MATH 106L: Lab calculus and functions II

Instructor: Spencer Nicholas Whitehead

Lecture times: MWF 12:00-12:50, W. Duke 08A

Lab times: TTh 12:00-12:50, W. Duke 08A

Office Hours: W 13:00-14:00, Classroom 132; Th 16:00-17:00, F 15:00-16:00, Zoom (link sent by email). One online office hour TBD.

Syllabus: See the course syllabus in PDF format here.

Contact: See a facesheet of course staff and contact info in PDF format here.

Grading: Detailed information on the grading scheme is available in PDF format here.

A more detailed breakdown of the grading scheme and FAQ is available here.

The rubric used to assess reports is available here (see above link for more grading info).

Course description: The notion of quadrature, the computation of the area of a complicated shape by filling it with smaller, better understood shapes, has been studied since the Greeks. In MATH 105L, you studied the calculus of differentials and learned how the behaviour of a (reasonable) function could be described locally by its derivative. Historically, this notion came almost 2,000 years after that of quadrature. It would be a further two centuries until James Gregory would publish a statement and proof of what is today known as the fundamental theorem of calculus. This theorem states roughly that quadrature, called integration today, is inverse to differentiation: the derivative of an integral of a function is the original function, and the integral of a derivative of a function is the original function—perhaps up to the oft-forgotten '+ C'.

The fundamental theorem of calculus cannot be described as anything short of astonishing. There is no reason a priori to believe that the operations of finding area and finding local linear approximations should be related to each other in any reasonable way, but the fundamental theorem of calculus provides exactly this relation. The fundamental theorem of calculus comprises perhaps the first example of what one might call a 'local-global result': by knowing local information about a function (its derivative at each point), one learns global information about the function (the area under its graph). This philosophy underlying the fundamental theorem of calculus is omnipresent in day-to-day life, if often invisible—as the sewist closes a sleeve with a stitch in the round or a ship circumnavigates the globe using only an atlas is present the fundamental theorem of calculus, providing rigorous mathematical justification to the idea that local approximations, sufficiently accurate, sum to the whole. While often exploited as just a computational tool, it would be no exaggeration to call the fundamental theorem of calculus one of the most beautiful in all of mathematics.

In MATH 106L we will develop the modern integral calculus from a particular kind of quadrature called a Riemann sum, and from this definition prove the fundamental theorem of calculus. After the fall break, we will begin the study of differential equations and methods of solution. Finally, after the Thanskgiving break (and throughout the labs) we will study specific examples of differential equations that motivated the historical development of calculus, such as Newton's laws of motion and models for population growth/decay, including the presence of forcing terms.

Best practices for writing mathematics: Through the course, you will have the chance in labs to write a number of group reports. While there is no universal style guide for mathematical writing, there are a number of accepted best practices, available in PDF format here. When writing reports, you are expected to follow the practices in this document.

Textbook problems: In the interest of fairness and accessibility, I will pull practice problems and enrichment reading from the free and openly available MIT OpenCourseWare Calculus text of Strang. Click here to access this textbook. The practice problems will be updated on a (roughly) week-by-week basis; for some weeks there may be no practice problems. Many Duke courses use recent editions of Hughes-Hallett; this is a resource you may consider purchasing or finding in the library, but is not required for this class. My personal favourite calculus texts are those of Michael Spivak. I highly recommend them for enrichment reading, but again do not require them for this class. See the syllabus for more information on titles, editions, etc.