Solutions to Homework 1
Due Friday, June 25, 2004.
Chapter 1.1.
Problem 1 solution. If is an even linear function then must be constant. If z (w, 1) = z (w, 1) = 0 then (w) (1) = (w) (1) = 0. The
rst part of the exercise explains that if is identically ze
Solutions to Homework 2
Due Tuesday, July 6, 2004.
Chapter 2.3
Problem 3 solution. If the series for ln(1+z ) and ln(1z ) both converge,
1+
then we can nd the series for ln 1z by term-by-term subtraction of
z
the two series:
ln
1+z
1z
= ln(1 + z ) ln(1 z
Solutions to Homework 3
Due Tuesday, July 13, 2004.
Chapter 3.1
Solution to problem 1. Less than 0.001: k = 11 works. Less than
0.000001: k = 15 works. (Note. I dont know why the author wants
to use Stirlings estimate, since one can do the exercise and ge
Solutions to Homework 4
Due Tuesday, July 20, 2004.
Chapter 3.3) 1, 2, 7, 9, 17, 19.
Solution to problem 1. Cauchys proof requires that i) f is dierentiable
at every point in [a, b] ii) f is bounded on [a, b] and iii) for any > 0 there
is a > 0 such that
Solutions to Homework 5
Due Tuesday, July 27, 2004.
Chapter 3.4) 2, 3, 5, 10, 11, 14, 15, 22, 23, 26.
Solution to problem 2. Here is a graph:
0.1
0.08
0.06
0.04
0.02
-0.1
-0.05
0.05
0.1
It looks like = 0.1 might work, but to be on the safe side we may as
Solutions to Homework 7
Due Wednesday, August 4, 2004.
Chapter 4.1) 3, 4, 9, 20, 27, 30. Chapter 4.2) 4, 9, 10, 11, 12.
Chapter 4.1.
Solution to problem 3. The sum has the form
a1 a2 + a3
with ak = 1/k . Since the ak are positive and decreasing, the seri
Solutions to Homework 8
Due Friday, August 6, 2004.
Chapter 4.3) 1f, 7f.
n+1
Solution to problem 1f. Its easy to see that | aan | =
a rational function of n.
(2n+1)2
(n+1)2
which is
Solution to problem 7f. From the above we see the radius of convergence i
Solutions to Homework 9
Due Friday, August 13, 2004.
Chapter
Chapter
Chapter
Chapter
4.4)
4.5)
5.1)
5.2)
3
9, 12
2, 3, 10, 11
4, 5.
Chapter 4.4.
Solution to problem 3. Let f (x) = x(ln 1 )3/2 . It is positive, decreasing,
x
and tends to 0 at innity and th
Math 650 Summer 2004 Test 1. Wednesday, July 7, 2004
Name:
1. Fouriers series for a function equal to 1 on the interval (1, 1) is given by
f (x) =
4
x 1
3x 1
5x 1
7x
cos
cos
+ cos
cos
+ .
2
3
2
5
2
7
2
a) Find f (3/2).
b) Find f (3/2).
Solution. Keeping
Topics to Review for Test 2
Chapter 3.1. Newton-Raphson method. Conditions that guarantee convergence.
3.2. Dierentiability. Why derivatives of innite sums may not be the
2
sum of the derivatives. Strange examples: x4/3 sin(1/x), e1/x .
3.3. Mean Value Th
Math 650 Summer 2004 Test 2 Solutions. Friday, August 6, 2004.
Name:
1. Use the Mean Value Theorem to prove that if f is continuous on [a, b],
dierentiable on (a, b) and f (x) > 0 for all x in (a, b), then f is strictly
monotonically increasing on [a, b].
Math 148 Test 1 Version A Solutions Autumn 2002
Name:
Instructor and class time:
This is the solutions to version A of the rst midterms.
h
1. (10 pts) Solve A = (b + c) for b
2
2A
c = b. Note: other versions had problems similar this one: Solve for d if
Notes for Math 148, Wed., Oct. 2
Homework questions:
1:002:05 p.m. class: A question was asked about #53 in section 1.1. The problem is easy because the
square has sides parallel to the coordinate axes. How do we solve the problem if the square is tilted?
Math 148 Quiz 1a Autumn 2002
Name:
Class time (circle one):
1:00 p.m.
2:30 p.m.
Please read all the questions carefully and show your work. Credit may not be awarded if answers are not
supported by work.
1. You have already invested $5,200 in a stock with
Math 148 Quiz 2a Solutions Autumn 2002
1. Let f (x) = 2x + 1. Find f (3 + 1).
Answer: f (3 + 1) = f (4) = 9.
2. A parabola has vertex at the point (5, 6) and a zeros at x = 7 and x = 3. Find the equation of the
parabola. (Hint: use the form y = C (x z1 )(
Solutions to problems from old Math 148 Final Exams
1. x = 4/3
2. a) f (2) = 8, f (x) = x4 x3 7x2 + 12
b) z = 2, 3
3. $236.20
4. Dimensions of sheet are 20 by 15 feet (dimensions of the box itself are: depth 2, width 11, length 16
feet).
5. a) 10 13 yards
Problems from old Math 148 Final Exams
These problems can be downloaded from the course home page:
http:/www.math.ohio-state.edu/courses/math148/
1. Solve x2 3x + 4 = x
2. Let f (z ) = z 4 + z 3 7z 2 + 12 and g (z ) = z 3 .
a. Find f (2) and f (x).
b. Fin