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 prob
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.
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
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 im
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 d
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 lim
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 de
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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 radiu
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 p
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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_ /
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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 _
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
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 th
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
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 o
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.
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. St
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 ser
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 wh
(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
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
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
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
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
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