# Rates of Change

See the design goals. See the matching Ontario Mathematics Curriculum expectations.

Elements to be fixed or checked:

- It looks like the same error in graphing exists here as it does with the graphing in the Polynomial interactive.The leading coefficient is creating the opposite graph, same as the polynomials interactive. (the -ive quadratic is opening up instead of down, etc.)
- There also appears to be an error with the calculation of the first and second derivatives. As an example, if the original function was y = 2(x-0)(x+2)(x-2) [this is the same as y = 2x^3-8x], the first derivative should be y' = 6x^2 - 8 and the second derivative should be y'' = 12x. The interactive shows the first derivative as y' = 6x^2 - 4 and the second derivative as y'' = -3x.
- It would be beneficial to be able to include or remove the 1st and 2nd derivative function graphs.
- Check videos 2-5.

- Consider the following:
- Sketch the graph of a function like y = x^2 and then to graph its first and second derivatives on the same set of axes.
- List as many relationships between these graphs as you can.
- Do you think these relationships you noticed in the interactive above hold true for other types of graphs, like trigonometric, exponential, and rational functions. Test your theories by graphing y = sin x and its first and second derivatives on the same axes.