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APPM 1360
Exam 1
Spring 2016
On the front of your bluebook, please write: a grading key, your name, student ID, your lecture number
and instructor. This exam is worth 100 points and has 6 questions on both sides of this paper.
• Submit this exam sheet with your bluebook. However, nothing on this exam sheet will be graded. Make
sure all of your work is in your bluebook.
• Show all work and simplify your answers! Answers with no justification will receive no points.
• Please begin each problem on a new page.
• No notes or papers, calculators, cell phones, or electronic devices are permitted.
1. (30 points, 10 points each) Evaluate the following integrals.
Z
sin θ
(a)
dθ
cos2 θ + cos θ − 2
Z
1
(b)
dt
2
t − 8t + 20
Z
1
(c)
dx
2
(x − 1)3/2
Z π/4
sec2 x
√ dx convergent or divergent? Fully justify your answer.
2. (10 points) Is
x
0
Z 4
3. (20 points) For this problem let I =
ln x2 dx.
2
(a) Estimate I using the trapezoidal approximation T2 . Simplify your answer using the approximations
ln 2 ≈ 0.7 and ln 3 ≈ 1.1.
(b) Now find the exact value of I.
(c) How many subintervals are needed to ensure that a trapezoidal approximation of I is accurate to
within 10−4 ? Simplify your answer.
4. (12 points) Consider the region enclosed by the line y = x − 1 and the parabola y 2 = 2x + 6.
(a) Sketch this region.
(b) Set up, but don’t evaluate, the integral to calculate the area enclosed by this region.
5. (16 points) Consider the region bounded by y = cos x and y = tan x from x = 0 to π6 .
(a) Sketch this region.
(b) Find the volume of the solid generated when this region is rotated about the x-axis.
TURN OVER - More problems on the back!
6. (12 points) Unrelated, short answer questions.
x2 + 5
. (Note: Do not try to solve for the
(x − 1)2 (x2 + 1)3
coefficients. This problem is just asking about the form of the partial fraction decomposition.)
1
(b) True or False and briefly explain your answer: Let f (x) = p and p > 0 be a constant. If
x
Z
Z
(a) Write the partial fraction decomposition of
∞
1
f (x) dx converges.
f (x) dx diverges, then
0
1
(c) Z
True or False and briefly explain your
Z ∞answer: If f (x) is continuous for all x ∈ [0, ∞) and if
∞
f (x) dx diverges, then so does
f (x) dx for all a > 0.
0
a
Some Trigonometric identities
2 cos2 (x) = 1 + cos(2x)
2 sin2 (x) = 1 − cos(2x)
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
Inverse Trigonometric Integral Identities
Z
du
√
= sin−1 (u/a) + C, u2 < a2
2
2
a −u
Z
du
1
= tan−1 (u/a) + C
2
2
a
Z a +u
du
1
√
= sec−1 |u/a| + C, u2 > a2
a
u u2 − a2
Midpoint Rule
Z
b
f (x)dx ≈ Mn = ∆x[f (x̄1 ) + f (x̄2 ) + · · · + f (x̄n )] where ∆x =
a
|EM | ≤
b−a
xi−1 + xi
and x̄i =
and
n
2
K(b − a)3
.
24n2
Trapezoidal Rule
Z
b
f (x)dx ≈ Tn =
a
∆x
b−a
K(b − a)3
[f (x0 )+2f (x1 )+· · ·+2f (xn−1 )+f (xn )] where ∆x =
and |ET | ≤
.
2
n
12n2