Investigation

### Looking at the quadratic $y=x^2$

How can we find the gradient of the line through the points with coordinates $(3,9)$ and $(3.5,12.25)$?

How can we find the gradient of the line through the points with coordinates $(3,9)$ and $(3.1,9.61)$?

How can we find the gradient of the line through the points with coordinates $(3,9)$ and $(3.01,9.0601)$?

How can we generalise this for any point $(x,x^2)$?

Can you use a spreadsheet to investigate what the tangent line gradient will be for $x=2$, $x=4$ etc…?

### Looking at the cubic $y=x^3$

Can you use a spreadsheet to investigate what the tangent line gradient will be for $x=1$, $x=2$, $x=3$ etc… on the cubic curve $y=x^3$?

If we take $x=2$, we have the point $(2,8)$ and we can choose a second close point on the curve $y=x^3$, $(2.01,9.261)$. This would give a gradient of

$\frac{y_2-y_1}{x_2-x_1}=\frac{9.261-8}{2.1-2}=12.61.$

How can we generalise this for any point $(x,x^3)$?