# Review of Calculus Principles

In order to *understand* dynamics, you are going to have to *understand* basic principles of integral and derivative calculus. For this reason, I have posted some notes on derivatives and integrals. When you understand the principles, you should be able to complete the practice problems I have included.

#### Reference Material

I’m going to offer you two different ways to review concepts of differentiation and integration. The first is a document that I wrote a few years ago. The second is a series of beautiful videos created by Grant Sanderson (3 Blue 1 Brown). You may choose either of these approaches, or both.

*The Document*

This document discusses how to think about the derivative and integral, based on the definitions of the operations. I have created video versions of parts of the calculus review in case you prefer to listen/watch the material, rather than read: Definition of Derivative; Check Your Understanding of Derivative; Check Your Understanding of Integral.

**The Videos**

Here are those gorgeous videos I mentioned. By illustrating how one would calculate the area of a circle using calculus-like thinking, Sanderson elegantly illustrates the Fundamental theorem of calculus.

In this second video, he illustrates the derivative by thinking about the “velocity” of a car. I do the same in my notes, but Sanderson’s is so much more beautiful. I just have two small beefs with the video. First, he uses the word “velocity” when he should say “speed.” Secondly, I disagree with the paradox he talks about. There’s nothing inconsistent about a moving object having zero velocity (motionless) for just an instant. If you want, you can stop watching after time 10:25.

And finally, a video on integration and its relationship with the derivative (i.e. the Fundamental theorem of calculus).

#### Take-Aways, Derivative

- A time derivative is a measure of how fast something is changing at some instant.
- If we have a plot of some function with respect to the dependent variable
*t*, then the derivative of the function with respect to*t*, evaluated at*t*=*t0*, is the slope of the tangent line at*t*=*t0*. - If something is
*increasing*with respect to time, then its time derivative is*positive*. - If something is
*decreasing*with respect to time, then its time derivative is*negative*. - When something is
*NOT changing,*even for an instant, its time derivative must be*zero*at that instant.

#### Take-Aways, Integral

- One can think of an integral as the opposite of derivative, an “anti-derivative.”
- One can think of an integral as a signed area under the curve.
- When the integrand (function you’re integrating) is
*positive*, the integral of that function should be*increasing*. - When the integrand is
*negative*, the integral of that function should be*decreasing*. - When the integrand is
*zero*, the integral should not change, at least for a brief moment. - For indefinite integrals, there is an arbitrary constant of integration.

#### Check Your Understanding, Derivative

To check your understanding of the derivative, I would like you to consider the plot below which shows a function x of a variable t. Given your understanding of the derivative, could you sketch (by hand) a plot of the derivative of function x with respect to t? Think about it. Try it!

To check your results, see the video here.

#### Check Your Understanding, Integral

Now consider the function in the figure below. This time I want you to sketch a plot of the integral of the function. You can set the initial condition at t=0 to zero if you wish. Think about it. Try it!

To check your results, see the video here.

#### Practice Problems

To help you understand differentiation and integration at a level that will help you understand dynamics, you should try answering these four practice problems.

#### Answers to Practice Problems

After you work out the practice problems you should check your answers. I have answers posted in two formats below. The PDFs are there if you want a quick look at the answers. I have also posted videos in case you want to hear my explanation.

**Video for Problem A:**

**Video for Problem B:**

**Video for Problem C:**

**Video for Problem D:**