**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.

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 x^{3} is what it is

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

**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 x^{2}

The derivative of x^{2}: The proof

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

Discovering the derivative of x^{3} by considering a cube

The derivative of x^{3}: How the usual proof relates to a cube

Discovering the derivative of x^{4} 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!*

## 3 Comments

Great job. Wish my tw eacher was as good.

Thanks!

Hi Dennis,

Thank you for being an awesome teacher. I had you at Stockton State College around 1983 or so. I still remember you knowing everyone’s name by the second class and the examples you used for exponential growth 2^50th. I use these in my own teaching. And I have the same knack for remembering names. Not sure if you were using a system?

I had been teaching Sustainability at a college in NJ for the past 10 years and now I teach it at a university in Thailand outside of Bangkok. Thank you for inspiring students to engage with Mathematics. I always try to head off the “I hate Math” kid into thinking that Math is important for the development of parts of the brain or in general the instrument sometimes musical that we call our bodies. And I now I feel more confident as to telling them what calculus actually is. Cheers man! Thanks for being a great teacher. I also see that you are a CCNY alum. I graduated CUNY BA, but CCNY was my home college. Many great memories of that school. It looks amazing now. It has been completely restored. When I was there in the 90s there were more than 80 languages spoken on campus. Amazing place. I went on to study International Affairs there and at the Univ of Pittsburgh.

All the best,

Drop me an email if you have the chance.

John (Jack) Daly

Jad327@live.com