Problem Points Score
M161, Test 3, Spring 2009
1ab
26
1c
14
Section:
2ab
16
Instructor:
2c
8
3
36
∑
100
Name:
Time: 75 minutes. You may not use calculators on this exam
d
sin(x) = cos(x),
dx
d
csc(x) = − csc(x) cot(x),
dx
d
1
arcsin(x) = √
,
dx
1 − x2
d
1
arccsc(x) = − √
,
dx
x x2 − 1
sin(2x) = 2 sin(x) cos(x)
tan2 (x) + 1 = sec2 (x)
d
cos(x) = − sin(x),
dx
d
sec(x) = sec(x) tan(x),
dx
d
1
arccos(x) = − √
,
dx
1 − x2
1
d
arcsec(x) = √
,
dx
x x2 − 1
ln xdx = x ln x − x + C
1 + cos(2x)
cos2 (x) =
2
∫
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
d
1
arctan(x) =
,
dx
1 + x2
1
d
arccot(x) = −
dx
1 + x2
sec(x)dx = ln ∣ sec(x) + tan(x)∣ + C
1 − cos(2x)
sin2 (x) =
2
∫
∞
f ′ (a)
f ′′ (a)
f (n) (a)
(x − a) +
(x − a)2 + ⋯ = ∑
(x − a)n
1!
2!
n!
n=0
n+1
∣x
−
a∣
Error term: If ∣ f (n+1) (x)∣ ≤ M, then ∣R n (x)∣ ≤ M
,
(n + 1)!
Taylor series of f (x) about x = a: f (a) +
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) 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.
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) sin(x) ⋅ cos(x). (Hint: Use a trigonometric identity!)
b)
1
(Consider an antiderivative first)
(2 − x)2
c)
√
x + 2.
2) The following infinite sums were obtained by evaluating Power Series for certain functions.
Using this fact, find the exact values for each sum. (Your answer should consist not just of a number,
but indicate the power series used.)
a)
3 3 3
3+3+ + + +⋯
2! 3! 4!
b)
2 2
2
2
+ 3 + 5 + 7 +⋯
3 3
3
3
c)
1−
32 33 34 35
+ − + −⋯
2! 3! 4! 5!
3) 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.)
sin t
is
t
t2 t4 t6
1 t t3 t5
a 1−
+ − +⋯ b
− + − +⋯
3! 5! 7!
t 2! 4! 6!
1 t t3 t5
t3 t5 t7
+ + + +⋯ e t− + − +⋯
d
t 2! 4! 6!
3! 5! 7!
A)
B)
a
e
A series expansion of
c
1+
t2 t4 t6
+ + +⋯
3! 5! 7!
2
The general term of the Taylor series for xe −x is
(−1)n x 2n
(−1)n+1 x 2n+1
(−1)n x 2n+1
b
c
d
(n + 1)!
n!
(n + 1)!
(−1)n+1 x 2n
n!
(−1)n x 2n+1
n!
C)
To approximate e x on [−1, 1] by a polynomial of degree n such that the
absolute error is not larger than 0.1, one needs to use a polynomial of degree n at
least
a
2 b 3 c 4 d 5 e 6
D)
A function f has the following Taylor series about a = 0:
∞
∑(−1)
n=1
n+1 2
n
n
x 2n
Which of the following is an expression for f (x)?
a
cos(2x 2 )
b
ln(1 + 2x 2 )
c
e −2x
2
d
sin(2x 2 )
e
1
1 + 2x 2
(x − 1)n
Determine the open interval, for which the series ∑
converges?
n
n=1
∞
E)
a
−1 < x < 3
b
0<x<2
c
−2 < x < 2
c
5⋅
d
1
2
3
2
e
R
1 − 522
1−5
e
52 ⋅
<x<
23
Calculate ∑ 5n
F)
n=2
a
5⋅
1 − 522
1−5
b
5⋅
1 − 523
1−5
1 − 524
1−5
d
52 ⋅
1 − 523
1−5
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?