M161, Final Exam, Spring
2011
Problem Points Score
1
40
2
40
Section:
3
40
Instructor:
4
40
Time: 120 minutes. You may not use calculators or other
electronic devices on this exam. No partial credit will be
given on multiple choice or box problems.
5
40
X
200
Name:
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
d
1
1
,
,
acos(x) = − √
atan(x) =
dx
dx
1 + x2
1 − x2
d
d
1
,
asec(x) = √
sec(x) = sec(x) tan(x),
x x2 − 1
Zdx
Zdx
ln xdx = x ln x − x + C
sec(x)dx = ln | sec(x) + tan(x)| + C
1 + cos(2x)
1 − cos(2x)
sin2 (x) =
2
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 0 (a)
f 00 (a)
f (n) (a)
2
f (a) +
(x − a) +
(x − a) + · · · =
(x − a)n .
1!
2!
n!
n=0
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
Error term: If |f (n+1) (x)| ≤ M , then |Rn (x)| ≤ M
|x − a|n+1
.
(n + 1)!
1. Multiple Choice. Write the letter of your answer in the box. No work
outside the box will be graded.
Suppose that z =
6 1 is a complex root of z 50 − 1.
50
X
zn.
Determine the value of
A.)
n=0
a
0
b
z 50
c
z
d
Cannot be determined
e
1
B.)
For a positive integer n, let Tn+1 (x) be the degree
n + 1 Taylor polynomial for f (x) centered at a. Then Tn+1 (x)
a is never a worse approximation of f (x) near x = a than Tn (x).
b is equal to f (x) in some neighborhood of x = a.
c is never equal to f (x).
d is always less than f (x) near x = a
e is always a better approximation of f (x) near x = a than Tn (x).
C.)
Determine the equation of the tangent line to the
curve cosh(1 − t) at t = 1.
a
e
y = −1
y=1
b
y=t
c
y =t+1
d
y = −t + 1
D.)
Which of the following complex numbers z satisfies
z
e = 1 − i?
√
√
√
a
2√
− i π4 b ln( 2)√+ i π4 c ln( 2) + i 3π
4
3π
π
d ln( 2) − i 4
e ln( 2) − i 4
2. Integrals
Z
A. Evaluate the integral
x+1
√
dx.
4 − x2
Z
B. Evaluate the integral
xcos(x)dx.
C. Does this integral converge or diverge? Explain and indicate all
tests used.
Z ∞
dx
.
2x3 − 1
1
D. Does this integral converge or diverge? Explain and indicate all
tests used.
Z ∞
2 + sin(x)
dx.
x
7
3. Functions and growth
A. Let y(t) be the concentration of Iodine-131 in a sample at time t
(measured in days). The half-life of Iodine-131 is 8 days. Find a
formula for dy
in terms of t. Do not solve the differential equation.
dt
B. Order the following functions from slowest to fastest growth rate:
4x , ln(5x), e3x , x3 + x.
slowest
fastest
Z
1
dx.
+1
(i) Which inverse trigonometric function is f (x)?
C. Let f (x) =
x2
(ii) Factor x2 + 1 using complex numbers.
(iii) Perform a partial fraction decomposition on
1
.
x2 +1
(iv) Find another formula for f (x) by integrating the answer to
part (iii). You do not need to show that C = 0.
4. Taylor series
Z
A. Let f (x) =
cos(x2 )dx.
(i) Find the degree 9 Taylor polynomial for f (x).
(ii) Is
R1
0
cos(x2 )dx greater than 9/10? Yes/No.
B. Let f (x) = e2x . The nth derivative of f (x) is f (n) (x) = 2n e2x . The
degree 3 Taylor polynomial of f (x) centered at 0 is
T3 (x) = 1 + 2x +
4x2 8x3
+
.
2
6
The Taylor series of f (x) centered at 0 is
T (x) =
∞
X
2n xn
i=0
n!
.
(i) What is the radius of convergence of T (x)?
(ii) Use Taylor’s error theorem to find an upper bound for the
error |R3 (x)|, which shows how closely T3 (x) approximates
f (x), on the interval [−1, 1], using the fact that e2 < 8.
5. Polar Coordinates
A. Set up and √
evaluate the integral necessary to find the length of the
curve r = 2eθ on the interval [0, π].
B. Match the equations with their corresponding graphs:
(a) r = 3 sin(2θ)
(b) r = 3 sin(θ)
(c) r = 3 + 3 sin(θ)
(d) r = 2 + 3 sin(θ)
(e) r2 = − sin(2θ)
(i)
(ii)
(iii)
(iv)
(v)