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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

168 Chapter 2 • The Derivative and Its Properties

DEFINITION Tangent Line
The tangent line to the graph of f at a point P is the line containing the
point P = (c, f (c)) and having the slope

f (x) − f (c)
m tan = lim (2)
x→c x − c
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provided the limit exists.

NOTE It is possible that the limit in (2) does The limit in equation (2) that defines the slope of the tangent line occurs so
not exist. The geometric significance of this frequently that it is given a special notation f (c), read, “ f prime of c,” and called
′
is discussed in the next section.
prime notation:
f (x) − f (c)
f (c) = lim (3)
′
x→c x − c
1 Find Equations for the Tangent Line and the Normal Line
to the Graph of a Function

R
AP EXAM TIP
THEOREM Equation of a Tangent Line
Problems on the exam often ask about the
If m tan = f (c) exists, then an equation of the tangent line to the graph of a
′
tangent line.
function y = f (x) at the point P = (c, f (c)) is
′
y − f (c) = f (c)(x − c)

RECALL Two lines, neither of which is The line perpendicular to the tangent line at a point P on the graph of a function f
horizontal, with slopes m 1 and m 2 , is called the normal line to the graph of f at P.
respectively, are perpendicular if and only if
1
m 1 = −
THEOREM Equation of a Normal Line
m 2
An equation of the normal line to the graph of a function y = f (x) at the
point P = (c, f (c)) is
1
y − f (c) = − (x − c)
f (c)
′
provided f (c) exists and is not equal to zero. If f (c) = 0, the tangent line is
′
′
horizontal, the normal line is vertical, and the equation of the normal line is x = c.

Finding Equations for the Tangent Line
CALC CLIP EXAMPLE 1
and the Normal Line
2
(a) Find the slope of the tangent line to the graph of f (x) = x at the point (−2, 4).
(b) Use the result from (a) to find an equation of the tangent line at the point (−2, 4).
(c) Find an equation of the normal line to the graph of f at the point (−2, 4).
(d) Graph f , the tangent line to f at (−2, 4), and the normal line to f at ( −2, 4) on
the same set of axes.

Solution
(a) At the point (−2, 4), the slope of the tangent line is
2
2
f (x) − f (−2) x − (−2) 2 x − 4
RECALL One way to find the limit of a ′
f (−2) = lim = lim = lim
quotient when the limit of the x→−2 x − (−2) x→−2 x + 2 x→−2 x + 2
denominator is 0 is to factor the
= lim (x − 2) = −4
numerator and divide out common factors. x→−2
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