AP Calculus AB and BC

Tangent Line vs Normal Line

Q: What is the normal line when the tangent is vertical?

Horizontal. The normal line is y = f(a), since perpendicular to vertical is horizontal.

The tangent line at a point has slope equal to the derivative there. The normal line is perpendicular to it, so its slope is the negative reciprocal of the derivative. Both pass through the same point on the curve.

Tangent line

Use when: You need the direction the curve is heading, or a linear approximation near the point.

Normal line

Use when: The question explicitly asks for the normal, or for a line perpendicular to the curve.

Side by side

	Tangent	Normal
Slope	$f'(a)$	$-\frac{1}{f'(a)}$
Relation	Parallel to the curve	Perpendicular to the curve
If $f'(a) = 0$	Horizontal	Vertical, $x = a$
Shared point	$(a, f(a))$	$(a, f(a))$

Everything carries over between them except the slope, which flips and changes sign. If the tangent slope is $\frac{2}{3}$ , the normal slope is $-\frac{3}{2}$ , and both lines use the same point in point-slope form.

The mistake

Using $f'(x)$ rather than $f'(a)$ . The slope of a specific line is a number, obtained by evaluating the derivative at the point of tangency.

Frequently asked questions

What is the normal line when the tangent is vertical?

Horizontal. The normal line is $y = f(a)$ , since perpendicular to vertical is horizontal.

In the CED: Unit 2: Defining the Derivative