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Math 1325 Review
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the limit, if it exists.
1) Find: limx 3
x - 3x2 - 3x
1)
A) 0 B) -13
C) 13
D) Does not exist
2) Let f(x) =
x2 - 16x + 4
if x > 0
x2 - 16x - 4
if x < 0
Find limx 0
f(x).
2)
A) B) -4 C) 4 D) Does not exist
3) Find: limh 0
f(7 + h) - f(7)h
for f(x) = -x + 1. 3)
A) 1 B) 0 C) -1 D) Does not exist
Provide an appropriate response.4) If the limit at infinity exists, find the limit.
limx
5x2 + 7x - 9- 6x2 + 2
4)
A) 0 B) -29
C) -56
D)
5) Find the vertical asymptote(s) of the graph of the given function.
f(x) =x2 - 100
(x - 9)(x + 3)
5)
A) x = -9 B) x = 10, x = -10 C) y = 9, y = -3 D) x = 9, x = -3
6) Solve the inequality and express the answer in interval notation: x2 - 4xx + 5
> 0. 6)
A) (4, ) B) (-5, 0) (4, ) C) (-5, 0) D) (-5, )
1
Find dy.7) y = 9x2 - 8x + 9 7)
A) 18x + 9 dx B) 18x - 16 dx C) 18x dx D) (18x - 8) dx
Provide an appropriate response.8) The revenue (in thousands of dollars) from producing x units of an item is modeled by
R(x) = 5x - 0.0005x2. Find the marginal revenue at x = 1000.8)
A) $4.00 B) $10,300.00 C) $104.00 D) $4.50
Solve the problem.9) A company is planning to manufacture a new blender. After conducting extensive market
surveys, the research department estimates a weekly demand of 600 blenders at a price of $50per blender and a weekly demand of 800 blenders at a price of $40 per blender. Assuming thedemand equation is linear, use the research department's estimates to find the revenue equationin terms of the demand x.
9)
A) R(x) = 80x - 20x2 B) R(x) = 20x +x220
C) R(x) = 80x - 20 D) R(x) = 80x -x220
Differentiate.10) Find f'(x) for f(x) = (2x - 4)(2x3 - x2 + 1). 10)
A) f'(x) = 12x3 + 30x2 - 10x + 2 B) f'(x) = 16x3 - 30x2 + 8x + 2C) f'(x) = 4x3 - 10x2 - 30x + 2 D) f'(x) = 16x3 - 10x2 + 30x + 2
11) Find y' for y =x2
2 - 5x11)
A) -15x2 + 4x(2 - 5x)2
B) -5x2 + 4x(2 - 5x)2
C) 5x3 - 10x2 + 4x(2 - 5x)2
D) 2x(2 - 5x)2
Find the derivative.
12) Find: dydx
88x7 - 10 12)
A) 448x67
8x7 - 10 B) 56x6
(8x7 - 10)7/8
C) 7x6
(8x7 - 10)7/8D) 8
78x7 - 10
2
Provide an appropriate response.13) Find f'(x) for f(x) = (x2 + 2)3 . 13)
A) f'(x) = 6x5 + 24x3 + 24x B) f'(x) = 6x5 + 20x3 + 24xC) f'(x) = 6x5 + 12x3 + 12x D) f'(x) = 3x5 + 24x3 + 24x
14) Find dydx
for y = ln (6x3 - x2) 14)
A) 18x - 26x3 - x
B) 6x - 26x2 - x
C) 18x - 26x2
D) 18x - 26x2 - x
Find the equation of the tangent line to the graph of the given function at the given value of x.15) f(x) = (x2 + 12)3/4; x = 2 15)
A) y =32
x B) y =32
x + 11 C) y =32
x + 5 D) y =34
x + 5
Provide an appropriate response.16) Find y' for y = y(x) defined implicitly by 3xy - x2 - 4 = 0. 16)
A) y' =3x -2y
4B) y' =
3y - 2x3x
C) y' =23
x D) y' =2x - 3y
3x
17) Assume x = x(t) and y = y(t). Find dxdt
if x2(y - 6) = 12y + 3 and dydt
= 2 when x = 5 and y = 12. 17)
A) -2013
B) -1330
C) 2013
D) 1320
3
Find the elasticity of the demand function as a function of p.
18) x = D(p) =600
(p + 2)218)
A) E(p) =2p
p + 2B) E(p) = 1200p(p + 2)
C) E(p) =1200p(p + 2)3
D) E(p) =2
p + 2
Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at theindicated values of p.
19) x = f(p) = 1500 - 5p2; p = 10 19)A) Inelastic B) Unit elasticity C) Elastic
Solve the problem.20) Given the revenue and cost functions R = 26x - 0.3x2 and C = 3x + 10, where x is the daily
production, find the rate of change of profit with respect to time when 20 units are producedand the rate of change of production is 7 units per day per day.
20)
A) $77.00 per day B) $149.00 per dayC) $156.80 per day D) $98.00 per day
4
Use the given graph of f(x) to find the intervals on which f (x) > 0.21) 21)
A) f (x) > 0 on (- , 2], f (x) < 0 on [2, )B) f (x) > 0 on (- , -2] [2, ), f (x) < 0 on [-2, 2]C) f (x) > 0 on [-2, 2], f (x) < 0 on (- , -2] [2, )D) f (x) > 0 on [-4, 4], f (x) < 0 on (- , -4] [4, )
Provide an appropriate response.22) Use the first derivative test to determine the local extrema, if any, for the function:
f(x) = 3x4 - 6x2 + 7.22)
A) local max at x = 0 and local min at x = -1 and x = 1B) local min at x = 0 and local max at x = -1 and x = 1C) local max at x = -1 and local min at x = 0 and x = 1D) local max at x = 1 and local min at x = 0
23) Use the first derivative test to determine the local extrema, if any, for the function:f(x) = 3(x - 4)2/3 + 6.
23)
A) f(x) has a local minimum at 6 B) f(x) has a local maximum at x = 4.C) f(x) has a local minimum at x = 4. D) f(x) has no local extrema
24) Find the critical values and determine the intervals where f(x) is increasing and f(x) is
decreasing if f(x) = 1 +3x
+2x2
.
24)
A) decreasing on (-4, 0); increasing on ( , -4) (0, )B) increasing on (-4, 0); decreasing on ( , -4) (0, )
C) increasing on -43
, 0 ; decreasing on , -43
(0, )
D) decreasing on -43
, 0 ; increasing on , -43
(0, )
5
Solve the problem.25) The annual revenue and cost functions for a manufacturer of grandfather clocks are
approximately R(x) = 520x - 0.01x2 and C(x) = 200x + 100,000, where x denotes the number ofclocks made. What is the maximum annual profit?
25)
A) $2,660,000 B) $2,760,000 C) $2,560,000 D) $2,460,000
Find the intervals where f''(x) < 0 ir f''(x) > 0 as indicated.26) f''(x) < 0 26)
A) (-1, 0) B) (-1, 0) , (1, ) C) (- , -1) D) (1, )
Provide an appropriate response.27) Find f"(x) for f(x) = (4x + 5)3 . 27)
A) f"(x) = 12x + 15 B) f"(x) = 4x + 5C) f"(x) = 24x + 30 D) f"(x) = 384x + 480
28) Find y'' for y = x4 - 8x1/2 28)
A) 4x3 -4x
B) 12x2 +2
x xC) 4x3 +
4x
D) 12x2 -4x
29) Find the inflection point(s) for f(x) =14
x4 - x3 + 6. 29)
A) (0, 6) and (2, -4) B) (0, 0) and (2, 2)C) (0, 6) and (2, 2) D) (0, 0)
Solve the problem.30) A company estimates that it will sell N(t) hair dryers after spending $t thousands on
advertising as given by:
N(t) = -3t3 + 450t2 - 21,600t + 1,100, 40 t 60
For which values of t is the rate of sales N'(t) increasing?
30)
A) 50 < t < 60 B) 40 < t < 60 C) t > 40 D) 40 < t < 50
6
Find the limit, if it exists.
31) Find limx +
x2
ex . 31)
A) 0 B) C) ex D) -
Solve the problem.32) The total cost, in dollars, of producing x cell phones is approximated by the function
C(x) = 2000 - 30x +x25
. Find the minimum average cost.
32)
A) The minimum average cost is $75 when x = 875 cell phones.B) The minimum average cost is $10 when x = 100 cell phones.C) The minimum average cost is $875 when x = 75 cell phones.D) The minimum average cost is $74 when x = 20 cell phones.
Provide an appropriate response.
33) Find the absolute minimum value of f(x) =ex
x3 for x > 0. Round your answer to three decimal
places.
33)
A) 0.7439 at x = 3 B) 3 at x = 0.7439 C) 1 at x = 2.718 D) 2.718 at x = 1
34) The percent of concentration of a acid absorbed in a new manufacturing process after x hr after
the acid has been mixed is given by A(x) =4x
x2 + 49 How long after the acid has been added is the
concentration a maximum? Round answer to the nearest tenth, if necessary.
34)
A) 4 hr B) 7 hr C) 4.9 hr D) 2.5 hr
Solve the problem.35) The annual revenue and cost functions for a manufacturer of zip drives are approximately
R(x) = 520x - 0.02x2 and C(x) = 160x + 100,000, where x denotes the number of drives made.What is the maximum annual profit?
35)
A) $1,520,000 B) $1,720,000 C) $1,620,000 D) $1,820,000
36) The average manufacturing cost per unit (in hundreds of dollars) for producing x units of aproduct is given by:C(x) = 2x3 - 42x2 + 288x + 12, 1 x 5At what production level will the average cost per unit be maximum?
36)
A) 1 unit B) 5 units C) 12 units D) 652 units
7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
37) A logo baseball cap manufacturer has a uniform annual demand of 25,000 caps. It costs$1 to store one baseball cap for 1 year and $500 to set up the plant for production of thelogo baseball caps. How many times a year should the company produce the caps inorder to minimize the total storage and set-up costs? (Assume that there are 250working days per year.)
37)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the integral.
38) (2x5 - 7x3 + 9) dx 38)
A) 6x6 -73
x4 + 9x + C B) 12
x6 -73
x4 + 9x + C
C) 13
x6 -74
x4 + 9x + C D) 6x6 -74
x4 + 9x + C
39) x3 - 7x
dx 39)
A) 14
x4 - 7x2 + C B) 13
x3 + 7 ln x + C
C) 13
x3 -7
x-2+ C D) 1
3x3 - 7 ln x + C
40) 5ex -1x
dx 40)
A) 5ex -2x2
+ C B) 5xex - ln x + C
C) 5ex -1
2x2+ C D) 5ex - ln x + C
8
Provide an appropriate response.
41) Find f(x) if f'(x) =3x5
and f 12
= 1. 41)
A) -45
x-4 + 13 B) -34
x-4 + 13 C) -34
x-4 + C D) -34
x-4 +54
Solve the problem.42) A company finds that consumer demand quantity changes with respect to price at a rate given
by D'(p) = -3500p2
. Find the demand function if the company knows that 829 units of the
product are demanded when the price is $5 per unit.
42)
A) D(p) =3500
p+ 129 B) D(p) =
3500p3
+ 129
C) D(p) =7000
p+ 829 D) D(p) =
3500p
+ 829
Find the integral.
43) e7x +4 dx 43)
A) 17
e7x + 4 + C B) (7x + 4) e7x + 4 + C
C) e7x + C D) 7e7x + 4 + C
44) ln 8xx
dx 44)
A) (ln 8x)28
+ C B) (ln 8x)2 + C C) (ln 8x)216
+ C D) (ln 8x)22
+ C
45) t53
1 + t6 dt 45)
A) 112(1 + t6)2
+ C B) 14
t6 (1 + t6)2/3 + C
C) 14
(1 + t6)2/3 + C D) 18
(1 + t6)2/3 + C
9
Solve the problem.46) A manufacturing company is ready to introduce a new product with a national sales
campaign. After extensive test marketing, the market research department estimates that sales(in millions of dollars) will increase at the monthly rate of S'(t) = 10 - 10e-0.2t for 0 t 24, tmonths after the national campaign has started. What will the total sales be five months afterthe beginning of the campaign if we assume zero sales at the beginning of the campaign?(Round the answer to the nearest million.)
46)
A) $2 million B) $49 million C) $1 million D) $18 million
Provide an appropriate response.47) Find the particular solution for the differential equation y' = 4x + 7; y(0) = -12. 47)
A) y = 2x2 + 7x - 6 B) y = 4x2 + 7x - 12C) y = 4x2 + 7x - 6 D) y = 2x2 + 7x - 12
Evaluate the integral.
48)e
116x -
5x
dx
(Express your answer in terms for e.)
48)
A) 8e2 - 13 B) 8e2 - 8 C) 8e2 - 5 D) 16e2 - 5
49)2
0
4x + 14x2 + 2x + 2
dx
(Round to three decimal places.)
49)
A) 1.040 B) 1.778 C) 1.199 D) 2.398
50)3
1(2x3 - 9x-2) dx 50)
A) 52.5 B) 34 C) 76 D) 58
Provide an appropriate response.51) Find the average value of the function g(x) = 32e0.04x over the interval [10, 30]. Round your
answer to two decimal places.51)
A) 1462.63 B) 73.13 C) 2.29 D) 45.71
52) Find the area bounded by f(x) = x2 - 4x - 5 and y = x + 1. 52)
A) 54 B) 3436
C) 236
D) 3016
10
Solve the problem.
53) The Lorenz curve for the income distribution in a certain country is given by f(x) =34
x2 +14
x.
I) Find the Gini index.II) Use the answer found in I) to determine if the income of this country is more equallydistributed, less equally distributed, or distributed the same as a second country having anindex of income concentration of 0.2.
53)
A) I) 0.33II) more equally distributed
B) I) 0.33II) less equally distributed
C) I) 0.25II) more equally distributed
D) I) 0.25II) less equally distributed
54) Find the total income produced by a continuous income stream in the first four years if the rateof flow is f(t) = 500e0.03t. (Round answer to the nearest dollar.)
54)
A) $2486 B) $2125 C) $18,792 D) $564
55) Find the equilibrium price and quantity,producers' surplus for p = D(x) = 71 -110
x and
p = S(x) = 35 +120
x.
55)
A) p = 47q = 240
B) p = 50q = 240
C) p = 47q = 180
D) p = 47q = 288
56) Find the consumers' surplus and producers' surplus for p = D(x) = 71 -110
x and
p = S(x) = 35 +120
x.
56)
A) CS = $14,160PS = $1440
B) CS = $15,160PS = $1440
C) CS = $2880PS = $1440
D) CS = $2880PS = $1660
Evaluate using integration by parts.
57) 2xex dx 57)
A) 2xex - 2ex + C B) 2ex - 2xex + C C) xex - 2ex + C D) 2ex - ex + C
11
58)4
26x ln x dx 58)
A) 9.48 B) 6.70 C) 55.2 D) 40.2
Find the indefinite integral using a table of integration formulas.
59) 1
x2 - 4dx 59)
A) ln x + x2 + 4 + C B) ln x + x2 - 4 + C
C) 14
ln 2 + x2 - x
+ C D) 14
ln x - 2x + 2
+ C
Provide an appropriate response.
60) Use the integral table to find x e3x dx . 60)
A) x e3x -e3x3
+ C B) x e3x3
+e3x9
+ C
C) x e3x3
-e3x9
+ C D) x e3x3
-e3x3
+ C
12
Answer KeyTestname: MATH 1325 FINAL REVIEW
1) D2) C3) C4) C5) D6) B7) D8) A9) D
10) B11) B12) C13) A14) D15) C16) D17) B18) A19) B20) A21) C22) A23) C24) C25) D26) B27) D28) B29) C30) D31) A32) B33) A34) B35) A36) B37) The company will minimize its costs by making 5000 caps five times during the year.38) C39) D40) D41) B42) A43) A44) D45) C46) D47) D48) A49) C50) B
13
Answer KeyTestname: MATH 1325 FINAL REVIEW
51) B52) B53) D54) B55) A56) C57) A58) D59) B60) C
14
