Find tangents and normals

Tangents and normals through a specified point

For $P(x_1,y_1)$ on $y=f(x)$, the tangent has the curve’s local direction at P; the normal passes through P perpendicular to that tangent. The source calls the tangent a line touching the curve. This does not mean that a tangent cannot cross the curve.

Finite tangent slopes

First check that the specified point lies on the curve. When $m=f^{\prime}(x_1)$ is finite, use the point and tangent slope:

$y-y_1=m(x-x_1).$

If $m\ne0$, the normal has slope $-1/m$:

$y-y_1=-\frac{x-x_1}{m}.$

The source remark gives equal slopes for parallel finite-slope lines, and negative reciprocal slopes for perpendicular lines.

Always substitute the specified point after finding the slope.

Horizontal and vertical tangents

The reciprocal formula requires a nonzero finite tangent slope. A horizontal tangent ($m=0$) has vertical normal $x=x_1$.

A vertical tangent, when it exists, has horizontal normal $y=y_1$; it is not represented by a finite derivative value. These special-case qualifications are independently authored.

Connect the method to the examples

Example 8 preserves all six printed calculator steps across pages 43–44, with an independent exact derivative check.

Self-Practice 2 adds analytical working to the source’s final-answer-only exercise.

Example 9 preserves the complete implicit differentiation and negative-reciprocal normal calculation.

Example 7 retains the preceding derivative sketch and its lecture attribution; it repeats Review 4, whose printed 3 marks do not appear in the lecture.