Math 135
Quiz 3 Sample Solutions
February 11, 2014
Show your algebra whenever calculations are done by hand.
1. What is the frequency (per unit time) of the function h(t) = 5 cos(14t)?
The frquency is the number of periods that occur in one unit of time.
Math 135
Quiz 2 Sample Solutions
February 4, 2014
Show your algebra whenever calculations are done by hand.
2
1. Let f (x) = x2x 9 3 . Sketch the graph of f . (Hint: After simplifying, make a sign chart
+2x
for f .) For which x is f not dened? Are any of
Math 135
Quiz 1 Sample Solutions
January 28, 2014
Show your algebra whenever calculations are done by hand.
1. Use algebra to nd the points of intersection of the line y = 1 x and the parabola
y = x2 1. Your answer should be two ordered pairs. (You may nd
MATH 135 Handout 6 Solutions
February 28, 2014
Solve the following problems using the product rule or the quotient rule for derivatives:
Find g (x) for g (x) = ex (x3 + 3x).
Use the product rule, then simplify:
g (x) = ex (x3 + 3x) + ex (3x2 + 3) = ex (x
MATH 135 Handout 5 Solutions
February 21, 2014
For each function (one on the front, one on the back), sketch the derivative of the function on
the second set of axes. Pay attention to where the derivative is zero, positive, and negative.
Figure 1: The gra
MATH 135 Handout 4 Solutions
February 19, 2014
For each of the following functions, use a limit of a dierence quotient to obtain the
solution.
Let h(x) = 2x3 3x. Find the slope of the tangent line to the graph of h at a = 2.
The dierence quotient for the
MATH 135 Handout 3 Solutions
February 5 , 2014
For each of the following functions, determine the interval(s) on which the function is
continuous.
Notice in the following problems, dierent letters are being used for the variables and the
functions. While
MATH 135 Handout 2
January 29, 2014
Figure 1:
Figure 1 shows the graph of a function. For each of the point x = 1, x = 1, and x = 2,
what are the one-sided limits and the two-sided limit? What do these limits tell us about the
continuity of f at these po
MATH 135 Handout 1 Solutions
January 24, 2014
Figure 1: Piecewise dened function.
Figure 1 shows the graph of a piecewise dened function.
Using function notation, write out a split denition formula for this function. (Hint: How
many pieces are there?)
Fi
MATH 135
First Hour Exam Sample Solutions
February 15, 2014
You may use your your calculator. Show your work, partial credit will be given.
1. (20 points) Short answer:
(a) A function y = f (x) has period , what is the period of f (3x)?
The period is . Mu
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Solutions to Final Exam Practice Questions
1. For each sequence of functions fn , determine the pointwise limit function f and whether ir not fn
converges uniformly to f on [0, 1].
(a) f
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 2 Practice Problems: Solutions
1. Suppose f : M R is continuous and A is a dense subset of M (cl(A) = M ). Decide whether each
statement below is true or false. If true, provide a p
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 1 Practice Questions: Solutions
1. Determine which of the following are metrics. For those that are metrics, prove it, and for those that
are not, demonstrate that one of the metric
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 7, Due Friday, April 29
1. (a) Show that if f is absolutely continuous, then f is uniformly continuous.
(b) Give an example of a uniformly continuous function that is not
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 6, Due Friday, April 1
1. Suppose f =
Show that
n=1
fn , where fn is a sequence of nonnegative measurable functions.
fn .
f=
n=1
2. Let fn be a sequence of L1 functions s
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 5, Due Friday, March 18
1. We proved in class that
f +g =
f+
g
and
cf = c
f
for nonnegative measurable functions f and g and nonnegative constants c. Using the
denition
f
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 4, Due Friday, March 4
1. Let A = Q [0, 1] and B = Qc [0, 1]. Show that A and B are measurable, and nd
m(A) and m(B ). Show that A is measurable and nd its measure.
2. Sh
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 3, Due Friday, February 11
1. Let Inv be the inverse operator Inv(X ) = X 1 , dened on the space of n n invertible
matrices. Show that Inv is dierentiable with derivative
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 2, Due Friday, February 4
1. Suppose U is a connected open subset of Rn and f : U Rm is twice dierentiable,
with (D2 f )p = 0 for all p U . What can you say about f ?
2.
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 1, Due Friday, January 28
1
1
1. Let T : R2 R3 be the linear transformation with matrix A = 1 1. Compute
1 1
2
T , and nd a vector in v R such that T (v ) = T |v |.
2. Fi
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Exam 2, Due Wednesday, April 13
Professor Levandosky
You have one week to work on the test. You may not consult with anyone (except me)
about the exam. You may use your notes, the
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 2 Solutions
1. For each pair of metric spaces A and B (with the usual Euclidean metric), either nd a
homeomorphism between them or prove that no homeomorphism exists between them.
(
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Exam 1, Due Wednesday, February 23
Professor Levandosky
You have one week to work on the test. You may not consult with anyone (except me)
about the exam. You may use your notes a
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 1 Solutions
1. Let M = R, d(x, y ) = |ex ey |.
(a) Show that d is a metric on M .
Solution. d(x, y ) = |ex ey | 0 for all x, y M . It is clear that d(x, y ) = 0 if
x = y . If d(x, y
Math 136: Calculus 2
Continuous Income Streams
Professor Levandosky
1. Suppose each of the following continuous income streams is invested in an account that
earns 3% annual interest, compounded continuously. Calculate the following for each
income stream
Math 136: Calculus 2
Fall 2012, Professor Levandosky
Geometric Series Worksheet
Recall the following basic facts about geometric series.
The sum of a nite geometric series is
n
ark = a + ar + ar2 + + arn =
k=0
a(1 rn+1 )
1r
The sum of an innite geometri
Math 136: Calculus 2
Worksheet 3, Fall 2012
Professor Levandosky
The center of mass of a planar region with constant mass density that lies below the
graph of a function f (x) over an interval [a, b] has coordinates (x, y ) given by
b
xf (x) dx
a
x = b
a
Math 136: Calculus 2
Worksheet 2, Fall 2012
Professor Levandosky
1. For each improper integral, (a) identify the point or points where the integral is improper and (b) evaluate the integral.
/2
sec2 (x) dx
(a)
0
1
(b)
1
1
1
dx
1 x2
ln(x) dx
(c)
0
2. Supp
Math 136: Calculus 2
Spring 2012
Worksheet 4: Center of Mass
Professor Levandosky
The center of mass of a planar region with constant mass density that lies below the
graph of a function f (x) over an interval [a, b] has coordinates (x, y ) given by
b
xf
Math 136: Calculus 2
Spring 2012
Worksheet 3: Improper Integrals
Professor Levandosky
1
1
dx in each of the following cases. Either show that the integral diverges,
p
0x
or show that it converges and nd its value.
1. Evaluate
(a) p < 1
(b) p = 1
(c) p > 1