Exam III Sample Questions (from OLD exams)
1. Which of the following is an antiderivative of f (x) = xex ?
(a) (x − 1)ex + C
1 2 x
x e +C
2
1
(c) − x2 ex + C
2
(d) (x + 1)ex + C
(b)
(e) None of these
2. A function f has derivative f 0 (x) = 2x + 3 and passes through the point (1, 2). What is f (x)?
(a)
(b)
(c)
(d)
(e)
y = 5x − 3
None of these
y = x2 + 3x + 2
y = x2 + 3x − 2
y = 2x2 + 3x − 3
3. Use areas to compute
ˆ
2
p
4 − x2 dx
0
(a) None of these
4
3
(c) 2
(b)
(d) 2π
(e) π
4. The graphs of f and g are shown below. If the curves intersect at the points (−1, −5), (2, 4), and (4, 8), what
is the area of the shaded region?
ˆ 4
ˆ 8
(a)
(f (x) − g(x)) dx +
(g(x) − f (x)) dx
−5
4
(b) None of these.
(c)
(d)
(e)
5. If f (4) = 10 and
ˆ
6
ˆ
4
(g(x) − f (x)) dx
−1
ˆ
ˆ
2
(f (x) − g(x)) dx +
−1
ˆ
4
(g(x) − f (x)) dx
2
8
(g(x) − f (x)) dx
−5
f 0 (x) dx = 5,what is f (6)?
4
(a) 15
(b) 5
(c) 50
(d) 2
(e) Insucient information to answer
6. Evaluate
ˆ
√
5
x3
p
25 − x4 dx
0
(a)
125
12
(b) None of these
125
6
8(125)
(d)
3
25(125)
(e)
6
(c)
7. Find the value of b such that the average value of f (x) = 2 + 10x on the interval [0, b] is equal to 6.
√
−2 + 124
(a) b =
10
(b) No value of b exists
(c) None of these
4
5
2
(e) b =
5
(d) b =
8. Approximate
ˆ
4
ln x dx using 3 equal subintervals and a left endpoint estimate.
2
(a) ln(2) + ln(3) + ln(4)
8
10
(b) ln(2) + ln
+ ln
3
3
(c) None of these
2
ln(2) +
3
2
(e) ln(2) +
3
(d)
9. Evaluate
(a)
(b)
(c)
(d)
(e)
ˆ
2
2
ln(3) + ln(4)
3
3
2
8
2
10
ln
+ ln
3
3
3
3
ˆ
6
|x| dx +
−2
None of these
14
10
6
2
6
4
|x| dx.
10. Evaluate each of the following (work by hand required):
(a)
ˆ 1
3
e + 2−
x
x
x
dx
(b) If marginal cost is given by C 0 (x) = 150 − .01ex and xed costs are $1000, nd the cost function C(x).
(c)
(d)
ˆ
1
(x2 + 2x + 3) dx
ˆ0
4x
dx
+1
x2
11.
(a) Find an upper estimate by hand for
ˆ
1
3
1
dx using 4 equal subintervals.
x
1
(b) Sketch the graph of f (x) = on the interval [1, 3] and illustrate your answer in part (a).
x
(c) Find the exact value of this integral.
12. Write and compute an integral to nd the area between the graphs of y = x2 − 2x and y = x.
13. The demand for tax software is given by p = D(x) = 45 − 18x2 , where x is in thousands of items and p is in
dollars. The supply equation is given by p = S(x) = 12x2 + 15x.
(a) Find the Producers' Surplus and Consumers' Surplus.
(b) Sketch a graph of the supply and demand curves on the interval [0, 1.5]. Shade in the part of the graph
that represents the Consumers' Surplus.
14. A person slams on the brakes of their car. Their velocity, in feet per second, is given in the following table:
time (sec) 0
1
2
3 4
vel (ft/s) 50 40 25 15 5
(a) Find a lower estimate of the distance traveled during the four-second interval.
(b) The data can be modeled by y = −11.5x + 50, where x is in seconds and y is in feet per second.Find an
upper estimate of the distance (in feet) the car has traveled during the four-second interval using 8 equal
subintervals.
(c) Using the model, nd the exact distance the car has traveled during the rst 4 seconds.