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Chapter 7 Videos — Differentiation

Introduction to Derivatives

Vi Hart's introduction to the derivative by explaining what the derivative is for a car ride---and then what the second and third derivatives correspond to.

The Paradox of the Derivative

3Blue1Brown has an 12-part video series on the "Essence of Calculus."  The derivative is discussed from many different perspectives.  The rest of the videos relating to derivatives are below.

Derivative Formulas Through Geometry

Here's the second derivative video in the aforementioned series.

Visualizing the Chain Rule and Product Rule

Here's the third derivative video in the aforementioned series.  It's on means to visualize the chain rule (Theorem 7.13) and the product rule (Theorem 7.11).

L'Hopital's Rule and the Epsilon Delta Definition

Here's the fourth derivative video in the aforementioned series.  Among other things, it's on l'Hopital's Rule (Theorem 7.27).

Higher order derivatives

Here's the final derivative video in the aforementioned series.

A SPHERE'S SURFACE AREA = 4 CIRCLES

The open question in the derivative chapter (page 256) asked about why if take the derivative of the πr² formula for the area of circle, do you get the formula for the diameter: 2πr.  And, more importantly, how deep does this correspondence go?  Related issues between geometric formulas are discussed in this video.

THE DEVIL'S STAIRCASE

The devil's staircase is a function which has a derivative of zero nearly everywhere, and yet somehow climbs from (0,0) to (1,1).  And here's the crazy thing:  It does so continuously.  This is the topic of Example B.8 in Appendix B, as well as of this video by PBS Infinite Series.

Chapter 8 Videos — Integration

Infinitesmals

This video, from Numberphile, discusses infinitesmals, which were originally used to define the integral, and prove theorems thereof.  This video also touches on some of the interesting history of this development, including the rivalry between Newton and Leibniz.

Integration and the Fundamental Theorem of Calculus

Check out this great video

What does area have to do with slope?

Check out this great video

MUSIC AND MEASURE THEORY

Section 8.8 was about the measure zero integrability criterion, and this idea of measure was investigated further in Example B.17 of Appendix B.  This idea of measure is important in real analysis and a connection between it and music is discussed in this 3Blue1Brown video.

THE AXIOM OF CHOICE

The final open questions of Chapter 2 (on page 63) deal with the axiom of choice.  This video talks generally about this axiom, as well as a specific application to the measure of sets, which was discussed in the zero case in Section 8.8.  This specific application was also discussed in Fact B.17 of Appendix B.

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