Estimating a gradient with a tangent

A tangent follows the local direction of a smooth curve at a chosen point. Draw it with a ruler, using the curve on both sides of the point to judge its direction.

The solid curve is \(y=x^2\); the dashed line is a tangent at \((2,4)\).

Choose two well-separated points on the tangent, not two nearby points on the curve. Estimate gradient using \(\dfrac{\text{change in}\hspace{0.5em}y}{\text{change in}\hspace{0.5em}x}\). Use the axis numbers, not the apparent angle on the page.

A tangent may cross a curve or meet it elsewhere. “It meets the curve only once” is not a definition of a tangent.