Exam 1 solutions (M126C)
1. Determine if the sequence converges or diverges. If it converges, nd its limit: an =
(ln(n)2
n
SOLUTION: Use lHospitals rule (twice):
(ln(n)2
2 ln(n) (1/n)
ln(n)
1/n
= lim
= lim 2
= lim 2
=0
n
n
n
n
n
1
n
1
lim
Therefore, the s
Exam 2 Review Solutions
1. State the Fundamental Theorem of Calculus: Let f be continuous on [a, b].
x
If g(x) =
f (t) dt, then g (x) = f (x).
a
b
f (x) dx = F (b) F (a), where F is any antiderivative of f .
a
n
b
2. Give the denition of the denite integ
Selected Solutions, Section 5,1
2. The purpose of this exercise is to be sure we have a little experience with estimating
area using rectangles. The midpoint rule is to evaluate the heights of the rectangles
at the midpoint of each rectangle.
Recall that
Final Exam Review
Calculus II
Sheet 3
1. Determine if the series converges (absolute or conditional) or diverges:
(a)
(1)n ln(n)
n
n=1
SOLUTION: We might rst check to see if it converges absolutely. It will not, by
the integral test. To check that the fun
Review Sheet 1 Solutions
n
i2 =
1. Prove by induction:
i=1
n(n + 1)(2n + 1)
6
SOLUTION:
Prove the rst case: If n = 1, then does 12 =
123
?
6
Yes.
k
k(k + 1)(2k + 1)
.
6
i=1
Use the assumption to prove that the statement is true if n = k + 1. In that case
Final Exam Review
Calculus II
Sheet 2
1. True or False, and give a short reason:
2n+3
(a) The Ratio Test will not give a conclusive result for 3n4 +2n3 +3n+5
TRUE. The ratio test fails for plike series (the limit will be 1). To show convergence, use a dir
Solutions to the Review Questions, Exam 3
1. A cross section of a tank of water is the bottom half of a circle of radius 10 ft, and is 50
ft long. Find the work done in pumping the water over the rim of the tank if it lled to
a depth of 7 feet (set up the
Selected Solutions, Section 5.4
10. Hint: Multiply it out rst,
1
v 5 + 4v 3 + 4v dv = v 4 + x4 + 2v 2 + C
6
v(v 2 + 2)2 dv =
12.
1 3
x
3
+ x + tan1 (x) + C.
18. The hint is that sin(2x) = 2 sin(x) cos(x). Using that,
sin(2x)
dx =
sin(x)
2 sin(x) cos(x)
dx
Selected Solutions, Section 5.2
1. This is good practice in taking left endpoints.
In this case, f (x) = 3 x/2, and the interval is [2, 14]. The Riemann sum using 6
rectangles will use:
Width of each rectangle: (14 2)/6 = 12/6 = 2.
The height of the rec
Review Problems: Chapter 11
1. What does it mean to say that a series converges (Im looking for the denition; be
sure you dene any notation you use).
2. Does the given sequence or series converge or diverge?
1
(a)
n
n=2 n
(b)
(e)
(6)
n1 1n
5
(j)
n=1
(f)
Review Solutions: Chapter 11
1. What does it mean to say that a series converges?
SOLUTION: We dene the nth partial sum sn as follows:
n
S 1 = a1
S 2 = a1 + a2
S3 = a1 + a2 + a3
Sn =
an
k=1
The partial sums Sn form a sequence (of numbers). The (innite) se
Quiz 2 Solutions
Part of this quiz was to see if you could follow the directions!
Solutions are written neatly, clearly and completely using your own paper (up to 10 pts)
Quiz is stapled (up to 10 pts).
Quiz turned in on time (up to 10 pts).
Here are t
Selected Solutions, Section 4.9
10. Note that e2 is a constant, so the antiderivative is e2 C
17. The antiderivative is 2 cos() tan() + C, but notice that the C can change because
sec() has a lot of vertical asymptotes, breaking up the real line. Therefor
/dh+// /0 WI/X/M n3 M 3
J: ._...... ¢ [/)»/D./x/
laul ("+03 /o/)c/ w
an; \ g
5; /((KM / *I m L) m /x/ z /0/x/
44*» (an! a '4
/0"/X/<I =7
To 'hu/ lwcvvcl, (VIM/Mr hv WIS: X3 l 9.
4+ nil/,0;
lo /0/4 n w N r 1
Z % 7 Z ,0 (75) PM) dwux Q,
' , _________
Selected Solutions, Section 5.3
4. This is a good exercise to understand the area function that we described in class.
(a) g(0) = 00 f (t) dt = 0, and g(6) = 06 f (t) dt = 0 by symmetry (it looks like there is
as much positive area as negative.
(b) Estima
6.5: Construction of the Dual
Here is the problem well work on:
minx 8x1 + 5x2 + 4x3
s.t. 4x1 + 2x2 + 8x3
7x1 + 5x2 + 6x3
8x1 + 5x2 + 4x3
3x1 + 7x2 + 9x3
= 12
9
10
7
x1 0, x2 URS, x3 0
First well take care of the sign restrictions so that all of our vari
Example 1, Section 4.16
The Leon Burnit Ad Agency is trying to determine a TV schedule for Priceler Auto.
Ad
HIM LIP HIW
Cost
Football
7
10
5
100, 000
3
5
4
60, 000
Soap Opera
Goals
40
60
35 600, 000
This means, for example, that the ad agency wants to re
Final Exam, Operations Research (Fall 13)
This is a take home exam. You may use your text, any class notes and anything on our class
website to help, and you may use LINDO, Maple and/or Matlab. You should not look for
solutions on the internet, or get hel
Solutions to Review Questions, Exam 1
1. If x = [1, 1, 0, 2]T and y = [2, 1, 0, 4]T , then compute the distance from x to y using
the (a) 1-norm, and (b) the innity norm.
SOLUTION:
xy
1
= |1 2| + | 1 + 1| + |0 0| + |2 4| = 1 + 0 + 0 + 2 = 3
xy
= max cfw_1
Review Solutions, Exam 2, Operations Research
1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the
dual, then.
HINT: Put the primal so that Ax b and the dual so that AT y c
SOLUTION: With the primal and dual in norma
Review Material, Exam 2, Ops Research
1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the
dual, then.
HINT: Put the primal so that Ax b and the dual so that AT y c
2. Show that the solution to the dual is y = (cT B
Review Material, Exam 1, Ops Research
The exam will cover material from Chapter 3 (we skipped 3.6, 3.7), up through section 4.8.
Background Material: Linear Algebra
Though these may not be asked explicitly, you should be able to do the following (and may
HW Set 3 Solutions
3.10, 7 Let st be the pairs of shoes made during quarter t.
Let it be inventory of pairs of shoes at the end of each quarter.
Let xt be the number workers getting quarter t o during each year. So, for example,
x1 would be working all qu
Homework Solutions to Section 3.1-2
These are the solutions to the problems not turned in: 3.1- 1, 5 and 3.2-1, 2, 4
3.1 Solutions
1. In this case, x1 , x2 are given to you- Normally, you would want to dene those terms rst.
Let x1 , x2 be the number of ac
Operations Research
Prof. D.R. Hundley
Whitman College
Fall 2013
DRHundley (WHI)
Math 339
10/16/2013
1 / 18
Introduction
We started this last time:
Were buying advertising time for HIW and HIM.
Let x1 , x2 be the number of ads purchased during a comedy sh
Matrix Notation and Simplex (6.2)
Prof. D.R. Hundley
Whitman College
Fall 2013
DRHundley (WHI)
Math 339
10/16/2013
1 / 14
Introduction
Consider the LP in standard form,
max z = cT x
st Ax = b
x0
where A is m n with rank m. We will re-partition these matri
Scientic Applications of LP
These notes will show how to interpret curve tting problems as linear programs. In order
to start, we need to dene dierent ways of measuring the error between an unknown set of
points in the plane:
cfw_(x1 , y1 ), (x2 , y2 ), (
Homework Solutions to Section 3.9
1. Dene variables rst. In this case, we need to know the hours to run each process per week,
and the barrels of stu being bought/sold. In this case, we might dene:
x1
x2
x3
g2
o1
o2
Hours of Process 1 run per week
Hours o