Problem Points Score
M161, Test 3, Spring 2008
1ab
24
Name:
1c
12
2
24
3
25
4
15
∑
100
Section:
Instructor:
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
asin(x) = √
,
dx
1 − x2
d
1
acsc(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
acos(x) = − √
,
dx
1 − x2
1
d
asec(x) = √
,
x x2 − 1
Zdx
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
atan(x) =
,
dx
1 + x2
1
d
acot(x) = −
1 + x2
Zdx
sec(x)dx = ln | sec(x) + tan(x)| +C
1 − cos(2x)
sin2 (x) =
2
Taylor series of the function f (x) about x = a:
∞
f 0 (a)
f 00 (a)
f (n) (a)
f (a) +
(x − a) +
(x − a)2 + · · · = ∑
(x − a)n
1!
2!
n!
n=0
Theorem (The Derivative Rule for Inverses) If f has an interval I as domain and f 0 (x) exists and
is never zero on I, then f −1 is differentiable at every point in its domain. The value of ( f −1 )0 at a
1
point b = f (a) in the domain of f −1 is given by ( f −1 )0 (b) = 0 .
f (a)
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
what steps were performed with series of what functions.
√ clear
a) cos x
b) f (x) =
Z x
sin(t)
0
t
dt. (You may assume, without proof, that lim sin(t)/t = 1.)
t→0
c)
1 + x2
−1
x
=
+
3
−1 + x
1 − x 1 − x3
2) The following infinite sums were obtained by evaluating Power series for certain functions.
Using this fact, find the exact values for each sum:
4
8 16 32
a) 1 − 2 + − + − + · · ·
2! 3! 4! 5!
3 3 3
3
b) 3 + − + − + · · ·
2 4 8 16
∞
3)
a) Determine the interval of convergence for the power series
(5x)n
∑ 3 , (ignoring the behavior
n=1 n
at the end points).
b) By looking at the first few terms of the Taylor series about x = 0, decide which of the following
functions is largest, and which smallest for x near 0. (Remember that for small x-values smaller
powers of x dominate over larger powers.) Explain your argument.
x
,
1−x
sin(x),
ex − 1
4) The goverment – due to an election year in spending mode – enacts a stimulus package that
puts $150 billion back into the economy. We assume that all the people who have extra money to
spend would spend 80% of it and save 20%. Thus, of the extra income generated by the package,
$150 · 45 billion = $120 billion would be spent again and so become extra income to someone
else. Assume that these people also spend 80% of their additional income (which would be $96 =
120 · 54 = 150( 45 )2 billion) and so on.
a) Let C = 150 be the amount of the stimulus package (in billion dollars) and r = 4/5 the spending
factor. Write down an infinite sum, involving C and r, for the amount of money added to the
economy.
b) Calculate the value for the infinite sum in a). (As you have no calculator, an expression as
fractions is fine; your answer should not involve infinite sums.)