Math 1325 – Final Exam Chapters 11, 12, 13, 14
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the partial derivative as requested. 1) fy(5, -6) if f(x,y) = 7×2 – 9xy 1)
A) 129 B) -54 C) -45 D) 39
Find the second-order partial derivative. 2) Find fyx when f(x,y) = 8x3y – 7y2 + 2x. 2)
A) 48xy B) -14 C) -28 D) 24×2
Solve the problem. 3) The profit from the expenditure of x thousand dollars on advertising is given by
P(x) = 930 + 25x – 4×2. Find the marginal profit when the expenditure is x = 9. 3)
A) 225 thousand dollars/unit B) 153 thousand dollars/unit C) 930 thousand dollars/unit D) -47 thousand dollars /unit
4) Find C(x) if C'(x) = 5×2 – 7x + 4 and C(6) = 260. 4)
A) C(x) = 5 3
x3 – 7 2
x2 + 4x + 2 B) C(x) = 5 3
x3 – 7 2
x2 + 4x – 2
C) C(x) = 5 3
x3 – 7 2
x2 + 4x – 260 D) C(x) = 5 3
x3 – 7 2
x2 + 4x + 260
5) The revenue generated by the sale of x bicycles is given by R(x) = 50.00x – x 2
200 . Find the marginal
revenue when x = 600 units.
5)
A) $12.00/unit B) $50.00/unit C) $56.00/unit D) $44.00/unit
6) The rate at which an assembly line worker’s efficiency E (expressed as a percent) changes with respect to time t is given by E'(t) = 70 – 6t, where t is the number of hours since the worker’s shift began. Assuming that E(1) = 92, find E(t).
6)
A) E(t) = 70t – 3t2 + 25 B) E(t) = 70t – 3t2 + 92 C) E(t) = 70t – 6t2 + 25 D) E(t) = 70t – 3t2 + 159
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Identify the intervals where the function is changing as requested. 7) Increasing 7)
A) (-2, -1) (2, ) B) (-1, ) C) (-2, -1) D) (-1, 2)
Determine the location of each local extremum of the function. 8) f(x) = -x3- 4.5×2 + 12x + 4 8)
A) Local maximum at 1; local minimum at -4 B) Local maximum at -4; local minimum at 1 C) Local maximum at -1; local minimum at 4 D) Local maximum at 4; local minimum at -1
Find the equation of the tangent line to the curve when x has the given value. 9) f(x) = 5×2 + x ; x = -4 9)
A) y = x 20
+ 1 5
B) y = 13x – 16 C) y = -39x – 80 D) y = – 4x 25
+ 8 5
Find the largest open interval where the function is changing as requested. 10) Increasing f(x) = x2 – 2x + 1 10)
A) (- , 0) B) (0, ) C) (- , 1) D) (1, )
Find dy/dx by implicit differentiation. 11) 2xy – y2 = 1 11)
A) x y – x
B) x x – y
C) y x – y
D) y y – x
Find the area of the shaded region. 12) 12)
A) 5 3
B) 3 C) 5 D) 23 3
Use the properties of limits to evaluate the limit if it exists.
13) lim x 6
x + 6 (x – 6)2
13)
A) 0 B) 6 C) -6 D) Does not exist
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14) lim x 0
x3 + 12×2 – 5x 5x
14)
A) 0 B) Does not exist C) -1 D) 5
Evaluate.
15) 34 x2
dx 15)
A) 34x + C B) 34 x
+ C C) -34x + C D) – 34 x
+ C
Find the integral.
16) 19 2 + 5y
dy 16)
A) 18 5
ln 2 + 5y + C B) 19 5
ln 2 + 5y + C
C) 19 ln 2 + 5y + C D) 18 ln 2 + 5y + C
17) 8x – 9x-1 dx 17)
A) 4×2 – 9 ln x + C B) 4×2 + 9 2
x-2 + C
C) 16×2 – 9 ln x + C D) 16×2 + 9 2
x-2 + C
18) x dx
(7×2 + 3)5 18)
A) – 1 56
(7×2 + 3)-4 + C B) – 1 14
(7×2 + 3)-6 + C
C) – 7 3
(7×2 + 3)-4 + C D) – 7 3
(7×2 + 3)-6 + C
19) 9z 3z2 – 7 dz 19)
A) z(3z2 – 7)3/2 + C B) (3z2 – 7)3/2 + C
C) 1 2
z(3z2 – 7)3/2 + C D) 1 2
(3z2 – 7)3/2 + C
Find the absolute extremum within the specified domain. 20) Maximum of f(x) = x2 – 4; [-1, 2] 20)
A) (-1, 3) B) (-2, 0) C) (1, -3) D) (2, 0)
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Assume x and y are functions of t. Evaluate dy/dt.
21) x3 + y3 = 9; dx dt
= -5, x = 2 21)
A) 20 B) 5 4
C) 4 5
D) – 20
Use the given graph to determine the limit, if it exists. 22)
lim x 0-
f(x) and lim x 0+
f(x).
22)
A) -1; 1 B) 1; -1 C) 1; 1 D) -1; -1
Find the derivative of the function. 23) y = (3×2 + 5x + 1)3/2 23)
A) y’ = (6x + 5)(3×2 + 5x + 1)1/2 B) y’ = (3×2 + 5x + 1)1/2
C) y’ = 3 2
(3×2 + 5x + 1)1/2 D) y’ = 3 2
(6x + 5)(3×2 + 5x + 1)1/2
24) y = ln (3×3 – x2) 24)
A) 3x – 2 3×2 – x
B) 9x – 2 3×3 – x
C) 9x – 2 3×2 – x
D) 9x – 2 3×2
Find the derivative.
25) y = e5x2 + x 25)
A) 10xe + 1 B) 10xe2x + 1 C) 10xex2 + 1 D) 10xe5x2 + 1
26) f(x) = 20×1/2 – 1 2
x20 26)
A) 10×1/2 – 10×19 B) 10×1/2 – 10×10 C) 10x-1/2 – 10×19 D) 10x-1/2 – 10×10
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Find the general solution of the differential equation.
27) dy dx
= x – 2 27)
A) x 2
2 – x + C B) x3 – 2x + C C) 2×2 – 2 + C D) x
2 2
– 2x + C
Evaluate f”(c) at the point.
28) f(x) = 3x – 4 4x – 3
, c = 1 28)
A) f”(1) = -56 B) f”1) = 7 C) f”(1) = 44 D) f”(1) = 32
29) f(x) = ln (4x – 3), c = 1 29) A) f”(1) = 1 B) f”(1) = 0 C) f”(1) = 4 D) f”(1) = -16
Find the largest open intervals where the function is concave upward. 30) f(x) = x3 – 3×2 – 4x + 5 30)
A) (- , 1) B) None C) (1, ) D) (- , 1), (1, )
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