Math 2300: Calculus II
Project 1: Integration of Trigonometric Functions
By using trig identities combined with u/du substitution wed like to nd anti-derivatives of the
form sinm x cosn x dx (for integer values of n and m). The goal of this project is for
Math 2300: Calculus II
Project 13: The SIR Model for Disease Epidemiology
This worksheet will analyze the spread of Ebola through interaction between infected and
susceptible people. Ebola is an infectious and extremely lethal viral disease that rst surfa
MATH 2300: CALCULUS 2
September 20, 2006
MIDTERM 1
I have neither given nor received aid on this exam. Name:
001 J. Newhall . . . . . . . . . . . (9am) 002 S. Preston . . . . . . . . . . . (10am) 003 K. Kearnes . . . . . . . . . . (11am)
004 S. P
CHAPTER 1
Functions
EXERCISE SET 1.1
1. (a) around 1943 (b) 1960; 4200 (c) no; you need the year's population (d) war; marketing techniques (e) news of health risk; social pressure, antismoking campaigns, increased taxation 2. (a) 1989; $35,600 (b)
Math 2300: Calculus II
Project 10: Numerical Integration using Power Series
Background for the example:
In probability it is important to be able to nd areas under the bell curve. The bell curve
is formally known as the normal distribution, and the functi
Math 2300: Calculus II
Project 6: Taylor Polynomials
1. Recall the idea of linear approximation, which we will use to approximate the numbers e.1
and e. Let f (x) = ex . We want to nd a linear function L(x) = C0 + C1 x whose value and
whose derivative at
Math 2300: Calculus II
Project 5: Volumes and arclengths
1. (a) Graph the region bounded by y = e
x,
y = e and the y-axis.
3
2
1
1
1
2
3
4
1
(b) Write an integral that will give the area of this region by slicing vertically.
1
ee
Solution:
x
dx
0
(c) Use
Math 2300: Calculus II
Project 3: Comparison of Improper Integrals
Developing function sense will be important in Chapters 9 and 10.
Problems 1-4 will help develop your numerical function sense.
1
1. Consider the improper integral
3
10
(a)
1
x2 ln(x)
3
10
Math 2300: Calculus II
Project 2: Techniques of Integration
1. The following integrals look similar, but are evaluated very dierently.What techniques work
on each one?
(a)
4
dx
9 x2
B
4
A
Solution: Partial fractions. 9x2 = 3+x + 3x , so A(3 x) + B(3 + x)
HOMEWORK 4 SOLUTIONS
MATHEMATICS 2300: CALCULUS I, FALL 2013
Section 7.7 Problem 40
If p = 1 then using the u-substitution u = ln x we have
e
ln x
ln x
dx = lim
dx
b x
x
(ln x)2 b
= lim
b
2 e
(ln b)2 (ln e)2
= lim
b
2
2
2
1
(ln b)
= lim
b
2
2
(ln b)2 1
=
Calculus 2
Homework Set 8 - Solutions
9.2 2
1
1 1 1 1
1 + + + + + =
2 3 4 5
1
n
n=1
This is the harmonic series, not a geometric series.
9.2 8
2
3
4
(x)n
1 x + x x + x =
n=0
The is a geometric series. The rst term is a = 1. The ratio between successive te
Calculus 2
Homework Set 8 - Solutions
9.2 2
1
1 1 1 1
1 + + + + + =
2 3 4 5
1
n
n=1
This is the harmonic series, not a geometric series.
9.2 8
2
3
4
(x)n
1 x + x x + x =
n=0
The is a geometric series. The rst term is a = 1. The ratio between successive te
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Calculus 2
Homework Set 9 - Solutions
1
For 9.4 13, use the comparison test to conrm the statements are true.
9.4 1.
n=4
1
diverges, so
n
n=4
1
diverges.
n3
1
1
1
For n 4,
=
> 0. The series
n3
n0
n
n=4
1
diverges since it is a tail of
n
the harmonic serie
HW 3 SOLUTIONS
Section 7.4
2
x
dx
9x2
46.) Find
Solution:
Let x = 3 sin , so dx = 3 cos d.
Substituting into our integral we get:
2
x
dx
9x2
=
2
(3 sin )
9(3 sin )2
2
3
sin (3 cos )d
cos2
9( 1
2
1
4
sin2
cos (3 cos )d
=3
sin 2) = 9( 1
2
1
2
9 sin
2
Series - summing it all up
Heres a list of all of the convergence tests for series that you know so far:
Divergence test (a.k.a. n-th term test)
Geometric series test
Integral test
p-series
Term-size comparison test (your book calls this the comparis
Math 2300 Calculus II University of Colorado
Final exam review problems: ANSWER KEY
2
1. Find fx (1, 0) for f (x, y) =
xesin(x y)
. 2
(x2 + y 2 )3/2
2. Consider the solid region W situated above the region 0 x 2, 0 y x, and bounded above by
2
the surface
Math 2300
Exam 1 topics
The exam will cover sections 7.1, 7.2, trig integrals, 7.4, 7.5, 7.7 and 7.8. Here is a list of topics that you
should have mastered for the exam:
Know the integrals of cos(x), sin(x), sec2 (x), sec(x) tan(x), x1 , ex , ax , xr ,
Math 2300 Calculus II University of Colorado
Final exam review problems:
2
1. Find fx (1, 0) for f (x, y) =
xesin(x y)
.
(x2 + y 2 )3/2
2. Consider the solid region W situated above the region 0 x 2, 0 y x, and bounded above by
2
the surface z = ex .
(a)
Math 2300, Spring 2014
Exam 2 topics
The exam will cover sections 8.1 - 8.4, 10.1, and 9.1 - 9.3. We will complete 9.3 on Monday, and the
remaining days will be spent reviewing.
Here are the topics that the exam will cover:
Chapter 8:
1. For any of the ap
REVIEW 1: Key
1. Find the distances of the point (2, 3, -1) to
(a) the xy-plane
(b) the point (1, 4, -3) p
Key: (a) | 1| = 1; (b) (1 2)2 + (4 3)2 + (3 (1)2 = 6.
2. Give an equation of the sphere of radius 2 centered at the point (1, 2, 3).
Key: (x1)2 +(y
11.1 SOLUTIONS
857
CHAPTER ELEVEN
Solutions for Section 11.1
Exercises
1. (a) = (III), (b) = (IV), (c) = (I), (d) = (II).
2. (a) (III) An island can only sustain the population up to a certain size. The population will grow until it reaches this
limiting
Harvey Mudd College Math Tutorial:
Special Trigonometric Integrals
In the study of Fourier Series, you will find that every continuous function f on an interval
[L, L] can be expressed on that interval as an infinite series of sines and cosines. For
examp
Integral Practice Problems
Evaluate the following integrals. Note: The last two pages are significantly more challenging.
Z
1.
e x dx =
Z
2.
cos x
p
2 sin2 x
Z
1
dx =
x ln x
Z
x5 +
3.
4.
3
x
x1
4
3
dx =
dx =
Z
5.
x sec x tan x dx =
Z
6.
x
dx =
x1
Z
sec x