Mathematics Teaching Portfolio
Bevin Maultsby
maultsby[at]umn.edu
The aim of this teaching portfolio is to summarize my experience as a teaching-focused
postdoctoral fellow at the University of Minnesota and as a graduate teaching instructor
at the University of North Carolina. It includes my teaching philosophy, several handouts
written by me from different classes I have taught, and student comments from evaluations.
Contents
1 Summary of Teaching Experience . . . . . . . . . . . . . . . . . . . . . . .
1
2 Evaluations from UMN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3 Student comments from UMN . . . . . . . . . . . . . . . . . . . . . . . . .
6
4 Evaluations from UNC, incl. comments
7
5 Sample Moodle Site
6 Sample Syllabus
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Sample Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Examples of student work
. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9 Examples of in-class group activities
. . . . . . . . . . . . . . . . . . . . . 29
10 Sample of in-class notes at UNC . . . . . . . . . . . . . . . . . . . . . . . . 41
11 Example of a “curve worksheet” at UNC . . . . . . . . . . . . . . . . . . . 43
12 Sample problem set from Math Club . . . . . . . . . . . . . . . . . . . . . 45
1
Summary of Teaching Experience
I am a Math Center for Educational Programs (MathCEP) postdoc in the School of Mathematics at the University of Minnesota. Most of my courses involve the University of Minnesota Talented Youth in Mathematics Program (UMTYMP), an accelerated five-year program for middle and high school students. As a professor in the program, I coordinate and
1
lecture for large honors-level university courses from the calculus sequence and linear algebra. Each lecture is supplemented by an hour-long recitation involving groupwork with a TA.
For these sessions, I design weekly worksheets and activities ranging from standard practice
problems to highly conceptual exercises that encourage the students to think creatively.
At UMN I also teach departmental courses. Last year I taught a second-semester differential equations course for math majors. In addition, I supervised two undergraduate
students for their senior projects.
I attended the University of North Carolina at Chapel Hill for both my undergraduate
and graduate studies; I began working for the mathematics department as an undergrad as
a TA and a help center tutor, and I have included some of this experience with my graduate
experience in this portfolio.
Since the University of North Carolina at Chapel Hill does not have a true engineering
school, mathematics classes during my time at the university typically had no more than 35
students and were taught by single instructors. As a graduate student I therefore gained a
considerable amount of teaching experience. In addition to being a TA for various classes
(both during my undergraduate years at Carolina as well as during my graduate years), I was
the primary instructor for 10 undergraduate courses. During my fifth year, I was recognized
for my teaching with the Linker Award, awarded to 1-2 graduate students per year.
In summary, I have been the instructor of record at UMN for
• UMTYMP Calculus I, Fall Semester 2014. This course corresponds to a first semester
of two-semester long single-variable calculus course. It begins at limits and covers
differentiation, including applications (e.g. optimization).
• UMTYMP Calculus II, Spring Semester 2016. This course corresponds to a first
semester of linear algebra: matrices, linear transformations, vector spaces and subspaces, orthonormal bases and matrices, orthogonal projections and transformations,
determinants, diagonalization, eigenvalues in R and C, eigenvectors, stability, dynamical systems.
• UMTYMP Calculus III, Fall Semester 2015. This course is an advanced multivariable
calculus course with emphasis on theory. It focuses on curves, surfaces, and differentiation.
• UMTYMP Calculus III, Spring Semester 2014. This is a continuation of the previous course, with a very thorough treatment of line integrals, multiple integration, and
the major fundamental theorems, culminating with forms and the Generalized Stokes’
Theorem.
• Math 4512: Differential Equations with Applications, Spring 2015, Spring 2016. Major
topics include first-order equations, first-order systems, linear and nonlinear systems,
forcing and resonance, numerical methods and Laplace transforms.
At UNC:
• Math 110: College Algebra, Summer 2011 and Summer 2013 (as part of the UNC
summer bridge program), Spring 2014
• Math 118: Selected Topics in Mathematics, Fall 2010 and Summer 2013
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• Math 231: Single Variable Calculus I, Spring 2011
• Math 232: Single Variable Calculus II, Spring 2013 and Fall 2013
• Math 233: Multivariable Calculus, Spring 2014
• Math 521: Advanced Calculus, Summer 2014. A rigorous proof-based course on real
numbers, continuity, uniform continuity, differentiability of functions of one variable,
uniform convergence, infinite series, power series, Riemann integration in one variable.
I have been a teaching assistant and grader for
• Math 118: Selected Topics in Mathematics, Summer 2008
• Math 233 and 233H: Multivariable Calculus (Honors), Summer and Fall 2008, Fall 2009
• Math 381: Discrete Mathematics, Spring 2010 (two sections)
• Math 548: Combinatorics, Spring 2009
• Math 551: Euclidean and non-Euclidean Geometry, Fall 2010
• Math 657: Qualitative Theory of Differential Equations, Spring 2012
In addition, I worked for eight semesters in the Math Help Center at UNC tutoring undergraduate students one-on-one. I also co-organized the UNC undergraduate mathematics
seminar, a role usually undertaken by faculty. In this seminar in the fall, I helped prepared
weekly practice sessions for the Putnam Exam, and in the spring gave a couple talks aimed
at exposing advanced undergraduates to mathematics outside of their courses.
During my first semester at UNC I took an intensive weekly teaching seminar for graduate
students. We were required to videotape ourselves teaching classes, and then watch and
analyze these videos together. This seminar covered many practical topics ranging from the
effectiveness of group work to how to handle cheating. As graduate students at UNC teach
so many courses as the instructor of record, such training is taken very seriously.
3
2
Evaluations from UMN
SRT Individual Report for MATH 4512 001 Diff Eqs w Applic (Bevin Maultsby) - Spring 2015
Instructor Items Carefully read each statement and select a response: - Frequency
1. The instructor was well prepared for class.
Statistics
Mean
Median
Standard Deviation
2. The instructor presented the subject matter
clearly.
Value
5.55
Statistics
6.00
Mean
+/-0.51
Median
Standard Deviation
3. The instructor provided feedback intended to
improve my course performance.
Value
Mean
Mean
5.53
Median
Median
6.00
Standard Deviation
Standard Deviation
5.70
6.00
+/-0.47
4. The instructor treated me with respect.
Statistics
Statistics
Value
Value
5.90
6.00
+/-0.31
+/-0.96
5. I would recommend this instructor to other
students.
Statistics
Mean
Median
Standard Deviation
Value
5.90
6.00
+/-0.31
4/9
Figure 1: This page and the next are evaluations of my teaching at UMN from Spring 2015
(20 out of 28 students responding).
4
SRT Individual Report for MATH 4512 001 Diff Eqs w Applic (Bevin Maultsby) - Spring 2015
Course Items Carefully read each statement and select a response: - Frequency
1. I have a deeper understanding of the subject
matter as a result of this course.
Statistics
Value
2. My interest in the subject matter was stimulated
by this course.
Statistics
Mean
5.45
Mean
Median
6.00
Median
Value
5.05
5.00
Standard Deviation
+/-0.69
Standard Deviation
+/-0.89
Standard Error (base on PSD)
+/-0.15
Standard Error (base on PSD)
+/-0.19
3. Instructional technology employed in this course 4. The grading standards for this course were
was effective.
clear.
Statistics
Value
Statistics
Mean
5.50
Mean
Median
6.00
Median
Value
5.68
6.00
Standard Deviation
+/-0.69
Standard Deviation
+/-0.48
Standard Error (base on PSD)
+/-0.15
Standard Error (base on PSD)
+/-0.11
5. I would recommend this course to other students.
Statistics
Mean
Median
Value
5.53
6.00
Standard Deviation
+/-0.70
Standard Error (base on PSD)
+/-0.16
7/9
5
3
Student comments from UMN
UniversityofMinnesotaStudentRatingofTeaching
Instructor:BevinMaultsby
Spring2015
Math4512
WrittenCommentsOnInstructor:
• Wroteexamplesontheboard.Dividedtimeintoworkingonboardand
workingonown
• Atonofexamples,foreverytopicwedidatleast3exampleswhichwasvery
helpful.Spokeveryclearlyandfullyexplainedeachtopic.
• Wasveryclearinpresentationofeachsubject.Therewasagoodbalanceof
examplesandtheorypresentation.
• Shebrokethesubjectmatterintoveryconciseandexplicitsectionsinaway
thatcouldbeeasilyunderstood.
• Presentedmaterialclearlyinlecture,wasveryfriendlyandapproachable,
veryhelpfulinofficehours.
• Postedcodetohelpwithlapsandhomework,allowedquestionsonthe
courseMoodlepage.Inclassexercises!!!Theyweregreat,andsowasBevin…
thathelped.
• Ienjoyedthegeneralsetupofthecourse.ProfessorMaultsbyorganizeditall
verynicely,withlabs,homework,andlecturesthatpreparedmeforother
partsofclass.ShewasalwayswillingtohelpwhenIneededit,andherbeing
personablemadelearningmoreenjoyable.Sheisgreattotalkto!
• Sopatient!Theinstructionwasclearandcomprehensible,thepacingwas
good,andthein-classlabswereabighelp.
• Thein-classlabswereactuallyreallyhelpful,itwasalsohelpfultohavethe
mathematicacode.Explanationsinclass/lectureswerealmostalwaysreally
informative.
• Clearexamples.Labsthathelpedusunderstandourhomework.
• Verydetailedcodeforsolutions.
• Nice.
• Shepreparedorganizedlecturescoordinatedwiththebookmaterial.She
wasgreataboutansweringanyquestionsIhadandbeingflexibleabout
meetinguptoassiststudents.Shewasawonderfulteacher!
• Sheisveryapproachableandhasaveryintuitiveteachingstyle.Doesn’tskip
stepsandisteachingatagoodpace.Verygoodteacher.
Figure 2: Written comments pertaining to the instructor of the course.
6
Question
4.265
4.163
4.408
4.102
4.531
3.898
4.062
Mean
Mean
Mean
Mean
Mean
Mean
Mean
4.306
Mean
4.020
3.653
Mean
Mean
3.408
Mean
1 (2.0%)
4.633
Mean
Not Applicable
4.143
Mean
4.340
2 (4.1%)
4.531
Mean
Mean
Not Applicable
4.245
Statistics
Mean
Report Generated on Sep 29, 2013 at 03:54:05 PM CDT
Page 1 of 2
She was always available to HELP!
It's definitely hard. Bevin did a good job on getting all the material covered but sometimes went to fast and didnt allow us to understand a topic before going onto the next one
fun times
Very tough course but she is a wonderful instructor. She is always there to help her students. She also makes sure her students understand the material (she honestly cares for her students). She
explains complex concepts in a way anyone can understand. Definitely deserves a raise and a Teaching Award.
Comments on overall assessment of this course.
This course really challenged me to think mathematically, and I really learned a lot. Furthermore, the instructor for this course was outstanding at helping me understand the material more clearly.
This was a very enriching course. Subject matter was difficult, but doable. The only issue I have is that I wish exams reflected our homework more. However, the teaching style was very effective and I
learned a lot.
Overall, I learned a great deal from this course.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
This course was very exciting to me intellectually.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
Overall, this course was excellent.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
Overall, this instructor was an effective teacher.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The course assignments helped me better understand the subject matter.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor evaluated my work fairly.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor showed concern about whether students learned the material.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor expressed ideas clearly.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor showed enthusiasm for the subject matter.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor showed enthusiasm for teaching this class.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor treated all students with respect.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructional techniques engaged me with the subject matter.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor provided me with helpful feedback on my performance.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
This course challenged me to think deeply about the subject matter.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor clearly communicated what was expected of me in this class.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
I was able to get individual help when I needed it.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
Responses: 50 out of 127 (39.4%)
Surveys filled out: 49
Course Evaluation
Terms: Fall 2010, Spring 2011, SS I 2011, SS II 2011, Spring_2013, SS1_2013
Instructor: Maultsby, Bevin L: maultsby
Note: This report contains a subset of the data for this period
4
Evaluations from UNC, incl. comments
Figure 3: This page and the next are evaluations are compiled evaluations from 6 UNC
courses.
7
8
Question
Report Generated on Sep 29, 2013 at 03:54:05 PM CDT
Is this a required course for you?
The workload was appropriate for what I gained from this class.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
Question
The instructor handled questions well.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor used examples that had relevance for me.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor used class time well.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
The instructor encouraged students to participate in this class.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
This course was designed to keep me engaged in learning.
1: Strongly Disagree - 2: Disagree - 3: Neither Disagree Nor Agree - 4: Agree - 5: Strongly Agree
Response
3.857
Mean
No
Yes
Page 2 of 2
10 (20.8%)
38 (79.2%)
4.250
3.755
Mean
Mean
4.667
4.408
Mean
Mean
4.388
Mean
Statistics
Great instructor! Great class !
Bevin is very approachable and a good lecturer. Her class is enjoyable and she manages to simplify complex course content so that it is easier to understand.
I appreciate how the instructor took her time and explained all the notes clearly. If somebody had a question she answered it in an excellent manner. By far my favorite course and one of the best classes
i've had while attending UNC.
Very enthusiastic instructor. Would be excellent after acquiring more teaching experience.
This was a challenging course, but Bevin was an excellent teacher. The homework/classwork adequately represented the test material, which is very important to me.
Bevin is by far one of the best teachers I've had at Carolina. When comparing her notes with the other TA's that my friends have, her notes show much more effort and are overall much more learningbased. That being said, Bevin really made me enjoy math. While I think the grading for this course is HIGHLY unfair, I do feel like she evaluated my work effectively. She speaks clearly, expresses
ideas perfectly, and does an overall great job at helping you learn. I wish I could take more math classes with her!
Fantastic course that presents math in a fun way! Great for fulfilling the math requirement.
The printed notes were VERY helpful, I liked that. It is always helpful with me in Math classes when the instructor explains things conceptually so that I understand the concepts well before I have to get
into complicated math. Bevin did a good job with this but more could be done. Overall, this class exceeded my expectations for calc 232 and was fairly painless!
The class wasn't challenging only because I have taken algebra in high school and was taking it as a refresher. Bevin is very nice and very welcoming and sets aside time to answer questions which I
appreciate. I think she should try to get more class participation and ask questions of students and have them try more problems on their own. However, I really enjoyed this class with her.
I hate Math, but Bevin made it slightly enjoyable.
The teacher was very willing to help with all questions.
Comments on overall assessment of this course.
Bevin was excellent at relaying math ideas; a lot of times you get TAs that don't quite know how to convey concepts, but I definitely understood Bevin.
The course was tough, and the TA was also tough.
UNC Fall 2013, Math 232 (Second Semester Single-Variable Calculus)
9
10
5
Sample Moodle Site
Course: MATH 4512 Differential Equations with Applications (...
https://ay15.moodle.umn.edu/course/view.php?id=9576
MATH 4512 Differential Equations with Applications (sec
001) Spring 2016
My Home ▶︎ MATH4512_001S16
ADMINISTRATION
Syllabus 4512
Course
administration
WebWork Link
Grades
Your homework assignments are on WebWork under
math-4512-s16. You are responsible for keeping track of due
dates. Your login name is your x500 (capitalized) and your initial
password (which you can change) is your student ID number.
Switch role to...
Return to my
normal role
UMN Mathematica
My profile settings
CSE should download the latest version of Mathematica (free). If
you are not CSE, see if you can get a free copy through the
university (see the link below). Then contact me if it is still not
possible.
NAVIGATION
My Home
Site home
Mathematica for non-CSE students link to CSE Labs
Site pages
If you are not a CSE student, you can create a "CSE Lab
Account" using the above link. This should allow you to obtain a
copy of Mathematica for free. Some of my non-CSE students
last year were not able to use the above link to set up a CSE lab
account, but were able to get an account by emailing
operator@cselabs.umn.edu directly.
My profile
Current course
My courses
LIBRARY
RESOURCES
Search in:
Books, articles, etc.
Forums and Announcements
Course Announcements
Go
Questions and Discussion
Access course
readings
Find articles and
books for MATH 4512
Labs
Instructions, LaTeX, Code.
1 of 4
3/1/16, 12:29 PM
Lab 1 - Occurs on 2/1/16. Due 2/12/16.
Figure 4: The course page for Math 4512. I try to make sure to provide students with
all the links and troubleshooting that they need to succeed in the course. I make general
announcements or post videos (e.g. of resonance) in the Course Announcements forum, while
students post questions in the other forum. As of March 1, there are 22 discussions in the
11
student forum.
Course: MATH 4512 Differential Equations with Applications
(...files and instructions
https://ay15.moodle.umn.edu/course/view.php?id=9576
Report
are now posted. If you need help
with the LaTeX look at the resources listed below, or post in the
forum.
Lab 2 - Occurs on 2/5/16. Due 2/17/16.
The lab report files are in the zipped folder. The instructions are
the same as for Lab 1 (put your pictures in the right place, then
add them to the document, etc). You are encouraged to email
me the PDF so that I can see the colors (printing may make
everything b&w).
Lab 3 - Occurs on 2/15/16. Due 3/2/16.
The lab report files are in the zipped folder. The instructions are
the same as for Lab 1 (put your pictures in the right place, then
add them to the document, etc). You are encouraged to email
me the PDF. Note: #8 (A=5; F0=7000; B0=10; tEND = 200;) was
moved to the very end of the report.
Lab 4 - Occurs on 3/4/16 (new date). Due 3/28/16.
Lab 5 - Occurs on 3/28/16. Due 4/6/16.
Lab 6 - Occurs on 4/6/16. Due 4/18/16.
Lab 7 - Occurs on 4/7/16. Due 5/4/16.
Handouts and Extra Notes
Lecture-slides-1-25
Mathematica Eigenstuff Calculator
You should know how to compute eigenvalues and
eigenvectors by hand. Once you have mastered that, feel free to
use this code for the WebWork, etc.
Linear systems notes
Test Information
General information, practice problems, keys
Midterm 1 - 2/22/16
2 of 4
3/1/16, 12:29 PM
Midterm 2 - 4/11/16
Figure 5: Additionally, I post all necessary lab handouts on Moodle and clearly indicate the
dates of any important activities.
12
6
Sample Syllabus
Math 232, Section 004
Calculus of Functions of One Variable II
Spring 2013
INSTRUCTOR: Bevin Maultsby
EMAIL: maultsby@live.unc.edu
OFFICE HOURS: Monday 3pm-4pm (MHC in Phillips 365), Wednesday 3pm-4pm (office),
Friday 1pm-2pm (MHC), or by appointment
OFFICE: Phillips 364
CLASS MEETINGS: Mondays, Wednesdays and Fridays, 2:00-2:50pm, Phillips 228
TEXT: Single Variable Calculus: Early Transcendentals, James Stewart, 7th Edition
PLACEMENT: In order to be eligible to register for this course, at least one of the following
must be true:
• You earned at least a 3 on the AB-AP exam.
• You earned at least a 3 on the AB subscore on the BC-AP exam.
• You earned a grade of C- or higher in Math 231 taken at UNC-CH or have 231 transfer
credit.
Note: Math 152 is not an acceptable prerequisite for Math 232.
COURSE OUTLINE: We will cover all or parts of Chapters 6-9 and 11. The major topics are techniques and applications of integration and infinite sequences and series. In the first
week, we will review some basic material from Calculus I and then move quickly on to Chapter 7.
CALCULATOR: You will need a basic scientific calculator (i.e. one that will do trigonometric, logarithmic and exponential functions) for this course. However, you will not be allowed
to use the graphing features on certain quiz, test and final exam problems. TI-89 and other
symbolic manipulators are not allowed.
LAPTOPS, ETC.: You are not allowed to use a laptop in class unless I specify otherwise.
Any student using a smartphone, iPod, or other device irrelevant to class will be asked to put
it away or leave. During tests, cell phones must off and out of sight.
ATTENDANCE: You are expected to be in class every day and to come prepared to learn
and work. There will be no make-up tests.
This is a challenging course for many students. Reading the textbook before attending lectures is a highly recommended study habit. It will be helpful to look at the example problems
in each section before (or while) working on homework problems. You are responsible for all of
the material covered in class or assigned as homework.
HOMEWORK: Online homework assignments will total 100 points. The website is
https://www.webassign.net/login.html
class key: unc 6656 6639
1
Figure 6: My typical syllabus. I try to address every major issue from conduct in class (e.g.
no laptops) to test dates and grading.
13
New Users
-Click on “I have a Class Key.”
-Enter the appropriate class key from above and
submit.
-Verify that you have the correct class and choose
“Need to Create. . .”
-Complete all boxes with an ∗ beside them with
your personal information.
-Click on “Create my account.”
-Click on “Log in.”
Previous Users
-Click on “I have a Class Key.”
-Enter the appropriate Course Key from
above and submit.
-Verify that you have the correct class and
choose “Already Have. . .”
-You can log in with your User Name, Institution, and Password.
From this point on you will sign in as a returning
user with your User Name, Institution, and Password. You have a 14-day grace period before you
need to enter your access code or purchase one from
WebAssign.
You have a 14-day grace period before you
need to enter your access code or purchase
one from WebAssign.
Note: The best way to succeed in this course is to do as many problems as you possibly can to
build up speed, confidence and understanding. Doing only homework exercises is considered a
bare minimum effort.
GRADING: All grades will be assigned according to a 10-point scale: 90-100 A, 80-90 B,
70-80 C, 60-70 D, Below 60 F. Your overall average must be at least 60% to earn a passing
grade for this course. Your course grade (with tentative test dates) will be determined as
follows:
• Online Homework: 100 points
• Test 1 on 9/13: 100 points
• Test 2 on 10/11: 100 points
• Test 3 on 11/11: 100 points
• Cumulative Final on 12/6 (4:00-7:00pm): 200 points
You must have an official examination excuse in order to schedule the final exam for a different
time. This excuse must be signed by a Dean (in Steele Building).
OTHER IMPORTANT DATES: As indicated on the university calendar, there is no class
on 9/2 (Labor Day), 10/18 (Fall Break), 11/27 & 11/29 (Thanksgiving).
RESOURCES: The Math Help Center is located in Phillips 365, open M-Th 1:00pm-6:00pm
and F 1:00pm-3:00pm. You are strongly encouraged to take advantage of this resource.
HONOR CODE:
All students are expected to conduct themselves within the guidelines of the UNC Honor System. All academic work should be done with the high level of honesty and integrity that this
University demands.
2
14
7
Sample Exam
Math 4512
Spring 2015
Exam 1
2/23/2015
Time Limit: 50 Minutes
Name (Print):
This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top of every page, in case the pages become separated.
You may not use your books, notes, or any calculator on this exam.
You are required to show your work on each problem on this exam.
The following rules apply:
• If you are applying a theorem, you must indicate this fact, and explain why the theorem
may be applied.
• Organize your work in a neat and understandable way. Work that is disorganized, hard to follow, or lacks clear reasoning will receive little or no
credit.
• Unsupported answers will not receive full
credit. An answer must be supported by calculations, explanation, and/or algebraic work to receive full credit. Partial credit may be given to
well-argued incorrect answers as well.
Problem
Points
1
16
2
24
3
24
4
20
5
16
Total:
100
Score
• If you need more space, request blank pages from
me. Clearly indicate when you have done
this and staple any work you wish to be
graded to your exam.
Do not write in the table to the right.
Figure 7: This was the first exam from a second-semester differential equations course.
15
Math 4512
Exam 1 - Page 2 of 6
2/23/2015
1. (16 pts) State any equilibrium solutions for the differential equation below. Then find the
general solution. You must show sufficient work.
dy
= t − 2ty
dt
Page 2
16
Math 4512
Exam 1 - Page 3 of 6
2/23/2015
2. (24 pts) Sketch both the direction field and the phase line for
dy
= ey (y 2 − 1).
dt
Be sure to label any equilibrium solutions as sinks, sources, or nodes. Plot three solutions on
the direction field.
Direction Field:
Phase Line:
y
y
t
Page 3
17
Math 4512
Exam 1 - Page 4 of 6
2/23/2015
3. (24 pts) The bifurcation diagram for the differential equation
wider points are sinks, the narrower points are sources).
dy
= fµ (y) is given below. (The
dt
1
1
µ
y
(a) (12 pts) Describe clearly and precisely what happens as the bifurcation parameter µ is
increased. (2-4 sentences should suffice.)
(b) (12 pts) Sketch a picture of what fµ (y) might look like (as a function of y) when µ = 1.5.
f (y)
y
Page 4
18
Math 4512
Exam 1 - Page 5 of 6
2/23/2015
4. (20 pts) Consider the system
dx
dt
dy
dt
= 2x + 10y
= 3x + y.
(a) (5 pts) Write this system in matrix form Y0 = AY.
(b) (5 pts) Find the eigenvalues of A.
The eigenvalues are
.
The eigenvectors are
.
(c) (5 pts) Find the eigenvectors of A.
(d) (5 pts) What is the general solution to this system?
Page 5
19
Math 4512
Exam 1 - Page 6 of 6
2/23/2015
5. (16 pts) Consider
dy
1
=
.
dt
y(2 − t)
(a) (6 pts) Solve the initial value problem y(1) = 2. Give your solution in explicit form.
(b) (6 pts) State the domain of definition of your solution from (a).
(c) (4 pts) Describe what happens to the solution as it approaches the limits of its domain.
Page 6
20
8
Examples of student work
UMN UMTYMP Calc 3 Fall Project
Project 1 - Group 32
Let
! !, ! =
1
,
!! + !!
!0,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! ! ! + ! ! sin
!, ! ≠ 0,0
.
!, ! = 0,0
We will prove that ! !, ! is differentiable at all points, even though !! !, ! and !! !, ! are
discontinuous at the origin.
For all points !, ! ≠ 0,0 in ℝ! , we have
!! !, ! = 2! sin
!= 2! sin
1
!!
+
!!
1
!! + !!
1
+ ! ! + ! ! cos
−
!
!!
1
cos
!! + !!
+
−2!
2 !! + !!
!!
!! + !!
!/!
.
Similarly, because ! !, ! is symmetric on ! and !, we have
!! !, ! = 2! sin
1
!! + !!
−
!
1
cos
!! + !!
!! + !!
Since these partial derivatives are continuous on the open set ℝ! ∖
.
0,0 , Theorem 3.5 tells us
that !(!, !) is differentiable for !, ! ≠ 0,0 .
Notice that the limits
lim
!,! → !,!
!! !, ! !!!!!!!and!!!!!!!
lim
!,! → !,!
!! !, !
do not exist: Along the line ! = !, we have
!
lim
!,! → !,!
!! !, ! =
lim
!,! → !,!
!! !, !
= lim 2! sin
!→!
1
!
2
−
!
!
which does not exist because as ! approaches 0, the term 2! sin
term
!
!
!
cos
!
!
!
2
cos
!
!
!
1
!
2
,
approaches 0, but the
oscillates between −1 and 1.
Thus, the limits
lim
!,! → !,!
!! !, ! !!!!!!!and!!!!!!!
lim
!,! → !,!
!! !, !
do not exist, so !! !, ! and !! !, ! are discontinuous at the origin.
Figure 8: This is the first draft of a group project in multivariable calculus (page 1 of 2).
21
We can calculate !! 0,0 using the limit definition of the partial derivative, which gives
! ℎ, 0 − ! 0,0
ℎ
1
ℎ! sin
−0
ℎ
= lim
!→!
ℎ
1
= lim ℎ sin
!→!
ℎ
!! 0,0 = lim
!→!
=0
by the Squeeze Theorem. Similarly, the limit definition gives
!! 0,0 = lim
!→!
! 0, ℎ − ! 0,0
= 0.
ℎ
Let the function ℎ:!ℝ! → ℝ be defined by
ℎ !, ! = ! 0,0 + !!! 0,0 + !!! 0,0 = 0.
Then ! !, ! is differentiable at 0,0 if and only if
lim
!,! → !,!
! !, ! − ℎ !, !
=
!, ! − 0,0
lim
!,! → !,!
! ! + ! ! sin
1
!! + !!
= 0.
Switching to polar coordinates gives
lim
!,! → !,!
! ! + ! ! sin
1
!! + !!
= lim! ! sin
!→!
1
=0
!
by the Squeeze Theorem. Thus, ! !, ! is differentiable at 0,0 .
Therefore, the function ! !, ! is differentiable at all points, even though !! !, ! and !! !, !
are discontinuous at the origin. This does not contradict Theorem 3.5, which implies that the
continuity of !! !, ! and !! !, ! on all of ℝ! is sufficient, not necessary for ! !, ! to be
differentiable on all of ℝ! . This function ! !, ! disproves the converse of Theorem 3.5, namely,
that the differentiability of ! at !, ! implies the continuity of the partial derivatives of ! in a
neighborhood of !, ! .
Figure 9: (page 2 of 2)
22
UMN Math 4512 Lab
Figure 10: A student lab from Differential Equations with Applications; names removed.
23
24
25
26
27
28
9
Examples of in-class group activities
UMN Math 4512 (Spring 2015, Spring 2016)
Section 1.4 Class Exercise
1/29/2016
dy p
= y, y( 2) = 1.
dt
1. Using Euler’s Method with t = 2, estimate the solution to the IVP and plot it along
with your answer to #1 below. What if we try to set t = 2 to go to the left?
Today we will study the initial value problem
y
t
2. Repeat Euler’s Method t = 1, estimate the solution to the IVP and plot it along
with your answer to #1 below. What if we try to set t = 1 to go to the left?
y
t
1
Figure 11: (See next page.)
29
3. Find all equilibrium solutions to y 0 =
p
y.
4. Using separation of variables, find the analytic solution to the IVP. Plot it on the
graphs on the first page.
5. Does your solution to #4 touch the t-axis? Explain why this phenomenon is not a
contradiction.
6. What is happening?
2
Figure 12: I designed this activity to give them a chance to actually do Euler’s Method in
class. Some interesting behavior arises: in #1 the solution leaves the domain, in #2 the
solution meets (and then stays on) the t-axis, where there is an equilibrium solution. This
behavior leads to some discussion of the existence and uniqueness theorems.
30
UMN Math 4512 (Spring 2015, Spring 2016)
Section 2.1 Class Exercise
2/12/2016
Consider the following two Predator-Prey systems of equations:
(A)
dx
= x(10 − x) − 20xy
dt
(B)
dx
xy
= 0.3x −
dt
100
xy
dy
= −5y +
dt
20
dy
= y(15 − y) + 25xy
dt
1. For each system, which variable (x or y) represents the predators, and which the prey?
How do you know?
2. Find all equilibrium solutions. Explain them in terms of the two species.
3. In one of these two systems the prey are elephants and the predators are mosquitos.
Thus it takes many predators to eat one prey, but each prey eaten has a tremendous
benefit for the predator population.
The other system has whales as predators and plankton as prey. Hence one predator
will eat many prey, and one individual prey is not a substantial benefit to the predator.
Which system is which?
1
Figure 13: This exercise is based on Blanchard, Devaney, and Hall, §2.1, #1-6. It is the type
of groupwork that leads students to having good conversations with each other, as #1 and
#3 are not computational questions.
31
UMN UMTYMP Calc 1 Fall Semester
UMTYMP Calculus I
Week 2
What is lim
x→0
In this problem, we will examine lim
x→0
sin x
?
x
sin x
using the unit circle and triangles, as pictured below.
x
B
D
1
A
x
C
1
1
1
sin x, area(4ABC) = tan x, and area(2ADC) = x, where
2
2
2
2ADC is the sector. (Hint: notice all of the radii have unit length.)
1. Verify that area(4ACD) =
2. Using some algebra with your expressions from #1, determine two functions f and g so that
f (x) ≤
sin x
≤ g(x),
x
3. Use the Squeeze Theorem to find limx→0+
4. Is
0 < x < π.
sin x
.
x
sin x
sin x
even, odd, or neither? Discuss how this can be used to find limx→0−
.
x
x
5. Find limx→0
sin x
.
x
Figure 14: I wanted the students to practice some trigonometry before using the Squeeze
Theorem to find the limit.
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UMN UMTYMP Calc 3 Spring Semester
UMTYMP Calculus III
Week 8
Green’s Theorem is All Squared Away
Let F(x, y) = (M (x, y), N (x, y)) be a C 1 vector field on R2 .
1. We are going to prove Green’s theorem in the special case where the domain D (with positively oriented
boundary curve C) is a rectangle parallel to the coordinate axes.
y
C3
d
C4
C2
c
C1
(a) Show
Z
M dx + N dy =
C1
Z
a
b
b
x
M (x, c) dx.
a
(b) Using part (a) as a guide, find
Z
M dx + N dy, j = 2, 3, 4.
Cj
(c) Compute
(b)?
ZZ
D
∇ × F dA (where ∇ × F is the 2D scalar curl). Is this equal to the sum of parts (a) and
(over)
Figure 15: The students were able to prove Green’s Theorem for a rectangle. This proof led
to a discussion of Green’s Theorem on other domains (see next page).
33
2. How would you approach the proof of Green’s Theorem for the region below? (Discuss, do not prove.)
y
x
Figure 16: After the previous page, the students discussed how they would approximate
other domains with rectangles. This activity reinforced the idea that much of calculus is
derived by approximating quantities with finer and finer partitions. I hoped the pun in the
title would help this message stick!
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UNC Math 118: Selected Topics in Mathematics
Names:
Group Quiz
1. We have listed the symmetries of the square as the group D4 , given by
D4 = {e, r, r2 , r3 , s, s ∗ r, s ∗ r2 , s ∗ r3 },
where e is the identity action, r is rotation by 90 degrees clockwise, s is reflection
across the vertical axis, the operation s ∗ r means “perform s, then r”, and r2 = r ∗ r.
(a) Perform the following motions. Write each one as an element from the above
group D4 . (3 pts)
i. r ∗ s
ii. r2 ∗ s
iii. r3 ∗ s
(b) Using the above (i.e. not using your square), write the following motions as an
element in D4 . (Show work) (2 pts)
i. r3 ∗ s ∗ r2
ii. r5 ∗ s3 ∗ r ∗ s ∗ r
2. We have listed the symmetries of the equilateral triangle, D3 = {e, r, r2 , s, s ∗ r, s ∗ r2 }.
(a) Perform the following motions. Write each one as an element from the above
group D3 . (2 pts)
i. r ∗ s
ii. r2 ∗ s
(b) Using the above (i.e. not using your triangle), write the following motions as an
element in D3 . (Show work) (2 pts)
i. r ∗ s ∗ r
ii. s ∗ r2 ∗ s ∗ r2
1
Figure 17: An in-class group exercise from Selected Topics in Mathematics. We did a unit
on patterns and symmetry, and I introduced a bit of group theory. This lab was designed to
help them better understand the dihedral group. Students seemed to really enjoy working
with the Rubik’s cube.
35
3. (a) Dn is the group of symmetries of a regular n-gon. Let i be some integer between
0 and n. As in 1(a) and 2(a), we may write ri ∗ s = s ∗ rk for some integer k.
Write k in terms of i and n. (3 pts)
(b) How many elements does Dn have? (3 pts)
4. With your Rubik’s cube, perform the following operations:
(a) Describe in 1-2 sentences what has happened to your cube. (3 pts)
(b) Return your cube to its identity state (and give it back to me). (2 pts)
Bonus: What country does Erno Rubik come from? (1 pt)
2
36
UNC Math 232: Single-Variable Calculus II
Names:
Applications Group Work
This is worth two homework grades. Only one copy per group is needed. Please write neatly
and show all work. Due at the end of class on Tuesday.
Newton’s Law of Cooling describes how a body (or a cup of coffee) changes temperature
over time relative to its ambient (surrounding) temperature. The differential equation is
dT
= k(A − T ),
dt
where
T
t
k
A
is
is
is
is
temperature
time
a positive constant, and
the ambient temperature (assumed constant).
1. Is this differential equation separable, linear, or both? Do both (a) and (b) if applicable.
(a) If it is separable, solve for T (t) using the separation of variables technique:
(b) If it is linear, identify P and Q, compute the integrating factor, and solve.
1
Figure 18: Students did this group work during a unit on basic differential equations in
Calculus II (Chapter 9 in Stewart’s).
37
2. If T > A is the body cooling down or warming up? Answer this by referring to the
dT
sign of
.
dt
3. Consider your answer(s) from 1(a)-(b). After lots of time passes (i.e. t → ∞), where
does T go? Does this make sense? Explain briefly.
4. As knowledgeable Math 232 students, you are summoned to Tudor Mansion: the butler
has been found murdered with a lead pipe in the conservatory. You arrive at 1:16am.
The house is kept at 72◦ , and the butler was a healthy 98.6◦ at the time of his passing.
You immediately check the butler’s temperature and find that it is 82◦ . An hour later,
it is 78◦ .
All party guests were in the ballroom the entire evening (with various candlesticks and
revolvers) with the following exceptions.
(a) Miss Scarlett, who went to the restroom at 10:45pm.
(b) Colonel Mustard, who stepped outside to smoke a pipe at 11:20pm.
(c) Mrs. White, who searched for a painting in the library at 11:35pm.
(d) Professor Plum, who checked out the math books in the study at 11:50pm.
Who is the murderer?
2
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Assume the world population was exactly 6 billion persons at the start of the day on
January 1, 2000, and was increasing at a rate of 120 million people per year at that moment.
Use this information to answer the following questions.
1. The law of natural growth is
dP
= kP,
dt
k is a positive constant.
This says a population’s growth rate is proportional to its size. Solve this equation to
find an equation P (t) (where P is in “billions of persons”) for the human population
t years after 2000.
2. Use P (t) to predict world population in the years 2020 and 2100.
(a) 2020:
(b) 2100:
3. What is the “doubling time” for the world’s population?
3
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4. Another model of population growth is the logistic model given by
P
dP
= kP 1 −
,
dt
M
where M is the carrying capacity of a population and k is a positive constant. In this
model, the rate of population increase is proportional both to the current population
and to the number of people for which there is room left on the Earth.
Assume again that at the start of 2000, the world’s population was 6 billion and
increasing at that moment at a rate of 120 million people per year.
According to the model, what happens in the following scenarios?
(a) P < M :
(b) P = M :
(c) P > M :
5. Use this information to sketch a direction field (it does not have to be too detailed!).
P
M
M/2
0
t
4
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10
Sample of in-class notes at UNC
6.1 Areas Between Curves
9/11/2013
Recall the Riemann sum expression for the area under a curve:
Now we are interested in the area enclosed between two curves:
f (x)
g(x)
a
b
The Riemann sum expression for this area is:
which we express as an integral using
So if a ≤ x ≤ b and 0 ≤ g(x) ≤ f (x), then the area of enclosed by x = a, x = b, f (x) and
g(x) is:
1
Figure 19: The first two pages from the student version of lecture notes on the area between
two curves in Calculus II. I give them the skeleton of my notes for them to fill-in. Having
these notes allows students to spend less time in class transcribing.
41
Ex 1. Sketch the region enclosed by the given curves and find its area.
y = 12 − x2
y = x2 − 6
Ex 2. Find the area of the shaded region
2
42
11
Example of a “curve worksheet” at UNC
Name:
Midterm 3 Curve Worksheet
If you complete this worksheet and return it to me by Tuesday, April 23rd, you can increase
your Midterm 3 grade. Let x be your original grade; your new grade would be 0.8x + 20
(round to nearest whole number), or x + 2 if your grade was already very high. Please print
front and back or staple. Unstapled papers will receive no points.
Show all work. All answers must be neat and entirely correct to receive credit.
Much of this material is easily found online. The goal is for you to better understand
some basic ideas in mathematics, and therefore it is in your best interest to figure these out
first (then look them up if you wish).
1. Complete the next 3 rows of Pascal’s Triangle.
Row
Row
Row
Row
Row
0
1
2
3
4
1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
Row 5
Row 6
Row 7
2. Expand and simplify
(a) (x + 1)0 =
(b) (x + 1)1 =
(c) (x + 1)2 =
(d) (x + 1)3 =
3. Based on Questions 1 and 2, expand the following (note: you should not distribute/foil!!)
(x + 1)7 =
1
Figure 20: If a test needs to be curved, I ask the students to complete a worksheet designed
to expose them to some mathematics they might not see elsewhere. The above is a curve
worksheet designed to help Calculus 1 students work toward the binomial formula on their
own. Several students said this exercise was enlightening.
43
4. Expand and simplify
(a)
(b)
(c)
(d)
(x + y)0
(x + y)1
(x + y)2
(x + y)3
=
=
=
=
5. Based on Questions 1-4, give
(x + y)7 =
n
6. The notation
is read “n choose k” and is computed as
k
n
n!
=
k!(n − k)!
k
Compute the following: (note 0! = 1)
7
(a)
=
0
7
(b)
=
1
7
(c)
=
2
7
(d)
=
3
7
(e)
=
4
7
(f)
=
5
7
(g)
=
6
7
(h)
=
7
7. Combining all your responses, complete the following formula:
(x + y)N =
N
X
k=
2
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·x
·y
12
Sample problem set from Math Club
Problem Solving Seminar: Games
1. (One-person game.) There are 1990 boxes containing 1, . . . , 1990 chips, respectively,
on a table. You may choose any subset of boxes and subtract the same number of chips
from each box. What is the minimum number of moves you need to empty all boxes?
2. Abigail and Brendan alternately place +, −, · into the free places between the numbers 1 2 3 . . . 99 100 . Show that Abigail can make the result (a) odd, (b) even.
3. Alice and Bob start with p = 1. Then they alternately multiply p by one of the numbers 2 to 9. The winner is the one who first reaches (a) p ≥ 1000, (b) p ≥ 106 . Who
wins, Alice or Bob?
4. Adam places a knight onto an 8 × 8 board. Then Barack makes legal chess move. Then
Adam makes a move, but he may not place it on a square visited before, and so on.
The loser is the one who cannot move any more. Who wins?
5. Aaron and Bernard alternately color squares of a 4 × 4 chessboard. The loser is the
one who first completes a colored 2 × 2 subsquare. Who can force a win?
6. Anna and Betty alternately draw diagonals of a regular 1988-gon. They may connect
two vertices if the diagonal does not intersect an earlier one. The loser is the one who
cannot move. Who can have a winning strategy?
7. (From last year’s Putnam.) Alan and Barbara play a game in which they take turns
filling entries of an initially empty 2008 × 2008 array. Alan plays first. At each turn, a
player chooses a real number and places it into a vacant entry. The game ends when all
the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero;
Barbara wins if it is zero. Which player has a winning strategy?
1
Figure 21: Example of a problem set I compiled from, e.g., previous Putnam exams to go
over with students in Math Club. Every week had a theme; this week’s theme was problems
that are presented as games.
45