MATH 151B
Mar. 8, 2010
Handout 13
Dierential Equations
We are used to dealing with dierential equations of the rst order, in other words orders of y and
by solving for y is only by techniques of integration. In this case we look at dierential equations
of
MATH 151B
Name:
Practice Exam 2: Volumes
Find the volume of the solid by rotating the following functions about the given axis using disk
method.
b
[f (x)]2 dx
V=
a
1. y = 9 x2 , y = 0, x = 2, x = 3 about the y-axis
2. y = ex , y = 0, x = 0, x = 1 about
MATH 151B
Name:
Practice Exam 2: Volumes
Find the volume of the solid by rotating the following functions about the given axis using disk
method.
b
[f (x)]2 dx
V=
a
1. y = 9 x2 , y = 0, x = 2, x = 3 about the y-axis
since y = 9 x2 , now x = 9 y
5
y2
9 y
MATH 150B WORKSHOP
Spring 2010
12:00-12:50 MWF
MH 453
Peter Ho
peter ho @ csu.fullerton.edu
408 - 887 - 2294
Attendance
This clas is oered for credit/no-credit and thus attendance will be taken. Make sure to arrive to
workshop as schedule above on time an
MATH 151B
Jan. 27, 2010
Derivatives
1.
d
(c) = 0
dx
2.
d
[cf (x)] = cf (x)
dx
3.
d
[f (x) + g (x)] = f (x) + g (x)
dx
4.
d
[f (x) g (x)] = f (x) g (x)
dx
5.
d
[f (x)g (x)] = f (x)g (x) + g (x)f (x)
dx
6.
g (x)f (x) f (x)g (x)
d f (x)
=
dx g (x)
[g (x)]2
7
MATH 151B
May 3, 2010
Handout 27
Areas and Lengths in Polar Coordinates
By showing that we can represent functions rst in polar coordinates we can compute the area by
method of
1
A = r2
2
where r is given as radius and is the angle measured from the cent
MATH 151B
Jan. 27, 2010
Handout 1
The Substitution Rule: If u = g (x) is a dierentiable function whose range is an interval I and
f is continuous on I , then
f (g (x)g (x)dx =
f (u)du
Remember: If u = g (x), then du = g (x)dx
Evaluate the the following pr
MATH 151B
Jan. 29, 2010
Handout 2
Trignometric Integrals and Substitutions
Some integrals in calculus involve trigonometric functions that are usually not easily solved by just
substitution. With trigonometric substitutions, we allow some functions to be
MATH 151B
Feb. 1, 2010
Handout 3
Trigonometric Substitution
In some cases where evaluating integrals come in the form of a2 x2 in which requires a trigonometric substitution. With use of trigonometric identities allows for this type of substitution making
MATH 151B
Feb. 3, 2010
Handout 4
Partial Fraction Decomposition
When dealing with some integrals where there are polynomials involved in a rational function, it can
be quite deceiving as one can see, and without diculty the idea of decomposing a rational
MATH 151B
Feb. 17, 2010
Handout 6 continued
lHospitals Rule: Suppose f and g are dierentiable and g (x) = 0 on an open interval I that
contains a. Suppose that
lim f (x) = 0
and
xa
lim g (x) = 0
xa
or that
lim f (x) =
0
0
There must be an indeterminiate
MATH 151B
Feb. 19, 2010
Handout 7
Review: Evaluate the following integrals.
1.
3.
ex
dx
1 e2x
xe2x
dx
(1 + 2x)2
2.
4.
1 tan x
dx
1 + tan x
x2
dx
(x + 2)3
5.
3
x+1
dx
3
x1
6.
cos3 x sin 2xdx
Challenge: 2010 Shibaura Institute of Technology entrance exam
1
MATH 151B
Mar. 1, 2010
Handout 10
Arclength
On the subject of Arclength, we are interested in nding the length of curves. Which gives rise
to an interesting mathematical idea that has applications in the real world and in other areas of
math, namely geome
MATH 151B
Mar. 3, 2010
Handout 11
Applications to Physics and Engineering
One of the greatest perks of learning calculus is to actually begin applying the use of calculus to real
world situations! Where calculus is highly dominant in the world of physics,
MATH 151B
Name:
Practice Exam 3: Series
Find the sum of the series
1
1
2n 3n
1.
n=0
By Geometric Series
n=0
n
2
3
2.
n=0
1
2
n
n=0
1
3
n
=
1
1
1
2
1
1
1
3
=2
3
1
=
2
2
1
(n + 1)(n + 2)
Remember rst term is a Geometric series and second term is harmonic se
MATH 151B
Name:
Practice Exam 3: Series
Find the sum of the series
1.
n=0
1
1
2n 3n
2
3
2.
n=0
n
1
(n + 1)(n + 2)
Determine the convergence or divergence of the series
3.
n=0
n!
en
n!(x 2)n
4.
n=0
Find the sum of the convergent series by using functions w
MATH 151B
Mar. 12, 2010
Handout 14
Linear Equations
In order to solve certain types of linear dierential equations we must have the equations in the
form of
dy
+ P (x)y = Q(x)
dx
where P and Q are continuous functions on a given interval. Also consider th
MATH 151B
Mar. 17, 2010
Handout 15
Sequences
Can be dened as a list of numbers in a denite order
a1 , a2 , a3 , a4 , .an , .
where we can nd a general case of terms to the an terms using sequences and just like functions,
sequences have some similar prope
MATH 151B
Mar. 22, 2010
Handout 16
Series
There is an innite series when we add terms of an innite sequence in the expression of
a1 + a2 + a3 + a4 + . + an + . or
an
n=1
meanwhile using the idea of adding an innite series leads to a new term of sequences
MATH 151B
Mar. 24, 2010
Handout 17
Series Continued
Find the convergence or divergence of the given series.
1.
n=0
3.
n=0
2n+2
3n
2.
n=0
1
2 + 3n 2
9n
4.
n=0
8
(n + 1)(n + 2)
2
3
n
n2
1
+ 3n + 2
Challenge:
The Fibonacci sequence is dened recursively by an
MATH 151B
Apr. 5, 2010
Handout 18
The Integral and Comparison Tests
In some particular cases we are unable to nd a specic sum of a given series. Since it would not
be easy to compute a given sum as we take limn sn , we will develop the use of series testi
MATH 151B
Apr. 7, 2010
Handout 19
Other Convergence Tests
With the understanding of our integral and comparison tests we can go ahead to look at other
types of ways we could test for convergence. Such as
The Alternating Series - A series whose terms are a
MATH 151B
Apr. 9, 2010
Handout 20
Absolute Convergence
Given any corresponding series say.
|an | = |a1 | + |a2 | + |a3 | + .
n=1
We have by denition a series
an that is called absolutely convergent if the series of absolute
values
|an | is convergent. Now
MATH 151B
Apr. 12, 2010
Handout 21
Power Series
A Power series is a series in the form
cn xn = co + c1 x + c2 x2 + c3 x3 + .
n=0
Where x is a variable ad the cs are constants or mostly called the coecients of the series. Meanwhile here is the general case
MATH 151B
Apr. 14, 2010
Handout 22
Representing Functions as Power Series
To refresh things up, lets recall from several sections ago by considering the function given by
f (x) = 1/(1 x) and by the geometric series
a
, |r | < 1
1r
arn =
n=0
and looking at
MATH 151B
Apr. 16, 2010
Handout 23
Taylor and Maclaurin Series
Just as with nding representations of power series functions we look at a more general case in
nding functions that can represent a power series.
Power Series Representation - If f has a power
MATH 151B
Apr. 19, 2010
Handout 24
Parametric Curves
We have been dealing so far functions with a variable x to describe planar curves such as y = f (x)
or inversely x = g (y ). Here we will introduce a new variable t that describes complex curves where
f