Problem Points Score
1
25
2
30
3
19
Section:
4
30
Instructor:
5
30
6
30
7
36
∑
200
M161, Final, Fall 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 cos(θ) + 1 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 the outer loop and outside the inner loop.
2) a)
b)
∫
∫ x ⋅ tan(x) dx (Note that tan(x)
√
9 − x2
dx
x2
2
2
= sec(x)2 − 1)
1
dx by partial fractions. Use the result to
2+1
express arctan(x) as sum of logarithms over the complex numbers.
3)
Using that x 2 + 1 = (x + i)(x − i), integrate
∫x
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.
a) 1 + 2x
b) sinh(x) =
1
1 x
(e − e −x ) = sin(ix)
2
i
5) For each of the following improper integrals determine, using a suitable comparison test, whether
the integral converges or diverges. (You do not need to calculate the values of convergent integrals.)
State clearly what comparison you are using, what function you compare with, and show that f < g,
respectively work out the limit of f /g. No points will be given if no work is shown, or if only an
argument of√polynomial degrees without explicit comparison is done.
∞
x +1
dx
a)
x3
1
∫
b)
∫
2
∞
1
dx
(ln(x))2
6)
Solve the following initial value problem using power series.
y ′ + x y = x,
Identify the function from the power series solution.
y(0) = 0
7) 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 this page (which you may tear off and do not need to hand in) will
not be scored.
It is strongly recommended that you work out the problems until the correct answer is uniquely
determined and don’t just try to guess – 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.)
ln(x − 2) < 0 if and only if
A)
a
x<3
B)
e
The arc length of r = 4 cos(θ) is
√
√
π/2
π/2
2 + (4 cos θ)2 dθ
(−4
sin
θ)
b
∫0 √
∫0 √ (−4 sin θ)2 + (4 cos θ)2 dθ
π
2π
(−4 sin θ)2 + (4 cos θ)2 dθ
∫0 √(−4 sin θ)2 + (4 cos θ)2 dθ d ∫0
2π
(−4 sin θ)2 + (4 cos θ)2 dθ
∫0
C)
The points of intersection for r = 1 + sin(θ) and r = 1 − sin(θ) are
a
c
b
0<x<3
2<x<3
c
x>2
d
e
x>3
√
a
b
d
D)
a
d
√
√
√
2
2
2
(0, 0), (1 − √2 , 1 + √2 ), (1 + √2 , 1 − √22 )
(0, 0), (1 − 22 , 1 − 22 ), (1 + 22 , 1 + 22 )
(0, 0),(2, 0) e (1, π2 ), (1, 3π2 )
(0, 0),(1, 0), (−1, 0)
What is the power series for sin(x) − x
∞
∞
∞
(−1)n (x)2n+1
(−1)n (x)2n
(−1)n (x)n
b ∑
c ∑
∑
(2n)!
n!
n=0 (2n + 1)!
n=0
n=1
∞
∞
n
2n+1
n
2n
(−1) (x)
(−1) (x)
e ∑
∑
(2n)!
n=1 (2n + 1)!
n=1
How many roots to z 5 = −1 have positive real part and imaginary part?
E)
a
c
0
1
b
c
2
3
d
e
4
F)
Let 3x 2 − 5x 3 + 7x 4 + 3x 5 be the fifth-degree Taylor polynomial for the
function f about a=0. What is the value of f (5) (0)?
a
−30
b
3 ⋅ 5!
c
0
d
-5
e
-15
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?