Tangents, Normals, and the Sign of the Derivative

Tangents, Normals, and the Sign of the Derivative

Equation of a tangent

A tangent is the straight line that touches a curve at a point and has the same gradient as the curve there. To find its equation at $x = a$: 1. Find the point: compute $y = f(a)$. 2. Find the gradient: compute $m = f'(a)$. 3. Substitute into $y - y_1 = m(x - x_1)$.

Worked example. Find the tangent to $y = x^2 - 3x + 1$ at $x = 2$. The point is $y = 4 - 6 + 1 = -1$, so $(2, -1)$. The derivative is $\frac{dy}{dx} = 2x - 3$, giving $m = 2(2) - 3 = 1$. The tangent is $y - (-1) = 1(x - 2)$, i.e. $y = x - 3$.

Equation of a normal

The normal at a point is perpendicular to the tangent there. Perpendicular gradients multiply to $-1$, so the normal gradient is $-\frac{1}{f'(a)}$. Using the same point above, the normal gradient is $-\frac{1}{1} = -1$, giving $y - (-1) = -1(x -