White
Problem Points Score
M161, Test 3, Fall 2009
1ab
20
1c
12
Section:
2
16
Instructor:
3
16
4
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
1
d
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
1
d
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)
2
(x − a) +
(x − a) + ⋯ = ∑
(x − a)n
Taylor series of f (x) about x = a: f (a) +
1!
2!
n!
n=0
n+1
∣x − a∣
Error term: If ∣ f (n+1) (x)∣ ≤ M, then ∣R n (x)∣ ≤ M
,
(n + 1)!
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) cos(3 x) − 1.
b) e x (1 + x)
c)
8x − 7
(Work on back of sheet)
(x − 2)(x + 1)
2) Consider the functions
e x ⋅ sin(x),
e x − 1,
1
− 1,
1 − 2x
x
By considering Taylor polynomials for these functions of small degree, determine for small positive
values of x, which function is largest, which second largest and so on. Explain your argument!
∞
3)
(−1)n x 2n
.
2n
2
n=0 2 (n!)
Consider the power series y(x) = ∑
a) Show that y(x) fulfills the differential equation x 2 y ′′ (x) + x y ′ (x) + x 2 y(x) = 0.
d4
y(x)∣ ≤ 1 on the interval [−2..2]. Determine the maximal
dx 4
2
error when approximating y(x) by its Taylor Polynomial of degree 3 (which is 1 − x4 ) on [−2..2].
b) You are given the information that ∣
4) 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 (which you may tear off and don’t need to return) 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 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.)
∞
∑ (−1)n
A)
n=0
functions?
a ex
b
xn
is the Taylor series about zero for which of the following
n!
ln(1 + x)
cos x
c
d
sin x
e
e −x
∞
What are all the values of p, for which the infinite series ∑
B)
n=1
converges?
a
p>1
b
p>2
c
p≥1
d
p>0
e
n
+1
np
p≥2
∞
Determine the interval, for which the series ∑ n!(x − 4)n converges?
C)
n=0
a
3≤x≤5
b
c
R
0≤x<8
d
−5 ≤ x ≤ −3
e
x=4
x2
D)
Which of the following is a power series expression for
?
1 − x2
a x2 + x3 + x4 + x5 + ⋯
b x2 − x4 + x6 − x8 + ⋯
c x 2 + 2x 3 + 3x 4 + 4x 5 + ⋯
d 1 + x2 + x4 + x6 + x8 + ⋯
e x2 + x4 + x6 + x8 + ⋯
E)
Which is the best of the following polynomial approximations to cos(2x)
near x = 0
a 1 − x 2 /2
b 1 − 2x + x 2
c 1+x
d 1 + x/2
e 1 − 2x 2
Find the value of the sum 1 + 23 + 49 + 278 + ⋯
F)
a
2
b
∞
c
5
2
d
11
3
e
3
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?