APPM 1350 Midterm #3 Fall 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 5 questions.
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Show all work! Answers with no justication will receive no points.
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Please begin each problem on a new page.
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No notes, calculators, or electronic devices are permitted.
1. (21 points) Evaluate the following integrals.
(a)
R1
03u
1+u2du
(b)
Rsin2(6θ+ 4) dθ
(Hint: You may nd the following identity useful:
cos(2θ)=1−2 sin2(θ)
.)
(c)
Rπ
0esec(2t)sec(2t) tan(2t)dt
2. (28 points) Find the requested information:
(a) Evaluate
Pn
j=1(2j+ 3)
. Your nal answer should be in terms of
n
.
(b) Use logarithmic dierentiation to nd
dy
dx
for
y=(3x+5)
10
3(2x2+9)
3
2
(5x−4)
7
5
. Find the expression for
dy
dx
but do not simplify your answer further.
(c) Find the total area between the curve
y= 4x3−4
and the x-axis on the interval
[0,2]
.
(d) Find the average value of
g(x)=4x3−4
on the interval
[0,2]
.
3. (14 points)
(a) Carefully state the Fundamental Theorem of Calculus, parts 1 and 2. Be sure to include
the hypotheses of the theorem in your answer.
(b) Find the linearization at
x= 1
for
f(x) = 4 + 2 Rx2+1
21
2+tdt
.
4. (21 points) Estimate the area under the curve
f(x) = x2
on the interval [1,4]
(a) using a Riemann sum. Use a partition with three subintervals of equal length, and
evaluate
f(x)
at the left end points of your subintervals.
(b) using the trapezoidal rule with three intervals.
(c) Use
ET
to give an upper bound for the error in the trapezoidal rule in part 4(b).
5. (16 points)
(a) Show that the function
f(x) = e3x+1 −2
is invertible for
x∈(−∞,∞)
.
(b) Find the inverse,
f−1(x)
, for
f(x)
in part (a).
(c) What is the domain and range for
f−1(x)
?