History and Applications of Calculus
A self-study course to be taken after an introductory course in calculus
This short course is intended to serve as a capstone to the mostly standard material you saw in your introductory single variable calculus course. It should
give you a glimpse of some of the early history of calculus, and show you some of the classical problems that motivated its development in the first place. As well as this, I hope you will work on some interesting problems by yourself, engage in some independent reading, and learn how to write mathematics using the LaTeX typesetting system.
The main content of the class consists of eight weeks of lectures which are posted to YouTube.
This web page contains some supplementary material. This consists of
recommended reading for each of the weeks of the term, almost all of which are taken from the delightful book Calculus Gems by George Simmons (which you will need to buy for yourself!). Then there are weekly homework sets, whch you are recommended not only to solve independently, but also to write up your solutions carefully using LaTeX.
On this page, you will find PDF summaries of the lectures,
PDF files containing the homework problems, and LaTeX templates to start you off in writing your solutions (these contain hints!).
For more instructions watch Lecture 0 on YouTube!
- WEEK ONE
- PDF summaries of the lectures for the week: 1-1 (review of differentiation and trig functions), 1-2 (first discussion of differential equations and the method of separation of variables), 1-3 (definite integrals and the Fundamental Theorem of Calculus)
- Recommended reading: Sections A.5 and B.6 from Calculus Gems (both about Archimedes)
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK TWO
- PDF summaries of the lectures for the week: 2-1 (Fermat's formula for area under the power function and other applications of FTC to calculating areas and volumes), 2-2 (arc length, surface area, and introduction to conic sections), 2-3 (parametric equations, polar coordinates and the cycloid)
- Recommended reading: Sections B.3 and A.13 from Calculus Gems (Archimedes' proof of the quadrature formula and the life of Fermat)
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK THREE
- PDF summaries of the lectures for the week: 3-1 (the cardioid and more about conic sections), 3-2 (the optical property of the parabola leading to the general polar equation for conic sections), 3-3 (crash course in Newtonian dynamics)
- Recommended reading: Section A.10 from Calculus Gems (Kepler and the man with the golden nose)
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK FOUR
- PDF summaries of the lectures for the week: 4-1 (Kepler's Laws and motion under a central force), 4-2 (Newton's Law of Gravity and his derivation of Kepler's First and Third Laws), 4-3 (More about planetary motion and some discussion of the substitution t = tan(x/2))
- Recommended reading: Section A.18 from Calculus Gems (Newton's life and scientific achievements)
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK FIVE
- PDF summaries of the lectures for the week: 5-1 (Pythagorean triples then rigorous definition of log and exponential functions), 5-2 (The binomial series), 5-3 (Power series definition of trig functions)
- Recommended reading: Section A.19 from Calculus Gems (Leibniz's life and scientific achievements)
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK SIX
- PDF summaries of the lectures for the week: 6-1 Iirrationality of e and pi), 6-2 (Hyperbolic functions), 6-3 (The catenary)
- Recommended reading: ()
- Homework problems: PDF
- LaTeX template for your homework solutions: TEX
- WEEK SEVEN
- WEEK EIGHT
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