infinitesimal 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 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

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 $x=0$ .*

*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