M161, Test 3, Spring 2007
Name:
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
sinh(x) = cosh(x),
dx
d
1
asin(x) = √
,
dx
1 − x2
d
1
acsc(x) = − √ 2
,
dx
x x −1
d
1
asinh(x) = √
,
dx
1 + x2
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
cosh(x) = sinh(x),
dx
1
d
acos(x) = − √
,
dx
1 − x2
d
1
asec(x) = √ 2
,
dx
x x −1
d
1
,
acosh(x) = √ 2
x −1
Zdx
ln xdx = x ln x − x + C
1 + cos(2x)
cos2 (x) =
2
Problem Points Score
1
16
2ab
16
2cd
16
3
18
4
16
5
18
X
100
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
d
tanh(x) = sech2 (x),
dx
1
d
atan(x) =
,
dx
1 + x2
d
1
acot(x) = −
dx
1 + x2
d
1
atanh(x) =
,
1 − x2
Zdx
sec(x)dx = ln(sec(x) + tan(x)) + C
1 − cos(2x)
sin2 (x) =
2
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
1
value of (f −1 )0 at a point b = f (a) in the domain of f −1 is given by (f −1 )0 (b) = 0 .
f (a)
1) Determine the Taylor polynomial of order 3 for cos(x) at a = π/4.
2) For each of the following series determine whether the series converges absolutely, converges conditionally or diverges. Justify your answer
∞
X
1
a)
3
n=1 n + n
b)
∞
X
n=1
(−1)n
sin(n)
n2
c)
d)
∞
X
1
(−1)n+1 √
n
n=1
∞
X
n2
n=1
2n
3) Determine the interval on which the power series
∞
X
n=0
the radius of convergence and check both endpoints.)
√
xn
converges. (I.e. determine
n2 + 3
4) a) Determine a power series representation for f (x) =
1
about x = 0. (Hint:
1 − x2
Geometric Series!)
b) Determine a power series for atanh(x) about x = 0. (Note that
1
d
atanh(x) =
.)
dx
1 − x2
5) a) Complete the definition for limit of a sequence:
The sequence {an } converges to the number L if for every positive ε there exists an integer
N such that
15n − 1
and let L = 5. For a given ε > 0 calculate
3n
N such that the condition in a) is fulfilled.
b) Now consider the sequence an =
c) Write a a proof that n→∞
lim an = 5.