SOC 200
01/21/2016
Karl Marx
1818 born capitalist economist
1848- revolution affected his writing and political analysiscommunist manifesto
1867- capital- economic critique of capitalism- why its problematic
and doom to fail due to its internal complicati
One-Sided Limits
*When we say that a limit exists (in other words we find the limit has a real-number answer), we are
saying that both the right sided and the left sided limits exist and are the same. If the right-side limit
does not equal the left-side l
Limits of rational functions as x approach infinity.
There is specific procedure for dealing with limits as the variable increases towards infinity in either
direction. Take the following example:
5x2 2 x 7
lim
x
3 x 2 10
As x gets larger and larger, it
Finding Extreme Values of Functions:
Finding extreme values of functions can take several steps. We may need to identify the domain of our
function, any domain end points, critical values, etc.
One important thing to keep in mind is that values that are N
Concavity of Functions:
Another important application of a derivative is to help us determine the concavity of a function. This
will represent the intervals where a function is concave up or concave down, as well as the precise point
where the changes in
Definite Integrals:
The use of a definite integral has a major purpose in calculus. Typically, calculus I can be broken down
into three major components: Limits, Derivatives and Integration. While limits and derivatives are major
parts of finding equation
Anti-derivatives:
The process of finding anti-derivatives is the reverse of finding derivatives. We saw this a little earlier in
the chapter when looking at some basic functions that share the same derivative and how we can write
the anti-derivative of th
Group: Name: Q|= M TMN g
Math 231 A. Fall, 2013. Worksheet 8. 10/3/13
1. The Hoover Dam near Las Vegas has penstock gates to control the ow of water. They
are circular, approximately 5 meters in radius, and are centered 90 melteras under water.
9 M
70 L u
Group: Name:
Math 231 Worksheet 13. March 6, 2015
1. Recallthattheharmonicseriesis1+4l+uu Let sn=1+%+-+'e
the nth partial sum.
a) What is lim 3"? 5 0 (WT. Y=1/X
5W GLIW "TFFOM EMB.
b) Draw a careful picture on the graph to the
right which illustrates that
Group: * E _* Name:
Math 231 Worksheet 12. Feb. 4, 2015
1.Sh thtthfll' 'alld' : . A
ow a e o ow1ng series 1verge 54% I/ lg:
i n2 15% N =AA:W_1_. :/ Bswmf' J
n2+1 -300 "1 9" Hz
$675,: Z ,_ -9; o_ / 8y MWQ. Ra 7L4"
"=0 .903 e ' a / S'CVls OLYVG/tjad .
Z
Group: Name:
-_._
Math 231. Worksheet 14. March 11, 2015
1. Use Comparison or Limit Comparison to show convergence or divergence. State your
choices of an and bu, on each problem. an , 64
ll
:00 1 _ I 4 L
(a) n: n2+n+1 CWWUM' 04nz+nrl r11 4;.de
i .L M
Midterm 1, Fall 2014: Solution Key
Name: Dandona, P.
UIN: 652863647
Exam code: BDDEAEEA
NetID: dandona2
Summary of answers:
Question
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Total
Correct Answer
E
D
E
C
A
B
B
B
A
B
A
B
B
*
*
*
*
Your Answer
C
B
A
C
A
B
B
Increasing and Decreasing Intervals:
The primary function of critical values is to use them to determine where a function is increasing or
decreasing. Unlike using the graphing function on a graphing calculator (which is limited to a specific
window and d
Constant Rule and Power Rule:
These first two derivative rules are the easiest and most widely applied. Starting with the derivative of a
constant term; As you have already seen in your studying of theses derivative rules, the derivative of a
constant is
Related Rate Problems:
Related Rate problems are applied word problems that involve more than one variable, however, the
one thing that all the variables have in common are that they are changing with respect to time. In other
words, all the variables are
Group:
Name:
Math 231 A. Fall, 2015. Worksheet 13. 10/27/15
Determine if the following series converge absolutely, converge conditionally, or diverge. Give
complete justication, and state which test or tests you are using.
1.
n=2
2.
n=1
3.
n=1
(1)n+1 ln n
Group:
Name:
Math 231 A. Fall, 2015. Worksheet 12. 10/22/15
1. a) Use the alternating series test to prove that that the series
n=0
6
b) Using a calculator, nd the partial sum s6 =
n=0
(1)n
converges.
n!
(1)n
to four decimal places. What is
n!
the maximum
Group:
Name:
Math 231 A. Fall, 2015. Worksheet 14. 10/29/15
1. In this problem we nd an approximation for
1
0
x arctan x dx.
a) For each function, write down the rst four terms of the power series. State the radius
of convergence as |x| < R.
1
1x
1
1 + x2
Group:
Name:
Math 231 A. Fall, 2015. Worksheet 15. 11/5/15
1. a) Use the Maclaurin series for cos x to nd the Maclaurin series for f (x) = x3 cos(x2 ).
b) Use part a) and the denition of Maclaurin series to nd the value of f (11) (0).
2. Find the Taylor s
Practice Exam Problems
Integration
What is the best change of variables to make to evaluate
R
x2
dx
x2 4
Solution: Let x = 2 sec(u). Then dx = 2 sec(u) tan(u)du and the integral becomes
R
cos(u)du which is easy.
Make the change of variables x = tan t in
(0()
300
v:-1-r~ u
d v :- ~u d u
~
-~
-J
c~V=_fn/v+ll
f
v -t-1
- I ()
I0I
= f~(~j
I~
I
-c-
'Azl +
II
r-C
r
0
-I (j ( ! ';
-+ ' (i cfw_
.g)
Joo
1/z
.
to11v
J arc r:a:J" \) J x
D
L
u-=- cos-'( x)
v-=xz1
d Vl .:- -
'
r
100
2Do
'6DD
(A rY7 - cfw_~-
Lj
tr-7~ -
Math 231E. Fall 2013. HW 10 Solutions.
Problem 1. Evaluate each of the integrals below.
Z
Z 2
x2
x 1
dx
a.
b.
dx
2
1x
x
Solution: (written by Jon George, TA).
1
dx
(1 + x2 )2
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Problem 2. Evalu
Math 231E. Fall 2013. Midterm 1 Practice Solutions.
Problem 1. Compute the derivatives of the following functions
a. f (x) = sin(x2 )ex
b. f (x) = arcsec(x)
c. f (x) = ln(1 + x2 + x4 )
Solution:
a. We use the product and Chain Rules:
d
d
(sin(x2 )ex ) =
(
Sequences and Series
Toolbox
1
Definitions
You should know the following definitions from 8.1 through 8.5:
Sequence (p.612) A sequence is any function whose
Convergent Sequence (p.613) The sequence cfw_an
n=n0 converges to
L if and only if
Factorial (
Practice Exam Problems
Integration
What is the best change of variables to make to evaluate
R
x2
dx
x2 4
R 3
Make the change of variables x = tan t in the integral 1 x3 dx
. What is the resulting t integral
1+x2
(including limits)? You need not evaluate
Table 1: Calculus Jeopardy Questions
Values
Trig. Integrals
Z
100
Simple Substitutions
Trig Substitutions
Z
2
Z
sin(x) cos (x)dx
cot(x)dx
0
Z
200
Z
3
2
sec(x) tan (x)dx
Z
300
4
x sin(x )dx
4
xex
dx
1 + e x2
Z
8
sin (x)dx
0
Z
400
sec(x)dx
1
Z
500
sec3 (x)d