Math 1220
Final
December 10, 2015
Please answer all the questions below. The value of every question is
indicated at the beginning of it. In addition to the tables provided, you may
only use scratch paper, a non-graphing calculator and a notecard. No cell
phones, notes, books or music players are allowed during the exam.
Name:
UID:
1. (10 points) Consider the function
Z x
√
sin t 1 + cos2 t dt,
f (x) =
0≤x≤π
0
(a) (4 pts) Justify that f (x) has an inverse over [0, π].
0
(b) (4 pts) Find f ( π2 ), f −1 f ( π2 ) and [f −1 ] f ( π2 ) .
(c) (2 pts) Write down the equation of the tangent line to the curve y = f −1 (x) at the
point x = f ( π2 ) (don’t compute the integral).
2. (10 points) Compute the following integrals
(a)
Z
ex (9 − e2x )3/2 dx
(b)
Z
5
tan
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x
2
dx
3. (10 points) (a) (2 pts) Write down the decomposition in partial fractions that you
would use to compute the following integral (you don’t have to find the constants)
Z 3
x + x2 + 2x + 2
dx
(x2 + 1)2
(b) (8 pts) Compute the integral
Z
x3 + x2 + 2x + 2
dx
(x2 + 1)2
Hint: All the constants in (a) are equal to 1.
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4. (10 points) Compute the limit
lim tan x · ln | sin x|
x→0
If you use l’Hppital’s rule, please indicate where you are doing it and also what indeterminate form you are trying to resolve.
5. (10 points) (a) (6 pts) Use integration by parts to compute
(b) (4 pts) Compute the integral
Z
1
−1
ln |x|
dx
x2
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R
ln x
x2
dx.
6. (15 points) Show whether the following series diverge, converge absolutely or converge
conditionally. Please indicate clearly what test you are using and make sure you verify
explicitly all the conditions that the test requires.
(a) (5 pts)
X ln n
n≥1
(b) (5 pts)
√
X
n≥1
2n
2n + 1
n2
(c) (5 pts)
X
sin n
(−1)n √
n n
n≥1
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7. (10 points) (a) (5 pts) Show that the series
(b) Show that the series
P
n≥1
ln n 2
n
1
n≥2 n[ln n]2
P
converges.
converges by comparing it with the series in (a).
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x
8. (10 points) (a) (4 pts) Find the Maclaurin series for the function x2 −3x+2
P
1
= n≥0 xn for |x| < 1.
Hint 1: Use partial fractions. Recall that 1−x
P
Hint 2: The answer is n≥1 (1 − 2−n )xn .
(b) (3 pts) What is the convergence set of the series that you found? Please do the
explicit computation, even though you should already know the answer...
(c) (3 pts) Find the Maclaurin series for the function
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Rx
t
0 t2 −3t+2
dt
9. (10 points) (a) (6 pts) Use any method you want to find the Maclaurin polynomial of
order 2 (i.e. through terms x2 ) for the function
√
f (x) = sin x 1 + x
P
P
x2n+1
In case you need this, recall that sin x = n≥0 (2n+1)!
and (1+x)p = 1+ n≥1 np xn .
(b) (4 pts) What is the error that we would make when approximating f ( π2 ) by the
Maclaurin polynomial of order 2? It suffices to write down the expression that you
would plug into your calculator.
√
cos x
Hint: Assume that f 3) (x) = − sin x 1 + x + 4(1+x)
3/2 .
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10. (10 points) Consider the polar equation r = 4 sin 3θ, whose graph is shown below. Find
the area of one leaf.
Hint: Recall that the area enclosed by a curve r = f (θ) is given by
1
A=
2
Z
β
[f (θ)]2 dθ
α
for appropriate α, β.
Figure 1:
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