January 2020 - Infinity is Really Big

January 27, 2020January 29, 2020

Integration by Parts (and Linear Motion)

Calculus 2, Lectures 3B through 4 (Videotaped Fall 2016)

Integration by parts is needed to compute the definite integral to find the area under this curve representing the speed of a linear motion. — Integration by parts is needed to compute the definite integral to find the area under this curve representing the speed of a linear motion.

You might say that integration by parts is an “unexpected” or “surprising” method of integration.

While the other main method, integration by substitution, can be thought … Read the rest

January 23, 2020January 23, 2020

Integration by Substitution (Method of Integration)

Calculus 2, Lectures 2A through 3A (Videotaped Fall 2016)

How to visualize integration by substitution for a definite integral. The starting and ending areas are the same. — The integral $\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx$ gets transformed to the integral $\displaystyle\int_{0}^{\pi/2}\cos(u)\, du$ under the substitution $u=x^{2}$ and $du=2xdx$ .

\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx — The integral $\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx$ gets transformed to the integral $\displaystyle\int_{0}^{\pi/2}\cos(u)\, du$ under the substitution $u=x^{2}$ and $du=2xdx$ .

In Calculus 1, the techniques of integration introduced are usually pretty straightforward. In fact, they are usually just memorized as basic facts about antiderivatives.

For Calculus 2, various new integration techniques are introduced, including integration by … Read the rest

January 14, 2020January 14, 2020

Applied vs Pure Mathematics

Calculus 2, Lecture 1 (Videotaped Fall 2016)

Population modeling is an important part of applied mathematics in calculus. It is a good example to start with in looking at the nature of applied vs pure mathematics. — Population modeling is an important part of applied mathematics in calculus.

Calculus is at the heart of much of modern mathematics and its applications. Its discovery by Newton and Leibniz sparked the scientific revolution and enlightenment of the 1600’s and 1700’s and brought about our modern world.

Throughout mathematical history, … Read the rest

January 8, 2020June 29, 2021

Introduction to Multivariable Calculus

Calculus 1, Lectures 18B through 20B

In an introduction to multivariable calculus, the graph of a function of more than one variable should be sliced to help you understand it and its partial derivatives. — The graph of a multivariable function can be sliced to help you understand it and its partial derivatives.

In some ways, multivariable calculus seems like a minor extension of single-variable calculus ideas and techniques. In other ways, it’s definitely a major step up in difficulty.

This post will mainly be an introduction to multivariable … Read the rest