Problem Points Score
1
50
2
30
3
30
4
30
5
30
6
30
X
200
M161, Final, Fall 2010
Name:
Instructor:
Section:
Time: 120 minutes. You may not use calculators or other
electronic devices on this exam.
d
sin(x) = cos(x),
dx
d
1
asin(x) = √
,
dx
1 − x2
d
1
acsc(x) = − √
,
dx
x x2 − 1
sin(2x) = 2 sin(x) cos(x)
d
d
cos(x) = − sin(x),
tan(x) = sec2 (x),
dx
dx
d
1
d
1
acos(x) = − √
,
atan(x) =
,
dx
dx
1 + x2
1 − x2
d
1
d
asec(x) = √
,
sec(x) = sec(x) tan(x),
2
x x −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
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 point b in the domain of f −1 is the
1
reciprocal of the value of f 0 at the point a = f −1 (b): (f −1 )0 (b) = 0 −1
.
f (f (b))
Taylor series of f (x) about x = a:
∞
X f (n) (a)
f 0 (a)
f 00 (a)
f (a) +
(x − a) +
(x − a)2 + · · · =
(x − a)n .
1!
2!
n!
n=0
Error term: If |f (n+1) (x)| ≤ M on an interval, then |Rn (x)| ≤ M
|x − a|n+1
.
(n + 1)!
1. Short answer questions. Put your answer in the box. No work outside
the box will be graded.
x
(a) Find sec(atan( )).
3
(b) Suppose that g(x) is a differentiable function that has an inverse
g −1 (x). Suppose the graph of g(x) passes through the origin with
slope 3. What is the slope of g −1 (x) at the origin?
(c) What is the limit of the sequence {an } if an =
1 − 2n
?
1 + 2n
−n n−1
(d) Evaluate the series Σ∞
3 .
n=1 4
(e) Let 3x2 − 5x3 + 7x4 + 3x5 be the fifth-degree Taylor polynomial
for a function f (x) centered at a = 0. What is the value of the
fifth derivative f (5) (0)?
2. Complex numbers.
(a) What are the functions f (θ) and g(θ) appearing as the real and
imaginary parts of eiθ = f (θ) + g(θ)i in Euler’s formula?
(b) If z is the complex number 3πi/4 and ez = a + bi, find a and b.
(c) Plot all the complex numbers z such that z 3 = −8.
4 4i
3i
2 2i
1i
-4
-4
-3
-2
-2
-1
0
-1i
-2 -2i
-3i
-4 -4i
1
2
2
3
4
4
3. Integrate the following functions:
Z
(a) x sin(x) dx.
Z
(b)
0
3
dy
p
9 − y2
.
4. (a) Graph the set of points whose polar coordinates satisfy the equation r = 2 + 2 sin(θ) for 0 ≤ θ ≤ 2π.
5
4
3
2
1
0
−5
−4
−3
−2
−1
−1
0
1
2
3
4
5
−2
−3
−4
−5
(b) Find the area of the region enclosed by the curve r = 2 + 2 sin(θ).
5. Here is some information about f (x) = ln(1 + x).
x2
Its degree 2 Taylor polynomial is T2 (x) = x − .
2
2
(3)
The third derivative of f (x) is f (x) =
.
(1 + x)3
Each part of this question can be done without doing the other parts.
(a) Use T2 (x) to approximate ln(1 + 12 ).
(b) What is the maximum value M of f (3) (x) for 0 ≤ x ≤ 1/2?
(c) Use Taylor’s error theorem to bound the error in part (a).
6. Multiple choice questions
No partial credit. You do not need to show work. It is strongly recommended that you work out the problems until the correct answer is
uniquely determined rather than trying to solve them by “intuition”
or “guessing”. Each correct anwer is worth 6 points, each incorrect
answer is 0 points. Do not check two answers on a question.
I)
a
II)
a
d
2x
dx ln(e )
2x
1/e
b
=
2/e2x
c
2x
d
1
e
2
If f 0 (x) = −f (x) and f (1) = 1 then f (x) =
1/2e−2x+2
b
e−x−1
c
e1−x
d
e−x
e
−ex
III)
Which of the following three series are convergent?
n2
A.
ln(n)
ln(n)
B. Σ∞
n=2
n2
n10
C. Σ∞
n=2 n
2
Σ∞
n=2
a
b
c
d
e
The series A,B and C are convergent
The series A and C are convergent, B is divergent
The series B is convergent, A and C are divergent
The series C is convergent, A and B are divergent
The series B and C are convergent, A is divergent