I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years back but only used it in a “mechanical” fashion, without having a clear understanding of the basic concepts).
Besides the formal abstract stuff (like the formal definition of the derivative, the derivatives of the elementary functions and the rules of differentiation), I thought to talk about more “popular” aspects concerning derivatives like:
- The concrete problems that motivate the concept of derivative (like the idea of instantaneous speed)
- Examples of nowhere differentiable functions or curves (like fractal curves or Weierstrass’ function)
- Simple geometrical and numerical applications of the derivative (constructing tangents to curves, approximating functions by first degree polynomials)
- A short histrory on how we arrived to the concept of derivative (? no ideas where to begin)
I would like to know your opinion on my approach, what other topics would you choose and what resources would you recommend me that would fit this context.
One humble suggestion:
Read chapter 1.8.2 of the famous "Feynman Lectures on Physics."
In order to introduce the concept of instantaneous speed, Feynman quotes this joke there:
The cop stops the lady and says: "Lady, you were going 60 miles per hour!" She says, "That is impossible, sir, I was travelling only for seven minutes..."
The conversation between the cop and the "blondy" is a perfect introduction to the concept of instantaneous speed.