Math 165 Section F1
Professor Lieberman
November 10, 2011
PRACTICE THIRD IN-CLASS EXAM
Carry out the solution of each problem: show steps of any required calculations and state
reasons that justify any conclusions. A short sentence is usually enough but answers without
any justification will receive no credit.
1. Write the sum
1 + 2 + 3 + · · · + 41
∑
in sigma notation. (That means that your answer should have a
in it.)
2. Fund the intervals on which the graph of y = f (x) is increasing if
∫ x
1−t
f (x) =
dt.
2
0 1+t
∑
3. (25 points) Calculate the Riemann sum ni=1 f (x̄i )∆xi for f (x) = x2 on the interval
[0, 2] if the interval is divided into 4 equal subintervals, and x̄i is the midpoint.
Formulas that may be useful:
n
n
n
∑
∑
∑
n(n + 1)
n(n + 1)(2n + 1)
1 = n,
i=
,
i2 =
.
2
6
i=1
i=1
i=1
4. Use the Substitution Rule for Definite Integrals to evaluate
∫ 1
(x + 1)(x2 + 2x)2 dx.
0