APPM 1340 Midterm #2 Spring 2009.
On the front of your bluebook, please write: a grading key, your name, student ID, and
section and instructor. This exam is worth 100 points and has 7 questions. Notice, that there are 110
possible points available, so it is possible to earn up to 10 extra credit points. Show all work! Answers
with no justification will receive no points. Please begin each problem on a new page. No Calculators.
No Notes.
1. (15 points) Express the definite integral !b
a
f(x)dx as a Limit of Riemann Sums.
2. (10 points)
(a) Express the sum
5
"
k=1
k(2k−1) without sigma notation and then evaluate the sum.
(b) Express the sum −1
2+1
4−1
8+1
16 in sigma notation. Start your summation with lower limit
k= 1.
3. (15 points) Mean Value Theorem
(a) State the Mean Value Theorem for Definite Integrals. Include both the hypotheses and the
conclusion of the theorem.
(b) Suppose f(x) is continuous and that !6
1
f(x)dx = 4. Find the average (or mean) value of f(x)
on the interval [1,6]. Show that f(x) attains its average (or mean) value on [1,6].
4. (20 points)
(a) State the Fundamental Theorem of Calculus parts 1 and 2. Include both the hypotheses and
the conclusion.
(b) Find dy
dx of the function y=!sin x
0
dt
√2−t2
5. (15 points) Find the total area of the region between the graph of y=x2−6x+ 8 and the x-axis on
the interval −1≤x≤3.
6. (20 points) Evaluate each of the following integrals.
(a) !4
9
1−√x
√xdx
(b) !π/2
π/4
cot θcsc2θdθ
(c) !1
0
54
√t
(1 + t5/4)2dt
(d) !π/4
0
(1 −sin 2r)3/2cos 2rdr
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