SL 5.4—Tangents and normals
- Syllabus
- First assessment 2021
- Objective
- —
- Level
- HL
A tangent gives local direction; a normal is perpendicular to it
At x=a, the tangent gradient is f'(a). If that gradient is non-zero, the normal gradient is −1/f'(a), because perpendicular non-vertical lines have product of gradients −1.
Find the point (a,f(a)), calculate the tangent gradient, then use point–gradient form. Handle a horizontal tangent separately: its normal is vertical and cannot be written with a finite gradient.
For y=x² at x=1, the point is (1,1), tangent gradient 2 and tangent y−1=2(x−1). The normal gradient is −1/2, giving y−1=−(x−1)/2.
A normal is not the negative of the tangent gradient. It is the negative reciprocal, and the vertical/horizontal special cases must be stated.