Things you might have noticed

This interactivity shows what you might have drawn. You can select the value of \(b\) and then zoom in to the graph at that point.

You have probably noticed that the graph looks like a straight line when you zoom in far enough.

  • What is the gradient of the straight line you end up with?

  • Comparing your results with those of other groups or with the interactivity, how does the gradient depend on the value of \(b\) you are using?

  • Does every graph look like a straight line when you zoom in far enough?

You can explore this more generally with the interactive graph we have provided.

After thinking about this problem, you might like to think about tangents or how you might develop these ideas to work out gradient functions for powers of \(x\).

You might also want to think about how this approach relates to the algebraic point of view explored in Binomials are the answer: the derivative there is found as the limit of the gradients of chords, using the algebraic expression \(\dfrac{(x+h)^2-x^2}{(x+h)-x}\).