
minimize its total storage and set-up costs?
22) ______
A) 5000 copies in 4 production runs B) 4000 copies in 5 production runs
C) 20,000 copies in 1 production run D) 10,000 copies in 2 production runs
Find the limit, if it exists.
23) Find: 23) ______
A) B) – C) 0 D) Does not exist
24) Find: for f(x) = -x + 1. 24) ______
A) 1 B) -1 C) 0 D) Does not exist
Provide an appropriate response.
25) If the limit at infinity exists, find the limit.
25) ______
A) ∞ B) 0 C) D) 1
Find average rate of change for the function over the given interval.
26) Find the average rate of change of y with respect to x if x changes from 3 to 5 in the function
26) ______
A) 11 B) 9 C) 22 D) 4
Solve the problem.
27) If an object moves along a line so that it is at y = f(x) = 8 at time x (in seconds), find the velocity at
27) ______
A) 160 ft/s B) 16 ft / s C) 8 ft / s D) 6 ft/sec
Use the given graph of f(x) to find the intervals on which (x) > 0.
28)
28) ______
A) (x) > 0 on [-3, ∞), (x) < 0 on (-∞, -3]
B) (x) > 0 on (-∞, -3] ∪ [ 1, ∞), (x) < 0 on [-3, 1]
C) (x) > 0 on [-1, ∞), (x) < 0 on decreasing on (-∞, -1]
D) (x) < 0 on (-∞,∞)
Find (x).
29) f(x) = 9 – 2 29) ______
A) 9 – 8 B) 9 – 8 C) 9 – 4 D) 9x – 8
30) f(x) = ln – 8 + 2 30) ______
A) – 8 + 4 B) – 8x + 4x
C) – 8 + 4x D) – 8 + 2x
Find for the indicated function y.
31) y = 3 – x 31) ______
A) 6 – B) 6 + C) 3 – D) 6 –
Find the equation of the line tangent to the graph of f at the indicated value of x.
32) f(x) = 8 + ln x; x = 1 32) ______
A) y = x – 9 B) y = x + 9 C) y = x + 7 D) y = x – 7
Differentiate.
33) Find f'(x) for f(x) = (5 + 4)(3 – 5). 33) ______
A) f'(x) = 150 + 84 – 75 B) f'(x) = 20 + 84 – 75x
C) f'(x) = 150 + 84 – 75x D) f'(x) = 20 + 84 – 75
34) Find y’ for y = 34) ______
A) B) C) D)
Find the derivative.
35) y = (x-2 + x)-3 35) ______
A) = B) =
C) = D) =
