In the previous activity, you may have noticed the following formula.[br][br]gradient function [math]\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}[/math] [br][br]The first part of this is read as the [b]limit as h tends to 0[/b], it means we are interested in what happens as h approaches zero.[br][br]The next part asks us to evaluate a function at different points and divide by [i]h[/i]. If we perform this calculation for the function [math]f\left(x\right)=x^2[/math] let us see what happens[br][br][math]f\left(x+h\right)=x^2+2hx+h^2[/math][br][br]So [math]f\left(x+h\right)-f\left(x\right)=2hx+h^2[/math][br][br]If we divide this expression by [i]h[/i] we get [math]2x+h[/math][br][br]But remember I am interested in what happens as [math]h\to0[/math], so this evaluates to [math]2x[/math][br][br]This is the gradient function for [math]f\left(x\right)=x^2[/math][br][br]We can perform the same calculations for [math]f\left(x\right)=x^3[/math][br][br][math]f\left(x+h\right)=x^3+3x^2h+3xh^2+h^3[/math][br][br][math]f\left(x+h\right)-f\left(x\right)=3x^2h+3xh^2+h^3[/math][br][br]Dividing by[i] h[/i] gives[br][br][math]3x^2+3xh+h^2[/math][br][br]and since [math]h\to0[/math][br][br]the gradient function is [math]3x^2[/math][br][br]As an exercise, you could try to perform the same process for different quadratic, cubics and other polynomials. You may even like to try some other functions.