- Galileo's Inclined Plane
- rolling a ball down the ramp
- picture shows where the ball after each moment.
- snapshots taken at constant rate
- We can figure out the instantaneous speed
- Equ is \(y = 32x^2\)
- Derivative is \(y = 32(x+d)^2 - 32x^2\)
- "Instantaneous speed as a function"
\[
\begin{split} y &= \frac{a(x+h)^2 -ax^2}{h} \\ &= \frac{a(x^2+h^2+2xh) -ax^2}{h} \\ &= \frac{ax^2 + ah^2 + 2axh -ax^2}{h} \\ &= \frac{\cancel{ax^2} + axh^2 + 2ah \cancel{- ax^2}}{h} \\ &= \frac{ah^{\cancel{h}} + 2ax\cancel{h}}{\cancel{h}} \\ &= \cancelto{0}{ah} + 2ax \end{split}\]