Problem Points Score
M161, Final, Spring 2008
1
20
2ab
30
2cd
30
3
30
4
30
5
30
6
30
X
200
Name:
Section:
Instructor:
Time: 110 minutes. You may not use calculators on this exam
d
d
sin(x) = cos(x),
cos(x) = − sin(x),
dx
dx
d
d
tan(x) = sec2 (x),
csc(x) = − csc(x) cot(x),
dx
dx
d
d
sec(x) = sec(x) tan(x),
cot(x) = − csc2 (x),
dx
dx
d
sinh(x) = cosh(x),
dx
d
1
asin(x) = √
,
dx
1 − x2
d
1
acsc(x) = − √
,
dx
x x2 − 1
d
1
asinh(x) = √
,
dx
1 + x2
sin(2x) = 2 sin(x) cos(x)
d
d
cosh(x) = sinh(x),
tanh(x) = sech2 (x),
dx
dx
1
1
d
d
acos(x) = − √
atan(x) =
,
,
2
dx
dx
1 + x2
1−x
d
1
d
1
asec(x) = √
,
acot(x) = −
dx
dx
1 + x2
x x2 − 1
d
1
1
d
acosh(x) = √
atanh(x) =
,
,
1 − x2
x2 − 1
Zdx
Zdx
ln xdx = x ln x − x + C
sec(x)dx = ln(sec(x) + tan(x)) + C
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
1 + cos(2x)
2
sin2 (x) =
1 − cos(2x)
2
Taylor series of the function f (x) about x = a:
∞
X
f 0 (a)
f 00 (a)
f (n) (a)
2
f (a) +
(x − a) +
(x − a) + · · · =
(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
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) a) Determine all complex numbers z such that z · z = 4.
b) For a complex number z 6= 1 such that z
100
= 1, calculate
99
X
n=0
geometric series.)
z n (Hint: This is a
2) Z Evaluate the following integrals. Show your work
1
√ · ln(x)dx (You may assume that x > 0.)
a)
x
Z
b)
x4
√
1
dx
x2 − 1
Z
c)
1
dx (Hint: After suitable substitution use partial fractions.)
x x+1
√
∞
1
tan
d) Does the integral
dx converge? (If so you do not need to determine its
x2
1
value.) Explain your argument! (Hint: To find a suitable comparison, observe that the
x3 2x5
+
.)
degree 5 Taylor polynomial for tan(x) about x = 0 is x +
3
15
Z
3) Match the Taylor series with the functions. Place the letter of the appropriate
series in the left column.
5x − 1
x
1−5x
A
B
∞
X
xn
5n n!
n=1
∞
X
(ln(5)x)n
n!
n=1
ln(1 + 5x)
C
∞
X
(−1)n+1 (5x)n
n
n=1
e
x/5
−1
D
∞
X
(5x)n
n=1
5
4) a) Sketch the polar curves r1 = 4 cos(θ) and r2 = 2 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 points where the curves r1 = 4 cos(θ) and r2 = 2 intersect.
c) Determine the area inside the curve r1 = 4 cos(θ) and outside the curve r2 = 2. (Be
careful to pick the right integration limits.)
5) Solve the following initial value problem using power series.
y 0 (x) + x · y(x) = x,
y(0) = 1
6) a) Complete the following 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
n2 + 1
b) Now consider the sequence an =
and let L = 0. For a given ε > 0 calculate N
n4
such that the condition in a) is fulfilled.
c) Using the definition of convergence from a) write a N —ε proof that lim an = 0. (No
n→∞
points will be given for methods that are not N —ε proofs!)