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Carefully constructed presentation to guide

you through complex material.

Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25
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Section 2.1 • Rates of Change and the Derivative 167
Alumni Profiles. Ryan Murphy hapter 2 opens by returning to the tangent problem to find an equation of the tangent
Each chapter Murphy Athlete Cline to the graph of a function f at a point P = (c, f (c)). Remember in Section 1.1
we found that the slope of a tangent line is a limit,
opens with a Iama f (x) − f (c)
professional
brief testimonial Photo provided by Ryan swimmer and m tan = lim x − c
x→c
four-time
from friends and Olympic gold medalist. The This limit is one of the most significant ideas in calculus, the derivative.
In this chapter, we introduce interpretations of the derivative, treat the derivative as
colleagues of knowledge of velocity and a function, and consider some properties of the derivative. By the end of the chapter,
acceleration that I learned in
the author team AP Calculus has helped me you will have a collection of basic derivative formulas and derivative rules that will be
used throughout your study of calculus.
R
describing how their better understand and develop
the strategy behind my
experience with AP ® backstroke. 2.1 Rates of Change and the Derivative
Calculus has shaped R OBJECTIVES When you finish this section, you should be able to: Objectives describe core
AP EXAM TIP
their professional BREAK IT DOWN: As you work through the 1 Find equations for the tangent line and the normal line to the graph of a knowledge and key skills.
function (p. 168)
lives. problems in this chapter, remember to follow Sullivan 2 Find the rate of change of a function (p. 169) 2023 9:25 Every section starts with
June 19,
04 apcalc4e 45342 ch02 166 233 3pp
these steps:
Step 1 Identify the underlying structure and 3 Find average velocity and instantaneous velocity (p. 170) objectives that are repeated
related concepts. 4 Find the derivative of a function at a number (p. 173)
Step 2 Determine the appropriate math rule 168 Chapter 2 • The Derivative and Its Properties as headlines to reinforce
or procedure. In Chapter 1, we discussed the tangent problem: Given a function f and a point P on its
Step 3 Apply the math rule or procedure. ideas. Objectives are linked
DEFINITION Tangent Line
graph, what is the slope of the tangent line to the graph of f at P? See Figure 1, where � T
Step 4 Clearly communicate your answer.
is the tangent line to the graph of f at the point P = (c, f (c)). to the worked examples
The tangent line to the graph of f at a point P is the line containing the
On page 231, see how we’ve used these steps The tangent line � T to the graph of f at P must contain the point P. Since finding
point P = (c, f (c)) and having the slope
R
to solve Section 2.2 AP Problem 12 on page the slope requires two points, and we have only one point on the tangent line � T , we and their clear, succinct
191.
reason as follows. f (x) − f (c)
Suppose we choose any point Q = (x, f (x)), other than P, on the graph of f . = lim explanations.
m tan
y y 5 f (x) Secant (Q can be to the left or to the right of P; we chose Q to be to the right of P.) The x→c x − c (2)
line
Q 5 (x, f(x)) line containing the points P = (c, f (c)) and Q = (x, f (x)) is a secant line of the graph
provided the limit exists.
of f . The slope m sec of this secant line is
T
f (x) − f (c) The limit in equation (2) that defines the slope of the tangent line occurs so
P 5 (c, f (c)) Tangent NOTE It is possible that the limit in (2) does (1)
m sec = this
line not exist. The geometric significance of x − c 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:
c x x Figure 2 shows three different points Q 1 , Q 2 , and Q 3 on the graph of f that are f (x) − f (c)
successively closer to the point P, and three associated secant lines � 1 , � 2 , and � 3 . The (c) = lim (3)
f
�
Easy-to-follow signposts. closer the points Q are to the point P, the closer the secant lines are to the tangent x→c x − c
Figure 1 m sec = slope of the secant line.
line � T . The line � T , the limiting position of these secant lines, is the tangent line to the. The line � T , the limiting position of these secant lines, is the tangent line to the
line � T
graph of f at
Headlines reinforce the order of Secant graph of f at P. P. 1 Find Equations for the Tangent Line and the Normal Line
y
Secant
y 5 f (x)
to the Graph of a Function
If the limiting position of the secant lines is the tangent line, then the limit of the
1 lines If the limiting position of the secant lines is the tangent line, then the limit of the
the coverage and the objectives. Q 1 2 3 slopes of the secant lines should equal the slope of the tangent line. Notice in Figure 2 2
slopes of the secant lines should equal the slope of the tangent line. Notice in Figure
that as the points Q 1 , Q 2 , and Q 3 move closer to the point P, the numbers x get closer, Q 2 , and Q 3 move closer to the point P, the numbers x get closer
AP EXAM TIP
Q 2 that as the points Q 1
R
to c. So, equation (1) suggests that
T to c. So, equation (1) suggests that
Q 3 T THEOREM Equation of a Tangent Line
P 5 (c, f (c)) Tangent Problems on the exam often ask about the �
If m tan
line tangent line. m tan = Slope of the tangent line to f at P= Slope of the tangent line to f at P = f (c) exists, then an equation of the tangent line to the graph of a
m tan
f (x) − f (c) function y = f (x) at the point P = (c, f (c)) is
f (x) − f (c)
as x gets closer to c c
= Limit as x gets closer to
= Limit of of
x
−
c x 3 x 2 x 1 x x x − c c y − f (c) = f (c)(x − c)
�
x
f
(
x f f (x) − f (c)) c
(
)
−
= lim
= lim
x
x x → c c x − c c
−
→
f (x) − f (c)
Figure 2 m tan = lim 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
x→c x − c provided the limit exists. is called the normal line to the graph of f at P.
provided the limit exists. m 1 and m 2 ,
horizontal, with slopes
respectively, are perpendicular if and only if
1
m 1 =−
m 2 THEOREM Equation of a Normal Line
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.
Easy-to-grasp titles.
Worked EXAMPLE titles CALC CLIP EXAMPLE 1 Finding Equations for the Tangent Line
and the Normal Line
underscore the objectives (a) Find the slope of the tangent line to the graph of f (x) = x at the point (−2, 4).
2
(b) Use the result from (a) to find an equation of the tangent line at the point (−2, 4).
and tell you the scope of (c) Find an equation of the normal line to the graph of f at the point (−2, 4).
the explanation. (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
RECALL One way to find the limit of a f (−2) = lim f (x) − f (−2) = lim x − (−2) 2 = lim x − 4
�
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
x
xi
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