Page 17 - 2024-calc4e-SE proofs-4e.indd

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

Section 2.1 • Rates of Change and the Derivative 167

hapter 2 opens by returning to the tangent problem to find an equation of the tangent
Murphy Ryan Murphy Cline to the graph of a function f at a point P = (c, f (c)). Remember in Section 1.1
Athlete
we found that the slope of a tangent line is a limit,
Photo provided by Ryan I am a m tan = lim f (x) − f (c)
professional
x − c
x→c
swimmer and
four-time
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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
knowledge of velocity and a function, and consider some properties of the derivative. By the end of the chapter,
acceleration that I learned in you will have a collection of basic derivative formulas and derivative rules that will be
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AP Calculus has helped me used throughout your study of calculus.
better understand and develop
the strategy behind my
backstroke. 2.1 Rates of Change and the Derivative

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AP EXAM TIP OBJECTIVES When you finish this section, you should be able to:
1 Find equations for the tangent line and the normal line to the graph of a
BREAK IT DOWN: As you work through the
function (p. 168)
problems in this chapter, remember to follow
these steps: 2 Find the rate of change of a function (p. 169)
3 Find average velocity and instantaneous velocity (p. 170)
Step 1 Identify the underlying structure and
related concepts. 4 Find the derivative of a function at a number (p. 173)
Step 2 Determine the appropriate math rule
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.
graph, what is the slope of the tangent line to the graph of f at P? See Figure 1, where ℓ T
Step 4 Clearly communicate your answer.
is the tangent line to the graph of f at the point P = (c, f (c)).
On page 231, see how we’ve used these The tangent line ℓ T to the graph of f at P must contain the point P. Since finding
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steps to solve Section 2.2 AP Problem 12
the slope requires two points, and we have only one point on the tangent line ℓ T , we
on page 191.
reason as follows.
Suppose we choose any point Q = (x, f (x)), other than P, on the graph of f .
y y 5 f (x) Secant
line (Q can be to the left or to the right of P; we chose Q to be to the right of P.) The
line containing the points P = (c, f (c)) and Q = (x, f (x)) is a secant line of the graph
Q 5 (x, f (x))
of f . The slope m sec of this secant line is
ℓ T
f (x) − f (c)
P 5 (c, f (c)) Tangent (1)
line m sec = x − c

c x x Figure 2 shows three different points Q 1 , Q 2 , and Q 3 on the graph of f that are
successively closer to the point P, and three associated secant lines ℓ 1 , ℓ 2 , and ℓ 3 . The
Figure 1 m sec = slope of the secant line. closer the points Q are to the point P, the closer the secant lines are to the tangent
line ℓ T . The line ℓ T , the limiting position of these secant lines, is the tangent line to the
graph of f at P.
y y 5 f (x) Secant
ℓ 1 lines If the limiting position of the secant lines is the tangent line, then the limit of the
ℓ
Q 2 ℓ slopes of the secant lines should equal the slope of the tangent line. Notice in Figure 2
1 3
Q that as the points Q 1 , Q 2 , and Q 3 move closer to the point P, the numbers x get closer
2
Q ℓ T to c. So, equation (1) suggests that
3
P 5 (c, f (c)) Tangent
line m tan = Slope of the tangent line to f at P
f (x) − f (c)
= Limit of as x gets closer to c
c x x x x x − c
3 2 1
x f (x) − f (c)
= lim
x → c x − c
f (x) − f (c)
Figure 2 m tan = lim
x→c x − c provided the limit exists.
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