Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
November 5, 2009
Quiz 4
1. [5 points each - 10 points total]
a. [5 points] Find the sum of the series
k=2
3k1
.
43k+1
This is a convergent geometric series since it is
=
1
12
k=2
3
4
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
November 19, 2009
Exam 3
1. [5 points each - 10 points total] Determine whether the following series converge or diverge:
n10 + 10
a. [5 points]
n!
n=1
Here we utilize the Ratio Test
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
November 19, 2009
Exam 3
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed bo
Computing Work via Definite Integrals
Fact: The work that a constant force, F, does when moving an object over a distance of
d is given by the formula W = F d .
However, most forces are not constant and will depend upon where exactly the
force is acting.
Computing Volumes via Definite Integrals
Basic Idea:
Given a solid S , we will use the area of a typical cross-sectional
slice of the solid to construct an appropriate definite integral.
Suppose the solid S is bounded between the lines x = a and x = b .
L
McCombs Math 232
Extra Practice
Chapter 6 and 7
Solution Key
Text Sections 6.16.2
1.
Use calculus to compute the exact a rea of the region in the first quadrant bounded between the curves
x = y 2 and y =
1
on the interval 1 ! x ! 4 .
x
Make sure that you
McCombs Math 232
Working with Power Series
1.
Given that the power series
!
" cn x n converges for x = !4 , and diverges for x = 6 .
n=0
Determine whether the following series converges or diverge.
Solution:
Note that
!
" cn x n is a power series centered
McCombs Math 232
Trigonometric Integrals
Trigonometric Integrals
1.
2
2
ODD power of cosine: Save a factor of cos ( x ) , use the sin ( x ) + cos ( x ) = 1 identity
and the substitution u = sin ( x ) .
2.
2
EVEN power of cosine: Half angle identity. cos (
McCombs Math 232
More Chapter 11 Practice Problems
Solution Key
Sections 11.5 11.8
1.
Use the Absolute Ratio Test to determine the interval of convergence for the
given power series. Make sure you check the endpoints.
"
2 n !1
n !1 x
!1
2 n !1 !
n =1
#( )
McCombs Math 232
More Chapter 11 Practice Problems
Solution Key
Sections 11.511.9
1.
Use the Absolute Ratio Test to determine the interval of convergence for the
given power series. Make sure you check the endpoints.
"
#( ) (
n =1
!1
n !1
x 2 n !1
2n ! 1
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
October 20, 2008
Exam 2
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed boo
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
October 20, 2008
Exam 2
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed boo
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 17, 2008
Quiz 2
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed b
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 17, 2008
Quiz 2
1. [10 points] Evaluate the integral
6x x2 8 dx.
x
We begin by completing the square to see that the integral is
1 (x 3)2 dx.
x
We then use the substitution
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
October 8, 2009
Quiz 3
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed book
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
November 5, 2009
Quiz 4
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed boo
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
October 8, 2009
Quiz 3
1. [10 points] Find the area of the region bounded by the curves
x + y = 2y 2 ,
y = x3 .
Note that these curves intersect at the points (0, 0) and (1, 1).
It i
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 3, 2008
Quiz 1
1. [10 points] Evaluate the integral
(ln x)2 dx.
1
Here we use integration by parts with u = (ln x)2 and dv = dx. Thus, du = 2 ln x x dx
and v = x. Thus,
(ln
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 3, 2008
Quiz 1
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed bo
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 24, 2008
Exam 1
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed b
Math 232 - Calculus of Functions of One Variable II (Metcalfe)
Fall 2009
September 24, 2008
Exam 1
Please show all of your work, and justify your answers completely. No credit will be given for
answers which are lacking supporting work.
This is a closed b
Trigonometric Substitutions
Solution Key
Basic Idea:
Method:
We can develop a procedure that will help us integrate complicated
radical expressions.
Rewrite the given integral using an appropriate inverse trig triangle.
Key Formulas:
(i)
To replace
a 2 x
Representing Functions via Taylor Polynomials and Taylor Series
Basic Idea:
We want to represent a non-polynomial function f ( x ) by carefully constructing an
appropriate polynomial Tn ( x ) . In particular, we want the polynomial and the original
functi
McCombs Math 232
Summary of Calculus I Facts
Math 232 assumes that you are already comfortable with the following facts and concepts:
1.
Product Rule, Quotient Rule and Chain Rule for derivatives.
2.
Power Rule for Antiderivatives.
2.
Basic Properties of
Application of Differential Equations
Basic Idea:
A differential equation is an equation that contains an unknown function
and one or more of its derivatives. To solve a differential equation means
to find a function that satisfies the equation.
Example:
Math 232 Final Exam Practice Problems
Answer Key
1
! x 2 + 1$
6
# x & dx = ( + 3
#e &
e
"
%
0
'
1.
Evaluate the integral.
2.
Find the exact area of the region bounded by y = e x ! 1 , y = x 2 ! x , and x = 1 .
11
Area = e !
square units
6
3.
()
()
Find th
McCombs Math 232
Practice Test 3
Key
Test Covers Sections 6.16.2, 7.17.4, 11.10, 11.11
1
x.
6
Set up the definite integral required to compute the area of the region R by
36 "
1%
integration with respect to x.
A = ( $ x ! x ' dx
0#
6&
Set up the definite
McCombs Math 232
Practice Test 3
Test Covers Sections 6.16.2, 7.17.4, 11.10, 11.11
1.
1
x.
6
Set up the definite integral required to compute the area of the region R by
integration with respect to x.
Set up the definite integral required to compute the a
Math 232 Final Exam Practice Problems
1
! x 2 + 1$
# x & dx
#e &
"
%
0
'
1.
Evaluate the integral.
2.
Find the exact area of the region bounded by y = e x ! 1 , y = x 2 ! x , and x = 1 .
3.
4.
()
()
Find the Taylor series expansion of f x = cos 3x centere
1.
Math 232 Practice Test 2
Chapter 11: Sections 49
Answer Key
Use the Absolute Ratio Test to determine the convergence interval for the
given power series. Make sure you check the endpoints.
"
(i)
#
( x +1)n !n!
3n
n =1
"
(ii)
( !1)n+1(3x )2 n
# ( 2 n) !