Math 31
Quiz 16 (to discount the disreputable Quiz 12)
July 18, 2013
credit given, work shown.
Determine if each of the following series converge or diverge.
1.
n 1
1
n
n
2.
1
n 1
n
n!
n 1
n 1
Math 31
Quiz 15
July 18, 2013
credit given, work shown.
1.
2.
1n
Determine if
is monotonic and if it is bounded (if so, give both the upper
n 1
and lower bound and, of course, justify your answer).
Determine if the following series converges or dive
Math 31
The Fun Quiz
1 November 20013
David Chayka
4
n
1
4
n n
Use our precise definition of convergence for sequences to prove the following statement: lim
Lets show that this particular choice for N works.
0, let ' s choose N
0, N an L whenever n N
Math 31
The Fun Quiz
1 November 20013
Use our precise definition of convergence for sequences to prove the following statement:
David Chayka
Lets show that this particular choice for N works.
Math 31
Quiz 14
July
18, 2013
Let . If the rate at which y changes with respect to x is jointly proportional to both z and the
inverse tangent of x, then write the corresponding differential equation and solve that differential
equation.
Math 31
Let z
=
Quiz 14
July 18, 2013
y 2 + y . If the rate at which y changes with respect to x is jointly proportional to both z and the
inverse tangent of x, then write the corresponding differential equation and solve that differential
equation.
Aside
Math 31
Take Home Quiz
2x
Solve the following differential equation.
4
26 October 2013
x3 2 x 2 2 x 12
David Chayka
dy
2 x5 3x 4 3x3 10 x 2 17 x 22 y ln 2 y ln y 1
dx
x2
2 x x 2 x 2 x 12 2 x 3 x 3 x 10 x 17 x 22
4
y
1
ln 2 y ln y 1
1
du dy
y
3
1
u u 1
Math 33
Solutions to Quiz II
July 11, 2013
Solve the following differential equation:
2x
4
x 3 2x 2 2x 12
dy
2x 5 3 x 4 3 x 3 10x 2 17x 22y ln2 y ln y 1
dx
Separating variables and integrating each side, we have that:
1
2x 5 3 x 4 3 x 3 10 x 2 17 x 22
d
Math 31
Take Home Quiz
1. Use Simpsons Rule to approximate
7 ln(2)
ln(2)
7 ln(2)
ln(2)
David Chayka
cosh( x)dx if we subdivided the appropriate interval into 6 subintervals
2. What value of n guarantees us that S n approximates
1.
19 October 2013
7 ln(2)
1.
Use S 4 to approximate
Quiz 1
Math 31
ln x
3
1
2
2
July 3, 2013
1 dx . Give both a completely simplified exact
value and an approximation rounded to the nearest 0.001.
2.
How accurate is the value you found in #1 to the actual value of the above
def
Math 31
Take Home Quiz
19 October 2013
1. Use Simpsons Rule to approximate if we subdivided the appropriate interval into 6 subintervals
2. What value of n guarantees us that approximates accurate to within 0.0001?
1.
2.
David Chayka
Math 31
Quiz (10012)
2013 July 3
1.) Use to approximate. Give both a completely simplified exact value and an approximation rounded
to the nearest 0.001.
2.) How accurate is the value you found in #1 to the actual value of the above definite integral? Giv
Math 31
Solutions to Quiz 8
Evaluate the following integral:
We will first let u e
and du
e
July 2, 2013
x
x xe 2
x
dx
x
1
2x
e x dx
Consequently, we have that
e
x
x xe2
x
1
ex
dx 2 2 x 2 x dx
1 e
1
2
du
1 u 2
1
1
2 2 2 du
1 u 1 u
ln 1 u ln 1 u C
Math 31
1.
Last Chance Quiz
Use the notion of geometric series to evaluate the following indefinite integral:
2.
July 25, 2013
ln 1 x
dx
x
Determine the radius and interval of convergence for the following:
4n 2x 3n
3n 2 1
n 0
Math
1.
2.
31
Additional Exercises Reqardinq Differential Equations Julv 11. 2013
A water reservoir that holds 100 million gallons supplies a city with 1 million
gallons a day. The reservoir is partly refilled by a spring that provides 0.9 million
gallons
Math 31
1.
Additional Exercises Regarding Numerical Integration
Approximate
2
0
July 11, 2013
x 2dx using L4 , R4 , T4 , M 4 , and S4 . Find the maximum error in
the last three estimates, and determine the exact area that is being
approximated.
2.
Into ho
Additional DE Exercises
A culture of bacteria (Algebrania) has a growth rate proportional to the tangent of the population
at time t where and . Write and solve the DE.
A population grows at a rate that is jointly proportional to the square of that popula
Math 31
I.
Tests for Convergence
July 11, 2013
Sequence:
1.
2.
II.
The sequence an n 1 converges if lim an L for some finite real number L
The sequence an n 1 diverges if lim an L for some finite real number L
n
n
Series:
1.
Special series:
a)
Consider
\ '\ \
\q
Math 31
Exam
T).r*r*1
.,/
*n*
\r-.,*
I
June 77, 2013
Show all of your work, completely simplify all answers, and give exact vatues only.
1.
(15 pts) Determine the length of the arc along the graph of y
4-
\f
rt
*x
J
*i9
*x
=
tfr) *
;
tu1x4
J*
th
Math 30
1.
Limits
February 28, 2013
Use the , definition of the limit to prove lim2x 3 7
x 5
Part 1: Find a that works:
Given 0, we need to find a s uch that :
2 x 3 7 whenever 0 x 5
2 x 10
when ever
"
2x - 5
whe never
"
2x-5
w henever
"
x-5
wh eneve
Math 31
Additional Homework Exercises
July 29, 2013
Use our analysis of intervals technique to clearly sketch the graphs of the curves
described by the following sets of parametric equations
1.
x t2 3
y 2t 3 54t
2.
x t2 4
y 4t 3 6t 2 24t
Math 31
Examples of Series
July 16, 2013
Determine if the following series converge or diverge:
Divergent series, geometric series, and p-series:
n
1.
n 1
n 1
2.
2 3n 1
1
5 n 2 n 2 3n
n 1
3.
n
1
n
n 2
4.
Estimate the error for
1
n
n 1
5.
How many ter
Basic Polar Curves Handout
Math 121-D. Benedetto
Recognize a few of the following basic polar curves. Try and understand the sketches.
r = sin
r = 2 sin
circle of radius
1
2
centered at 0, 1
2
circle of radius 1 centered at (0, 1)
Note: These circles
SERIES Review of Tests for Convergence / Divergence
Math 31 - Buchwald
1. Test for Divergence: If lim an 0 , then the series
n
a
diverges.
n
n =1
2. Common Series we see frequently (we often use them for comparisons):
a. Geometric Series: ar n1
n =1
If r