- A differentiable function is locally linear
- This is an alternate definition of differentiability. It will be used to prove at least three theorems, and the lecture shows that the idea generalizes to higher derivatives and to functions of several variables
- Adding zero; multiplying by 1 — a powerful proof technique
- This rather trivial statement leads to lots of proof opportunities. At least four different theorems will be proved using the idea. Some of the zeros or ones will be rather complicated expressions.
- Integration and differentiation are not different subjects
- You can’t teach everything at once, so courses are designed teach derivatives and then integrals. The Fundamental Theorem of the Calculus is always taught, so that students know the subjects are connected. But they are much more tightly woven than that, and students don’t realize it. At least four different theorems will be presented where something about derivatives is proved using integrals, or vice versa.
- The binomial theorem
- Useful in lots of ways. At least three theorems will be proved which are easier to do with the binomial theorem than without it.
- Approximations are your friends
- Although we tend to think that mathematical problems have “exact” solutions, most problems can be solved only approximately. Also, it turns out that approximation techniques are embedded throughout calculus and analysis as well as other areas of mathematics. So knowing more about approximation is important. Several examples will show various uses of approximation:
- quickly getting an idea of what the picture is
- developing a “clue” about what kind of answer to be looking for
- finding ideas about how to solve the real problem
- The lecture will also show that it is frequently possible to estimate the error between your approximation and the true value, even though you do not know the exact answer.
- The mean value theorem for derivatives
- Endlessly useful for proving more advanced results. This lecture offers three different proofs, one a bit surprising, along with a sampling of problems which can be solved using the mean value theorem.
- The exponential functions are differentiable (a three lecture set)
- Not usually covered in first year calculus. Because it presents some difficulties, some books honestly say that the topic is beyond the scope of the book. A number of calculus books are propagating a grossly erroneous proof. This should not be inflicted on innocent students. Then when people get to advanced calculus, it is assumed they know this material. Thefore, the entire subject falls through the cracks.
For this series we have rounded up four proofs, each entirely different from all the others. It is instructive to see how many different ways there are to grab hold of the same problem, and the students gain an understanding of the richness of the tools that calculus provides.
- Lecture 1
- Lecture 2
- Lecture 3
Basic definitions: sequence approximations of irrational numbers; definition of exponential functions; uniqueness of definition and continuity of exponential functions.
Sketch of proof #1; review of definition of Riemann integral; full presentation of proof #2.
Presentation of proof #3 and proof #4; comparison of the strengths and weaknesses of each approach.