In the last investigation you conjectured that the gradient of a tangent to the curve at any point of the graph [math]y=x^2[/math] was [math]2x[/math]. [br][br]To justify this conjecture we will consider the gradient of a chord from the point [math]\left(x,x^2\right)[/math] on the graph of [math]y=x^2[/math] to a point that is a distance h further along the x-axis as shown on the graph below.
[b]a)[/b] What do you notice about the gradient of the chord as h decreases?[br][b]b) [/b]What will happen as h approaches zero? [math]h\longrightarrow0.[/math]
[b]a) [/b]Explain why an expression for the gradient of the chord between the two points [math]\left(x,x^2\right)[/math] and [br][br][math]\left(x+h,\left(x+h^2\right)\right)is:[/math][br][br][math]\frac{\left(\left(x+h\right)^2-x^2\right)}{h}[/math][br][br][b]b) [/b]What would happen if you let [math]h\longrightarrow0[/math]?
Expand and simplify your expression. What happens now as [math]h\longrightarrow0?[/math]
How does your answer to this compare to the conjecture you made for the gradient of [math]y=x^2[/math] in the first Geogebra investigation?