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
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 cha
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
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, the
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 deriva
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
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
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 the
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 f
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 pe
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 whet
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
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
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
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
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
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
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
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 (D
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 ,
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
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 the
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 anyo
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
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 conti
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 + +
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]
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
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
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