Making Sense of Calculus

A Calculus Retrospective

Many people I speak with who have taken calculus remember vaguely some of the techniques they learned but say they never really understood what calculus was all about. The videos in this section are aimed at these people. We focus on the big ideas behind calculus so that you can see the reasons why things work the way they do. The first few videos in this series are appropriate even for people who’ve never had calculus. Subsequent ones require you to remember just a few basic things that you learned when you took calculus.

 

What IS Calculus, anyway?

Why we care about rates of change (coming)

A surprising use of rates of change (coming)

The area of a circle and the Fundamental Theorem of Calculus

• Part I – The derivative of the area of a circle is the circumference
• Part II – The “formula” for the circumference is no such thing
• Part III – Area of a circle and the Fundamental Theorem of Calculus
• Part IV – Spheres, squares, cubes and blobs

Why the derivative of x³ is what it is

 

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Believable Calculus: Learning Calculus So That It Makes Sense

If you never learned calculus before, or if you want a refresher on the details, the videos in this section are for you. We focus on the big picture, on understanding where each topic fits into the whole, and on being able to “see” why each concept is believable. So much of calculus is built on things you already know; we want to make sure that you see that.

Note: Unfortunately, the first few videos in this section haven’t been produced yet. You’ll have to know a few basics about derivatives in order to watch the first video listed below.

 

Introduction (coming)

Visualizing the derivative of x²

The derivative of x²: The proof

• Reminder: Why the definition of the derivative is what it is (coming)

Discovering the derivative of x³ by considering a cube

The derivative of x³: How the usual proof relates to a cube

Discovering the derivative of x⁴ by considering a tesseract (coming)

The Product Rule

• The derivative of a product is not the product of the derivatives
• Visualizing the derivative of a product
• The official proof of the Product Rule
• Some technical points about limits and continuity

The Power Rule for positive powers (coming)

Discovering the derivative of x^-1 by considering a special rectangle

Discovering the derivative of x^-1: more explanation

Visualizing the derivative of x^-2

The Power Rule for negative powers

Visualizing the derivatives of x^1/2 and x^1/3

Visualizing the derivative of x^-1/2 (an exercise)

The Power Rule for roots (coming)

Visualizing the derivatives of x^2/3 and x^3/2

The Power Rule for fractional exponents (coming)

…and much more to come!

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