Final MAC2233 A
(7 pts) 1. Given y 2 x2 + 3x = 3y 3 , y is given by:
2xy 2 +3
A. 9y 2 2yx2
2xy 2 +3
2xy 2 3
B. 9y 2 +2yx2
2xy 2 +3
C. 9y 2 2yx2
D. 9y 3 +2yx2
2xy 2 +3
E. 9y 2 2y 2 x2
(7 pts) 2. Given h(x) = (4x2 3x ln x)3 , h (1) is equal to:
A. 120
B. 80
C. 240
(7 pts) 3. The denite integral
A. 11
6
B. - 11
6
2
2
D. 180
E. 360
5x3 6x dx is equal to:
5
C. 6
D. 7
6
E. 0
(7 pts) 4. An investor places $3000 in an account which compounds quarterly at a standard rate
of 8%; how long will it take for his investment to double?
ln
A. 4 ln 1202 years
.
ln
B. 3000600002 years
ln 1.
D. 4ln 23 years
ln
E. lnln.2 years
1 02
ln 6000
C. 12,000 ln 1.02 years
(7 pts) 5. Given f (x) = 2xex , which of the following statements are true:
I. f (x) 0 for all x
II. the function is decreasing for x > 1
III. the function is concave upward on (2, +)
IV. there is an inection point at x = 2
A. only III
B. only II and III
C. only II and IV
D. only II, III, and IV
E. I, II, III, and IV
(7 pts) 6. The area of the region completely enclosed by the graphs of f (x) = x2 3x and
g (x) = x is given by :
A. 1
3
4
B. 3
C. 5
3
D. 2
3
7
E. 3
(7 pts) 7. The maximum value of the function f (x) = x3 2x2 4x + 4 on the interval [0, 3] is
given by:
A. 1
B. 4
C. 4
D. 6
E. 8
(7 pts) 8. Approximate the area under the graph of g (x) = 4x + 8 over the interval [1, 2] by
using a Riemann sum, subdividing [1, 2] into six subintervals of equal length, and choosing the
right endpoint of each subinterval as the sample point. The approximation is equal to:
A. 20
B. 30
C. 15
D. 24
E. 42
(7 pts) 9. A coee pot in the form of a circular cylinder of radius 4 in is being lled with water
at a constant rate. If the water level is rising at a rate of 0.4 in/sec, how fast water is owing into
the pot? (Note the volume of a cylinder is given by V = r2 h):
A. 5.6 in3 /sec
B. 6.4 in3 /sec
C. 12.8 in3 /sec
D. 4.8 in3 /sec
E. 2.4 in3 /sec
(7 pts) 10. The average rate of change of the function f (x) = 4x ln x on the interval [1, e] is
given by:
4
B. e1
A. 4
C. e4e1
4(e+1)
D. e1
E.
4e(e+1)
e1
(7 pts) 11. The function g (x) has vertical asymptotes at x = 1 and a horizontal asymptote at
y = 2; is concave upward on (, 1) (1, +) and concave downward on (1, 1); and is increasing on (, 1) (1, 0) and decreasing on (0, 1) (1, ). Which of the following is equal to g (x)?
2
A. (x2x1)2
2(x2 +3x+2)
B. (x1)(x+1)
2x
C. (x1)(x+1)
2
D. x2x 1
3
(7 pts) 12. The equation of the tangent line to the function f (x) =
A. y = 5x + 4
B. y = 7x 4
C. y = 4x + 8
2
E. x2x 1
2
x(2x+4) at x = 4 is given by
D. y = 3x + 12
E. y = 6x
(7 pts) 13. Let x and y be any two positive numbers such that xy = 80; the smallest possible
value of the sum 4x + 20y is given by:
A. 80
B. 120
C. 160
D. 180
E. 200
(7 pts) 14. Which of the following applications of the chain rule is (are) correct:
f (x)
d
I. dx ln(f (x)) = f (x)
d
II. dx ef (x) = f (x) ef (x)
d
III. dx [f (x)]n = n[f (x)]n1 f (x)
A. only III
B. only II and III
C. only I and III
D. only I and II
E. I, II, and III
(7 pts) 15. A calculator manufacturer has an estimated marginal prot given by
P (x) = 0.5x + 100
where the units for P (x) are dollars/calculator/month when the production level is x calculators
per month. If the xed cost for producing and selling these calculators is $6,000/month, the maximum monthly prot is:
A. $5,000
B. $4,000
C. $6,000
D. $4,500
E. $5,500
Solutions:
1. A
13. C
2. C
14. E
3. E
4. A
15. B
5. D
6. B
7. C
8. C
9. B
10. C
11. E
12. B