Calculus Archives - Infinity is Really Big

February 25, 2020February 25, 2020

The Fundamental Theorem of Calculus

Calculus 2, Lectures 5A through 6 (Videotaped Fall 2016)

The Fundamental Theorem of Calculus implies that the area under the graph of the speed function gives the distance traveled function.

The Fundamental Theorem of Calculus is often split into two forms in textbooks.

These forms are typically called the “First Fundamental Theorem of Calculus” and the “Second Fundamental Theorem of … Read the rest

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

November 30, 2019November 30, 2019

Infinitesimal Calculus and Calculus Rules

Calculus 1, Lectures 16A through 18A

Infinitesimal calculus can be used to derive the derivative of the sine function.

Did you know that Newton and Leibniz did not know the precise definition of a limit?

Instead, they approached calculus in an intuitive way. Today, this intuitive method is called infinitesimal calculus. It is based on the concept of … Read the rest

October 29, 2019October 29, 2019

Differentiable Functions and Local Linearity

Calculus 1, Lectures 12 through 15B

An infinitely oscillating continuous and differentiable function. — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

x=0 — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

In Steven Strogatz’s excellent book, Infinite Powers, there is a big emphasis on an idea he calls The Infinity Principle.

The concepts of a continuous function and of a differentiable function are … Read the rest

October 15, 2019October 15, 2019

Limit Definition, Continuity, and Derivatives

Calculus 1, Lectures 8A through 11

The limit definition of the derivative leads to the graph of a function with a hole (a removable discontinuity). The limit still exists, based on the precise epsilon delta definition of a limit. — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

DQ(h)=\frac{(2+h)^{2}-8}{h} — The graph of the difference quotient $DQ(h)=\frac{(2+h)^{2}-8}{h}$ for computing the derivative $f'(2)$ when $f(x)=x^{3}$ has a removable discontinuity (a hole) at $h=0$ . The limit still exists so that $f'(2)=12$ .

Graphs with holes (missing points) might seem like anomalies — like they are mere curiosities hardly worth studying. However, they arise at the heart of differential calculus.

… Read the rest

September 26, 2019September 26, 2019

Transformations of Functions

Calculus 1, Lectures 6A through 7B

Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.

\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1 — Visual for the fact that $\log_{2}(2x)=\log_{2}(x)+\log_{2}(2)=\log_{2}(x)+1$ in terms of transformations of functions.

In the 1600’s, algebra and geometry got “married” to create a new subject called analytic geometry. Insights in either branch led to insights in the other. This has been very fruitful over the years.

The symbiosis between these subjects can be especially … Read the rest

September 20, 2019September 26, 2019

The Derivative of y = x^2

Calculus 1, Lectures 4A through 5B

The derivative of x^2 at an arbitrary value of x is 2x. This is the slope of the tangent line at that point. — The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

y=f(x)=x^{2} — The derivative of $y=f(x)=x^{2}$ at an arbitrary value of $x$ is equal by $2x$ . This is the slope of the tangent line at that point and is approximated by $2x+\Delta x$ , which are slopes of secant lines.

Lectures 4A through 5B of my series of Calculus 1 lectures have a lot of content. I will summarize the

… Read the rest