Problem Points Score
1
25
2ab
30
2c
19
Section:
3
30
Instructor:
4
30
5
30
6
36
∑
200
M161, Final, Spring 2009
Name:
Time: 110 minutes. You may not use calculators on this exam
d
1
1
1
d
d
asin(x) = √
acos(x) = − √
atan(x) =
,
,
,
dx
dx
1 + x2
1 − x2
1 − x 2 dx
d
d
sec(x) = sec(x) tan(x),
tan(x) = sec2 (x),
sin(2x) = 2 sin(x) cos(x)
dx
dx
ln xdx = x ln x − x + C
sec(x)dx = ln(sec(x) + tan(x)) + C
1 + cos(2x)
1 − cos(2x)
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
sin2 (x) =
2
2
∫
∫
Taylor series of the function f (x) about x = a:
∞
f ′ (a)
f ′′ (a)
f (n) (a)
f (a) +
(x − a) +
(x − a)2 + ⋯ = ∑
(x − a)n
1!
2!
n!
n=0
Multiple Choice Answer Block
A a
b
c
d
e
D a
b
c
d
e
B a
b
c
d
e
E a
b
c
d
e
C a
b
c
d
e
F a
b
c
d
e
1)
Sketch the polar curve r = 2 sin(2θ) in the following diagram:
5
4
3
2
1
0
−5
−4
−3
−2
−1
−1
0
1
2
3
4
5
−2
−3
−4
−5
b) Determine the area inside this curve. (You may — and should — use without proof that this
curve has a 4-fold symmetry.)
2) a)
b)
∫
ln(x)
dx
x4
(3x − 1)
∫ 9 − 2x + 3x dx
2
c)
∫
x3
√
dx
1 − x2
3)
a) Determine all complex 3rd roots of 8i.
b) Determine a complex number z such that e z = −1 + i
4) Determine, for example by appropriate manipulation of known power series, or by calculating
the Taylor series about a = 0, power series for the following functions centered at 0. Collect sums
of series into a single infinite sum.
Your solution should make clear what steps were performed with series of what functions. If
you use “+⋯” notation without indicating the generic form of terms, you should give at least the
first four nonzero terms of the series to indicate the pattern of coefficients.
x
1
dt(= arctanh(x)) for ∣x∣ < 1
a)
0 1 − t2
∫
b)
3x − 3
. (Hint: Partial Fractions)
(x + 1)(x − 2)
5)
Solve the following initial value problem using power series.
y ′ + 2x y = 2x,
Identify the function from the power series solution.
y(0) = 2
6) The following multiple choice problems will be graded correct answer only. You do not need
to show work, and no partial credit will be given. Record your answer in the answer block on the
front page. Answers given on these pages will not be scored. You also may tear off these pages and
do not need to hand them in.
It is strongly recommended that you work out the problems until the correct answer is uniquely
determined and don’t just try to solve them by “intuition” or “guessing” – doing so is likely to result
in a wrong pick.
Each correct anwer is worth 6 points, each incorrect answer is counted as 0 points. (Unanswered
questions are 1 point, questions in which more than one answer is ticked are considered to have been
answered wrongly.)
A)
a
B)
An equation for the tangent to the graph of y = arcsin(x/2) at the origin is
x − 2y = 0
b
x−y=0
c
x=0
Let g be the function given by g(x) =
y=0
d
∫
0
x
e
πx − 2y = 0
sin(t 2 )dt for −1 ≤ t ≤ 3. On
which of the following intervals is g one-to-one.
√
√
a √
−1 < x < 0 √b 0 < x < 2 π c 0 < x < π
e
π<x<2 π
d
1 < x < π2
Which of the following three improper integrals are convergent
C)
A.
∫
1
∞
x 1.001
dx
ln(x)
B.
∫
1
∞
ln(x)
dx
x 1.001
C.
∫
1
∞
x 10
dx
2x
Integrals A,B and C are convergent
Integrals A and C are convergent, B is divergent
Integrals B is convergent, A and C are divergent
Integrals C is convergent, A and B are divergent
Integrals B and C are convergent, A is divergent
a
b
c
d
e
a
Which of the following statements about the convergence of
1
dx is true?
x ln(x)
It converges, because lim F(x) is finite for an antiderivative F(x) of the integrand
b
c
1
It converges by limit comparison with x 5/4
It diverges, because lim F(x) = ∞ for an antiderivative F(x) of the integrand
d
e
It diverges by limit comparison with x1
We cannot decide on convergence using methods from the course
D)
∫
2
∞
x→∞
x→∞
If 1 =/ z is a complex number such that
E)
z 20
18
= 1, then ∑ z n =
n=0
a
F)
a
c
−1
b
−
1
z
c
0
d
1
e
1
z
Which of the following is a power series expression for
x2
?
1 − x2
1 + x 2 + x 4 + x 6 + x 8 + ⋯ b x 2 + 2x 3 + 3x 4 + 4x 5 + ⋯
x2 + x3 + x4 + x5 + ⋯ d x2 + x4 + x6 + x8 + ⋯ e x2 − x4 + x6 − x8 + ⋯
If you are done and have time left, check your answers on all the problems. Is in each problem
clear, what your answer is? Did you tick the correct boxes on the multiple choice questions?