Math 139A, Fall 2021
Applications of Mathematics

Instructor                   John E. McCarthy
Class                           Tu 11.00-11.50 in Lopata 103

                                    No class October 12 (Fall Break); we will have class November 23

JM Office                  105 Cupples I
JM Office Hours       M 4.00-4.50, Tu 10.00-10.50,  Th. 2.30-3.20, and by appointment
                   In office and via Zoom: Meeting ID: 932 2571 1839


Math 131, preferably taken concurrently


Mathematics can often seem intimidatingly abstract.
“Why do we need to know this?” and “What is this good for?” are common
questions, which sometimes are not adequately answered. 
It is all very well to say that mathematics is needed for cell-phone design, or
to make ultrasound images, or for Google to calculate page-rank; but explaining
exactly how it is used in any of these applications takes a great deal ot time.

The purpose of this course is to give examples of how mathematics can be used to
understand real world problems. It is aimed at students who are also enrolled in Calculus I, Math 131,
so we will start with problems that only need pre-calculus to solve, and work up to ones that use

Do I need to be a math wiz to take this course?
No. The course is for students who are curious about how mathematics is used,
and want some inkling of its scope.


Here is a tentative schedule. The first seven do not use calculus, the next six do.
We may change some of these topics.

  1. Dimensional Analysis. How to guess plausible formulas.
  2. The mathematics of convoys. Are they a good idea? What are the pros and cons? Reading for class on Sep 6
  3. Fibonacci Numbers. See this site for pictures
  4. The golden ratio
  5. Fractals I :  Coastlines
  6. Fractals II : what is dimension? 
  7. Linear regression I. Application: the Gutenberg-Richter law of earthquake magnitudes. How can we speak of a twenty thousand year event? Earthquakes
  8. Linear Regression II: How to find the best line. Regression to the mean Application: Metabolic rate versus animal size Metabolic Rates
  9. Network capacities. Braess’s paradox – building extra roads can increase congestion.
  10. Fractals III: Blood flow.
  11. Should we all have the same mitochondrial DNA? Galton’s approach to surnames.
  12. How genes spread through populations.
  13. Sigmoid curves for populations, and the logistic equation.
  14. SIR model of infectious diseases.

Basis for Grading

Grading will be based on classrooom participation, and a term paper.
The topic of the term paper should be chosen in consultation with the Instructor.
A preliminary draft should be handed in October 26th. This will be graded and returned,
and a final version should be handed in December 7th.

Classroom Participation (contribution to discussion): 20%

Attendance is mandatory, unless you need to miss class due to illness or COVID exposure. In this event, please send me confirmation of the need to miss class from Habif.

Term Paper – first draft: 30%
Term Paper – final draft: 50%


You must come to class every day, and participate in class discussions, unless you are ill or quarantining.
I expect you to read the corresponding section in Korner’s book. I may call on you at
any time to answer a question.

Texts           The Pleasures of Counting, by T.W. Korner  (Cambridge, 1996).
                    This is a lovely book. If you get bogged down in some section, it is okay to move on.

COVID-19 Health and Safety Protocols                 Exceptions to course attendance policies, expectations, and requirements because of a COVID-19 diagnosis, symptoms consistent with COVID-19, or exposure to a person with a confirmed or suspected COVID-19 diagnosis that requires quarantine or isolation will be made in collaboration between the student and instructor. In these cases, please notify your instructor as soon as possible to discuss appropriate accommodations. While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community. The full set of University protocols can be found at This includes:

• Completing a self-screening using the WashU COVID-19 Screening app every day before coming to campus or leaving your residence hall room. If you do not receive a green check and pass the screening, you are not permitted to come to campus or leave your residence hall room. You must contact the COVID Call Center (314-362-5056) or the Habif Health and Wellness Center (314 935-6666) immediately. Note: In addition to the symptoms listed in the screening tool, everyone also should pay attention to symptoms that are new or different for you, including things like headache and congestion, particularly in combination with diarrhea. These can also be signs of COVID-19. Call the COVID Call Center or Habif to report these symptoms.

• Complying with universal masking. All individuals on campus must wear disposable masks or cloth face coverings while occupying indoor public settings, including: multi-person offices, hallways, stairwells, elevators, meeting rooms, classrooms and restrooms. Masks are encouraged but not required for outdoor activities, particularly at large events or in crowded settings. Students with disabilities for whom masked instructors or classmates create a communication barrier are encouraged to contact Disability Resources ( or talk to their instructor for assistance in determining reasonable adjustments. Adjustments may involve amplification devices, captioning, or clear masks but will not allow for the disregard of mask policies.

 • Maintaining physical distancing as needed. While distancing requirements have been removed for vaccinated students, those who are not fully vaccinated are strongly encouraged, for their own health, to maintain a distance of 6 ft from others in the classroom. If you are not able to be vaccinated or have conditions that may put you at increased risk of failed immunity and classroom activities would bring you in frequent proximity to other students, contact your instructor to discuss alternatives.

• Practicing healthy personal hygiene, including frequent handwashing with soap and warm water for at least 20 seconds and/or using hand sanitizer with at least 60% alcohol.

University-Wide Policies

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 Reasonable Accommodations for Disabled Students. Washington University in St. Louis supports the rights of enrolled students to a full and equal educational opportunity and, in compliance with federal, state, and local requirements, is committed to reasonable accommodations for individuals with documented disabilities. Disabled students for whom accommodations may be necessary must be registered with, and provide their instructors official notification through, WUSTL’s Disability Resources ( Once established, responsibility for disability-related accommodations and access is shared by DR, faculty, and the student. Please contact Disability Resources at 314.935.5970 or

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 To report an emergency: Danforth Campus: (314) 935-5555 School of Medicine Campus: (314) 362-4357 North/West/South and Off Campus: 911 then (314) 935-5555

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In all academic work, the ideas and contributions of others must be appropriately acknowledged and work that is presented as original must be, in fact, original. Faculty, students and administrative staff all share the responsibility of ensuring the honesty and fairness of the intellectual environment at Washington University in St. Louis.

For additional details on the university-wide Undergraduate Academic Integrity policy, please see:

Resources for Students:

Disability Resources At Washington University we strive to make the academic experience accessible and inclusive. If you anticipate or experience barriers based on disability, please contact Disability Resources at 314.935.5970,, or visit our website for information about requesting academic accommodations. See: Confidential Resources for Instances of Sexual Assault, Sex Discrimination, Sexual Harassment, Dating Violence, Domestic Violence, or Stalking The University is committed to offering reasonable academic accommodations (e.g. a no-contact order, course changes) to students who are victims of relationship or sexual violence, regardless of whether they seek criminal or disciplinary action. If a student needs to explore options for medical care, protections, or reporting, or would like to receive individual counseling services, there are free, confidential support resources and professional counseling services available through the Relationship and Sexual Violence Prevention (RSVP) Center. If you need to request such accommodations, please contact RSVP to schedule an appointment with a confidential and licensed counselor. Although information shared with counselors is confidential, requests for accommodations will be coordinated with the appropriate University administrators and faculty. The RSVP Center is located in Seigle Hall, Suite 435, and can be reached at or (314) 935-3445. For after-hours emergency response services, call (314) 935-6666 or (314) 935-5555 and ask to speak with an RSVP Counselor on call. See: RSVP Center.

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 Mental Health Services Mental Health Services’ professional staff members work with students to resolve personal and interpersonal difficulties, many of which can affect a student’s academic experience. These include conflicts with or worry about friends or family, concerns about eating or drinking patterns, and feelings of anxiety, depression, and thoughts of suicide. See: Additionally, see the mental health services offered through the RSVP Center listed above.

WashU Cares WashU Cares specializes is connecting students to mental, medical, financial and academic resources by using supportive case management. We seek to empower students to be successful through life’s challenges and to have ownership of their own experiences. Our services are designed to support Danforth Campus students. If you feel concerned about a student who may need help connecting to resources we accept referrals from all students, faculty, and staff. If you are concerned about a student, you can file a report here: and a WashU Cares Case Manager will reach out to you to get more information about your concern.

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Gephardt Institute State and local elections can have direct and immediate impacts on our communities and will be happening here in St. Louis and around the country this year. Make sure you stay current on upcoming elections at all levels by registering as a voter, requesting an absentee ballot, or getting election day reminders. You can do all this at for any of the 50 states and Washington D.C. If you are ineligible to vote, you can still participate by referring your friends who are eligible to the TurboVote link to register and by engaging your peers in local issues. If you have any questions about the voting process in the United States, please visit

©2024 John E. McCarthy, Ph.D.