Section 7.4
Advanced Integration Techniques: Partial Fractions
The method of partial fractions can occasionally make it possible to nd the integral of a
quotient of rational functions. Partial fractions gives us a way to break a fraction into smaller
comp
Math 1152 Final Study Material
OSU, Spring 2013
The following materials are included in this PDF:
Course topics overview
Suggested problems list
Formula sheet you will receive during the final exam
Quizzes #1 #10 with solutions
Exams #1 #3 with solutions
Section 6.2
Determining Volumes by the Shell Method
Depending on the shape of the solid, it might be dicult to use slicing or the disc/washer
method to evaluate its volume. For instance, neither method would be conducive to nding the
volume of the shape b
Name: goiw JV: (Ln/«g
Exam 1
Calculus II, §10
February 9, 2012
Instructions: Clearly show all your work and boa: your nal answer. No calculators allowed.
1. (10 points) Use logarithmic differentiation to nd the derivative of y = (3:2 sin (106$.
: is»,
Section 9.1
Polar Coordinates
Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are
measured along the corresponding axes. However, this is not the only way to represent points in
the plane; in this section we will
U3
Name: 3" l l 0 W 8.
Exam 2
Calculus II, §10
March 8, 2012
Instructions: Clearly ShOW all your work and boa; your nal answer. No calculators allowed.
1. Evaluate the following integrals:
2
Inst
1 ' 13 md
(a) ( 5p01n s)/1 372 $
L l 2' l 2'
«I J v IWX
Cl QM C
Name:
Exam 3
Calculus II, §10
April 19, 2012
Instructions: Justify each of your answers by clearly showing all your work and your nal answer.
No calculators allowed.
1. (a) (5 points) Does the sequence converge? If so, what is its limit?
m,
Section 10.1
Three-Dimensional Coordinate Systems
The plane is a two-dimensional coordinate system in the sense that any point in the plane can be
uniquely described using two coordinates (usually x and y, but we have also seen polar coordinates
and ). I
Section 7.7
Improper Integrals
Two dierent types of integrals can qualify as improper. The rst type of improper integral
(which we will refer to as Type I) involves evaluating an integral over an innite region:
Alternatively, we may need to evaluate an in
Section 8.3
In section 8.1, we saw that a nondecreasing sequence converges if and only if it has an upper
bound. We may occasionally be able to use this fact to our advantage when attempting to determine
whether or not a series converges. In particular, s
Section 8.8
Taylor Series
Given a function f (x), we would like to be able to nd a power series that represents the
function. For example, in the last section we noted that we can represent ex by the power series
ex = 1 + x +
x2 x3 x4
+
+
+ .,
2!
3!
4!
an
Section 8.2
Series
We can use innite sequences to create a series by adding the terms of the sequence. A series
is a sum of the form
an = a1 + a2 + a3 + a4 + .
i=1
The number an is the nth term of the series.
On rst inspection, the sum above really doesnt
Section 8.5
The Ratio Test and the Root Test
The Ratio Test is a convergence test that is often helpful when we wish to determine the
convergence or divergence of a series whose terms involve factorials in terms of n or terms raised
to the power n:
Theore
Math 1620
Practice Test 3
March 29, 2012
Name:
You must show ALL of your work in order to receive credit.
If your scratch paper shows work that leads to your solution,
please turn in in inside the test. Otherwise, dispose of it yourself.
1
1. Does the seq
1. The sequence converges to 0 since
( )
2n
1 2 n 1
lim
= lim
= 0 = 0.
n 3n+1
n 3
3
3
2. The series is geometric. Rewrite it as
1 ( 2 )n 1 ( 2 )n1
=
3 3
3 3
n=0
to see that a =
1
3
n=1
and r = 2 . So the series converges to
3
1
a
= 3
1r
1
2
3
= 1.
3. Th
Section 6.3
Determining Curve Length
In this section, we will learn how to determine the length of a curve. For instance, we might
want to determine the length of the curve below:
To think about the meaning of curve length, imagine laying a piece of strin
Series
P series
converges if P >1
diverges if P 1
Integral
only convergent if
0 < f(x) <
Comparison An Bn
if An diverges then Bn diverges
if Bn converges then An converges
Limit Comparison Lim n of = L will converge only if 0 < L <
proves they have th
Section 6.4
Areas of Surfaces of Revolution
When we revolved a two-dimensional gure about a line, we created a three-dimensional solid.
Each solid has a skin, or two-dimensional surface. In this section we will learn how to nd the
surface area of the surf
Section 7.1
Advanced Integration Techniques: Integration by Parts
We may dierentiate the product of two functions by using the product rule:
d
f (x)g(x) = f (x)g(x) + f (x)g (x).
dx
Unfortunately, nding an antiderivative of a product is not so straightfor
Section 7.2
Advanced Integration Techniques: Trigonometric Integrals
We will use the following identities quite often in this section; you would do well to memorize
them.
sin2 x = 1cos(2x)
2
cos(2x) = 1 2 sin2 x
sec2 x = 1 + tan2 x
cos2 x = 1+cos(2x)
2
co
. - EXAM 3 MATH 1620 03-09-2009
(11 7pts (a) State sandwich theorem for sequences:
,3 /
(b) Find limit as n->infinity of a_n foi' (i) a_n= 1620"-n (ii) (-1)"n*n\(-1)
" . «\/ .m in), -r - , ,. 7
sf 2/
. l 'i \. , W
(miH _\ i C
I 1 i) i
8 pts (a) State the
Fall 2013 Calc II Exam 1 Review
Ross Kilgore
October 2013
7.1, Number 15: Evaluate the integral:
(ln x)2 dx.
Solution: First use integration by parts:
u = (ln x)2
1
du = 2 ln x x dx
So
(ln x)2 dx = x(ln x)2 2
dv = dx
v=x
1
ln x xdx = x(ln x)2 2 ln xdx.
x
Instructions: By signing this exam, you agree to adhere to the policies of MU re
garding academic integrity. No calculators are allowed. There are ten problems worth
ten points each. To receive full credit for a problem, you must show all of your work.
An
Double Angle Formulas
Sin(2) = 2SinCos
Cos(2) = Cos Sin
Cos(2) = 2Cos 1
Cos(2) = 1 2Sin
Tan(2X) = 2TanX / (1 - tan^2X)
Half Angle Formulas
Sin x = (1-cos(2x) / 2
Cos x = (1+cos(2x) / 2
Sin = 1 - Cos
Cos = 1 - Sin
Tan = Sec 1
[e]' = (x)' e
[a]' = a ln(
Math 1620
Practice Test 3
March 29, 2012
Name:
You must show ALL of your work in order to receive credit.
If your scratch paper shows work that leads to your solution,
please turn in in inside the test. Otherwise, dispose of it yourself.
1
1. Does the seq
Section 10.4
The Cross Product
In this section we will dene another kind of multiplication on vectors; however, unlike the dot
product, which returns a real number, the cross product returns another vector.
In order to determine the cross product of a pai