MATH 101 SAMPLE MIDTERM 2
For full credit, please show all work. Time: 50 minutes.
Books, notes, calculators, or other aids are not allowed.
1. (18 marks, 6 for each part) Evaluate the integrals:
Z
(a) xe−2x dx
Z
(b)
x
Z
(c)
p
9 − x2 dx
ex dx
e2x − 3ex + 2
2. (12 marks, 6 for each part) (a) Evaluate the integral
Z
1/2
0
dx
,
x ln x
or show that it diverges.
Z
(b) Use the comparison test to decide whether the integral
∞
(x3 + x)−1/2 dx is conver-
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gent.
3. We want to evaluate numerically the integral
Z
2
1
sin x
dx
x
using the Midpoint Approximation.
(a) (4 marks) Write out (but do not evaluate) the approximation M5 .
(b) (6 marks) Find an n such that Mn approximates the actual value of the integral with
accuracy 10−5 or better. (The error estimate for the Midpoint Rule is K(b − a)2 /24n2 ,
where K = max |f 00 (x)| on [a, b].)
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