7.1-7.7
On 7.5, income stream is not included
On 7.6, only Newton's cooling is included
8.1-9.4
inverse trigonometric functions and their derivatives are also included.
The following topics are NOT co
Study Guide for Final Exam
Brian Van Koten
December 7, 2013
Half of the questions on the nal exam will be related to topics discussed
after the second midterm; the other half will cover the rest of th
Jon Paprocki
1
Problem solutions
(1) Evaluate
e x
dx
x
Let u =
x. Then du =
dx
,
2 x
e
and we have that
x
x dx = 2
eu du = 2eu + C = 2e
x
+C
(2) Let m, n be positive nonzero integers. Compute the po
MATH 31B: SELECTED HW SOLUTIONS
JOHN SUSICE
Section 7.3
Find the derivative.
Problem 41.
3
y = (ln (ln x)
Solution. Dierentiate using the Chain Rule:
d
d
3
y=
(ln (ln x)
dx
dx
2 d
ln (ln x)
= 3 (ln (l
Math 31B/4
Fall 2013
Exam 1
October 21, 2013
70 points
Name (Print):
This exam consists of 5 pages (including this cover
page) and 2 problems. Please check to see if any pages
are missing.
ID #:
TA:
do statement
while (condition);
It does the statement at least once before checking conditions.
int k = 0
do
cfw_cout < Hello < end1;
k+;
while (k<3);
cfw_ cout <BLABLABLA;
-for (initialization; sta
if (s.size() !=0)
/s: Hello
S: ABC
S:321
s[o] = toupper(s[o]);
/S is now Hello
S is now ABC
S is now 321
if (s[k] = E | s[k] = e)
if (tolower(s[k]) = e)
s = toupper(s); /Error! wont compile. s is a st
string s = Hello;
char c = s[1] / initialized to e
e
\t -> tab char
\n new line \
\
if (t[K] = E | e) /Wrong! But will compile and do the wrong thing
char c = x; /ERROR
string c = x; /ERROR
#include <
MATH 31B - SECTION 1
MIDTERM #2
FEBRUARY 20, 2015
Name
Student ID
Discussion Section
Problem 1
/30
Problem 2
/20
Problem 3
/30
Problem 4
/20
Total
/100
Problem 1. Evaluate the following integrals.
x
Math 31b : Midterm 1, Spring 2012
Professor Curran
Each problem is worth 10 points.
1. Evaluate the following limits:
(a)
lim
p
p
1+t
1
t
t
t!0
(b)
lim (cos x)1/x
2
x!0+
2.
(a) Evaluate the indenite i
MATH 31B, Integration and Innite Series, Lecture 3, Spring 2015
Exterior Course Website: http:/www.math.ucla.edu/heilman/31bs15.html
Prerequisite: MATH 31A, with a grade of C- or better.
Course Conten
Math 31B
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due June 5, at the beginning of class.
Assignment 10
Exercise 1. Find the Maclaurin series of
(x2
Math 31B
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due April 17, at the beginning of class.
Assignment 3
Exercise 1. A 70 kg skydiver jumps out of a
Math 31B
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due May 15, at the beginning of class.
Assignment 7
Exercise 1. For each of the following sequenc
Quiz 4
Tuesday, November 5, 2013
Name:
Student ID:
Math 31B/4
TA:
Problem 1 (10 points). Let f (x) = cos(x), and let Tn (x) be the nth Taylor polynomial for
f centered at a = . Find Tn .
4
Solution: F
Exam Review 4
Brian Van Koten
December 5, 2013
Here we present three solutions to one of your harder homework problems:
one by the limit comparison test, one by the comparison test, and one by the
int
Math 31B, Practice Midterm 1 Solutions
Ian Coley
October 21, 2014
Solution to 1.
(a) We should use a logarithm to make things easier for us. Set
lim xsin x = L.
x0+
Then
ln
lim xsin x
x0+
= lim ln(xsi
S C I11
E.
Deduce that since
p W
- Wg
= lim
R-+w
N = 1 if w0is interior to P and that N = 0 if wo is exterior to P. Thus show that the
mapping of the half plane Im z > 0 onto the interior of P is one
402
THESCHWARZ-CHRISTOFFEL
TRANSFORMATION
Hence w = A' sin-' z
CHAP. I I
+ B. If we write A' = l/a and B = b / a , it follows that
z = sin(a w - b) .
I
This transformation from the w to the z plane sa
CHAP. I I
images are to be the vertices of the polygon and where
a)
The vertices are the points wj = f ( x j ) ( j = 1,2, . . . , n - 1) and w, = f ( ? . The
function f should be such that arg f ' ( 2
362
APPLICATIONS CONFORMAL
OF
MAPPING
CHAP. I 0
with only the x and y coordinates. Since T does not vary with the coordinate along
the axis perpendicular to the xy plane, the flow of heat is, then, tw
5. Write the complex potential for the flow around a cylinder r = ro when the velocity V
at a point z approaches a real constant A as the point recedes from the cylinder.
6. Obtain the stream function
CHAP. I 0
The analytic function
F ( z ) = 4(x? y)
+ ilCr(x9 Y )
is called the complex potential of the flow. Note that
or, in view of the Cauchy-Riemann equations,
F'(z) =
Y) - i#@, Y).
Expression (2)
372
APPLICATIONS
OF CONFORMAL
MAPPING
CHAP. 10
Suggestion: This problem can be transformed into the one in Exercise 4.
(-)
tanh y
2
Ans. H = - arctan
.
tan x
Jr
DHZO
H= 1
H=Oz
2
X
.
FIGURE 142
9. Deri
A RELATED
PROBLEM
SEC. 102
367
Since the denominator here reduces to sinh2 y - cos2 x, the quotient can be put in the
form
2 cos x sinh y - 2(cos x/ sinh y )
= tan 2a,
1 - (cos x / sinh Y ) ~
sinh2 y
SEC.
99
TRANSFORMATIONS
OF BOUNDARY
CONDITIONS
357
Now, according to equation (3), the level curve H(x, y) = c in the z plane can
be written
and so it is evidently transformed into the level curve h (
Howl? LI I6
-1; vi
‘¥(z)=e e
60
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Z;CX+15)+{= Cx~d+‘)+ mg
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