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Sullivan
06 apcalc4e 45342 ch04 284 331 3pp
July 13, 2023
9:44
285
Section 4.1 • Interpreting a Derivative
I
Anna
n Chapter 2, we interpreted the derivative in several ways:
Engineer
• Geometric: If y = f (x), the derivative f � (c) is the slope of the tangent line to the
AP Calc filled a graph of f at the point (c, f (c)).
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college math credit, • Physical: When s = f (t) represents the position of an object moving on a line, the
allowing me to derivative f � (t 0 ) is the velocity of the object at time t 0 , and f �� (t 0 ) is the
start taking acceleration of the object at time t 0 .
classes related to my engineering • Analytical: If y = f (x), the derivative f � (c) is the instantaneous rate of
major. It taught me the skills to change of f at the number c.
succeed in later math & science In this chapter, we deepen our understanding of these interpretations of the
courses and was great practice for derivative and begin to delve into other applications of the derivative.
the real-life problem solving that
I do daily as an engineer.
Sharpen the strategies needed to
4.1 Interpreting a Derivative
tackle AP ® problems. OBJECTIVES When you finish this section, you should be able to:
1 Interpret a derivative as the slope of a tangent line to the graph of a function
(p. 285)
2 Interpret a derivative as an instantaneous rate of change (p. 286)
3 Interpret a derivative as velocity or acceleration (p. 287)
Suppose f is defined over some open interval containing the number c. Recall that the
average rate of change of f from c to any number x �= c in the interval is given by
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
�y f (x) − f (c) x �= c
�x = x − c
where �y is the change in y and �x is the change in x.
The derivative of f at c, f � (c), if it exists, equals the limit of the average rate of
NEED TO REVIEW? The derivative and
interpretations of the derivative are discussed change of f as x approaches c. That is,
in Section 2.1, pp. 168–173.
f (x) − f (c)
f (c) = lim x �= c
�
x→c x − c
An average rate of change measures the change in f from c to x; the derivative
measures change at c.
Stepped-out approach. 1 Interpret a Derivative as the Slope of a Tangent Line
The authors have AP EXAM TIP to the Graph of a Function
R
constructed a simple BREAK IT DOWN: As you work through the EXAMPLE 1 Interpreting the Derivative as the Slope of a Tangent Line
problems in this chapter, remember to follow
these steps:
Sullivan 06 apcalc4e 45342 ch04 284 331 July 13, 2023 9:44 (a) Find the derivative of f (x) = x cos x + x−1.
step-by-step framework 3pp
that you can use when Step 1 Identify the underlying structure and (b) Find the slope of the tangent line to the graph of f at the point (0, f (0)).
related concepts.
(c) Find an equation of the tangent line to the graph of f at the point (0, f (0)).
solving the AP ® Practice Step 2 Determine the appropriate math rule Solution
or procedure.
300 Problems . This AP ® Exam Step 3 Apply the math rule or procedure. (a) Use the Product Rule.
Chapter 4 • Applications of the Derivative, Part 1
Step 4 Clearly communicate your answer.
Tip also connects you On page 329, see how we’ve used these steps f (x) = x cos x + x−1
R
to solve Section 4.2 AP Practice Problem 9(a)Practice Problem 9(a)
to the BREAK IT DOWN on page 300. f (x) =− x sin x + cos x + 1(x) =− x sin x + cos x + 1
f
� �
PAGE feature at the end of the Free-Response Question
294 8. Two roads cross at right angles. A police officer sits in
chapter, where the authors
PAGE
a car 65 m east of the crossing and observes a car speeding 295 9. A roofer’s 13-meter ladder is placed against the wall of a
have applied these steps
northbound at 84 m/s. At what speed (in meters per second) building with its base on level ground. The top of the ladder
to one of the section-level
is the car distancing itself from the police officer 5 seconds slips down the wall as the bottom of the ladder slips away from
problems. the crossing?
after it passes the building at a constant rate of 5 m/s.
(a) At what rate is the top of the ladder moving when it is 5 m
(A) 166.024 m/s
from the ground?
(B) 83.012 m/s (b) At what rate is the area of the triangle formed by the ladder,
the wall, and the ground changing when the top of the
(C) 84 m/s
Sullivan 06 apcalc4e 45342 ch04 284 331 3pp July 13, 2023 9:44 ladder is 5 m from the ground?
(D) 95.859 m/s (c) If θ is the angle formed by the ladder and the ground, what
329
Review
Exer
cises
Chapter
Chapter 4 • Review Exercises 329 is the rate of change in θ when the top of the ladder is 5 m
4 •
Break It Down Preparing for the AP R R R Exam
Preparing for the AP Exam
from the ground?
Let’s take a closer look at AP Practice Problem 9 part (a) from Section 4.2 on page 300.
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9. A roofer’s 13-meter ladder is placed against the wall of a building with its base on level See the BREAK IT DOWN on page 329 for a stepped out solution to
ground. The top of the ladder slips down the wall as the bottom of the ladder slips away
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from the building at a constant rate of 5 m/s. AP Practice Problem 9(a).
(a) At what rate is the top of the ladder moving when it is 5 m from the ground?
There are rates changing with respect to time: the rate the top of the ladder slips
Step 1 Identify the underlying There are rates changing with respect to time: the rate the top of the ladder slips
structure and related
down the wall, and the rate the bottom of the ladder moves away from the wall. This
Retain Your Knowledge down the wall, and the rate the bottom of the ladder moves away from the wall. This
is a related rates question.
concepts.
Step 2 Determine the appropriate When possible, the first step in solving a related rates problem should be drawing aa
When
should
in
solving
a
related
drawing
problem
rates
possible,
first
be
the
step
picture (see the figure in Step 3). The ladder is 13 m long. Next, identify the
math rule or procedure. picture (see the figure in Step 3). The ladder is 13 m long. Next, identify the
h Questions
Multiple-Choice variables needed for the given situation. Let h denote the height of the top of the
ladder from the ground and let x denote the distance the bottom of the ladder is from
dh
the wall. As the ladder slides down the wall, equals the rate at which the height
d x + 1 dx dt
1. f Wall = is changing and dt equals the rate at which the distance from the wall is changing. Methodical problem solving.
3. Find an equation of the line tangent to the graph of
dx 2x 5 − 1 h Ladder We are told that dx = 5 m/s and are asked to find the rate of change of the
13 m
1
dt In the BREAK IT DOWN feature at the end
dh −1
of each chapter, you can follow a detailed
− 3 x height dt when h = 5 m. function f (x) = 3 sin (2x) at x = .
(A) Ground x 4
(− 2x − 1) 2
Step 3 Apply the math rule Together, the ladder, the height h, and the base x form a right triangle. solution to one of the section-level AP ®

1
1
12
π
or procedure. Using the Pythagorean Theorem, we have h 2 + x 2 = 13 2 . π 6 Practice Problems using the steps outlined
x + 1 1 When h = 5 m, we find x 2 = 169 − 25 = 144, so x = 12 m. (A) y − = √ x − (B) y − = √ x −
4
2
4
3
3
(B) f � · The distance x and the height h are each changing over time t, so we will take 2 at in the chapter opening AP ® Exam Tip .
2x − 1 (2x − 1) 2 the derivative of the above expression with respect to t to obtain an equation
involving the derivatives of h and x. 2 1 4 1
d d π π
(h 2 + x 2 ) = (13 2 )
x + 1 3 dt dt (C) y − 6 = √ x − 4 (D) y − 2 = √ x − 4
(C) − f � · dh dx = 0 3 3
2x − 1 (2x − 1) 2 2h dt + 2x dt
When h = 5 m, x = 12 m and dx/dt = 5 m/s. After dividing out the 2’s, we have
dh Free-Response Question
x + 1 3 5 dt + (12)(5) = 0
(D) f � · dh
2x − 1 (2x − 1) 2 dt =−12 4x + 1 if x < 1
Step 4 Clearly communicate your (a) When x = 5 m, the height of the ladder is decreasing at a rate of −12 m/s. 4. f (x) = 5 if x = 1
answer.
2
2. The functions f and g are both differentiable in an open 2x − 8x + 11 if x > 1
interval containing 6. The table below shows the values of (a) Determine whether f is continuous at x = 1.
x
vi
xxvi g, f , and g for x = 6. © 2024 BFW Publishers PAGES NOT FINAL
�
�
x f,
(b) Find f (2).
�
For Review Purposes Only, all other uses prohibited
(c) Determine whether f (1) exists. Justify the answer.
Do Not Copy or Post in Any Form. �
�
f (6) f (6) g(6) g (6)
�
− 8 0 3 2
01_apcalc4e_45342_fm_i_xxix_3pp.indd 26 10/11/23 2:39 PM
Use the table to find the derivative of the

3 3 f (x)
function h(x) = for x = 6.
g(x)
4 10 4
(A) − (B) (C) (D) 4
3 9 3