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Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25
190 Chapter 2 • The Derivative and Its Properties
85. Tangent Lines and Derivatives Let f and g be two functions, 87. A function f is defined for all real numbers and has the following
each with derivatives at c. State
AP® Exam prep every step of the way. the relationship between their three properties:
tangent lines at c if: f (a + b) − f (a) = kab + 2b 2
1 f (1) = 5 f (3) = 21
(a) f � (c) = g � (c) (b) f � (c) =− g � (c) �= 0
g � (c) for all real numbers a and b where k is a fixed real number
independent of a and b.
Challenge Problems
(a) Use a = 1 and b = 2 to find k.
86. Let f be a function defined for all real numbers x. Suppose f has
the following properties: (b) Find f � (3).
(c) Find f � (x) for all real x.
f (u + v) = f (u) f (v) f (0) = 1 f � (0) exists
88. A function f is periodic if there is a positive number p so
that f (x + p) = f (x) for all x. Suppose f is differentiable. Show
(a) Show that f � (x) exists for all real numbers x. that if f is periodic with period p, then f � is also periodic with
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
(b) Show that f � (x) = f � (0) f (x). period p.
Sullivan 06 apcalc4e 45342 ch04 284 331 3pp July 13, 2023 9:44 Preparing for the AP Exam
R
AP Practice Problems
R
Immediate reinforcement of skills. 312 Chapter 4 • Applications of the Derivative, if x ≤ 1 R
Multiple-Choice Questions Part 1
�
x 2 − ax
Preparing for the AP Exam
PAGE
Each section covering content that may appear on AP Practice Problems ax + b if x > 1 , where a and b 183 5. If f (x) =|x|, which of the following statements
PAGE
183
1. The function f (x) =
about f are true?
R
are constants. If f is differentiable at x = 1, then a + b =
the exam includes a comprehensive selection of AP ® PAGE Multiple-Choice Questions (B) −2 (C) 0 (D) 2 PAGE II. f is differentiable at 0.
I. f is continuous at 0.
(A) −3
III. f (0) = 0.= x 3 + 1 at x = 1 is used
304
Practice Problems . Practicing all year will pay off when 1. Let f be a function for which f (2) = 6 and f � (2) =−3. 304 6. The line tangent to the graph of f (x) (B) III only
PAGE
2. The graph of the function f , given below, consists of three line
If the line tangent
to approximate f (x) near 1. Which number below is the
180 to the graph of f at 2 is used to
(A) I only
approximate a zero of f , then the approximation is
segments. Find lim
you are more confident taking the exam. (A) 0 (B) 4 (C) 6 (D) 12 h→0 f (3 + h) − f (3) . greatest value of x that results in an error less than or equal to
(C) I and III only
(D) I, II, and III
h
0.5?
PAGE
6. The graph of the function f shown in the figure has horizontal
(B)
PAGE 3 √ y (A) 1.30 186 1.35 (C) 1.40 (D) 1.45
304 2. For small, positive values of h, 8 + h is best 6 tangent lines at the points (0, 1) and (2, −1) and a vertical
approximated by (0, 4) 304 PAGE 7. A linear approximation L is used to approximate f (x) = √ x,
tangent line at the point (1, 0). For what numbers x in the open
4
interval (−2, 3) is f not differentiable?
h h h h at c, c > 0. The approximation
(A) 4 − (B) 4 + (C) 2 − (22, 2) (D) 2 + (A) always underestimates the true value of f at c.
12 12 12 2 12
πx (6, 0) (B) always overestimates the true value of f at y c.
Continual practice. 304 3. A linear approximation to f (x) = x sin 24 22 2 + x 2 2 4 6 x (C) sometimes overestimates the true value of 4 f at c.
PAGE
at x = 3 is
Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25 (D) does not provide enough information to determine whether
the true value of f at c is over- or underestimated.
At the end of each section, you are (A) y = 5x + 6 (A) −1 (B) − 2 9 3 9 (C) − 3 2 (D) The limit does not exist. 2
(B) y = 5x −
PAGE
8. Suppose y = f (x) is a differentiable function. The table below
(C) y = 7x − 9
304
(D) y = 7x +
x
continually prompted to reinforce prior 304 PAGE 4. Using the line tangent to the graph of f (x) = if xe x + 2 at 0, gives values of f and f � for select numbers Chapter 2 • Review Exercises 231
22 x in the domain
2
⎧
⎪ ⎨ x 2 − 25
22 f (3.1).
of f . Use a linear approximation to approximate
x �= 5
x − 5
PAGE
186value of f ( − 0.3) is
Break
knowledge through the Retain Your the approximate 3. If f (x) = ⎪ ⎩ It Down x = 5 x − 3 0 1 Preparing for the AP Exam
R
5
if
3
5
(A) 2.3
(D) − 2.3
(C) 1.7
(B) 1.3
which of the following statements about f are true?
4
3
(A) −1 only − 1
Knowledge AP ® Practice Problems , 304 PAGE 5. If f � (x) = 2xe x 2 −1 − 3π sin(πx) and f (1) = 4, at AP Practice Problem 12 from Section 2.2 on page 191. − 1 4 − 2 − 2 (B) −1 and 1 only
f (x)
1
R
3
Let’s take a closer look
4
(D) −1, 0, 1, and 2
(C) −1, 0, and 2 only
f � (x)
I. lim f exists.
approximate f (1.03) using a
x→5 linear approximation.
Oil is leaking from a tank. The amount of oil, in gallons,
12.
which present three multiple-choice (A) 4.06 (B) 5.06 III. f is differentiable at x = 5. (A) 4.1 (B) PAGE in the tank is given by G(t) = 4000 − 3t 2 , f (1 + h) − f (1) =−3.
II. f is continuous at x = 5.
7. Let f be a function for which lim
(C) 4
where t,0
(D) 3.94≤ t ≤ 24 is the number of hours past midnight.
186 − 2.3
(D) − 1.7
(C) 0.1
h
h→0
f f
Which of the following must be true?
questions and one free-response (A) I only (a) Find G � (5) using the definition of the derivative. Which of the following must be true?
(B) I and II only
I. f is continuous at 1.
(b) Using appropriate units, interpret the meaning of G � (5) in the context of the problem.
(C) I and III only
(D) I, II, and III
II. f is differentiable at 1.
question drawing from concepts covered Retain Your Knowledge Step 1 Identify the underlying structure and the The problem is asking for G’(5), a derivative.
III. f � is continuous at 1.
PAGE
4. Suppose f is a function that is differentiable on the open
186 Questions
Multiple-Choice
(A) I only
interval (−2, 8). If f (0) = 3, f (2) =−3, and f (7) = 3,
related concepts.
in prior sections and chapters. 1. The function f (x) = e x + x has an inverse function g. 3. If f (x) = x 4 ln(2x 2 + 3), then f � (1) (B) II only
which of the following must be true?
(C) I and II only equals
(D) I, II, and III
Determine the appropriate math rule or
Step 2
Because we want the derivative of G at a number, we use
Then, g � (1) = I. f has at least 2 zeros. 4 PAGE Form (1) of the Definition of a Derivative (see p. 179).
procedure.
8. At what point on the graph of f (x) = x 2 − 4 is the tangent line4 is the tangent line4 is the tangent line
1 II. f is continuous on the closed interval [−1, 7].(A) ln 5 + 4 180 ln 5 (B) 4 + 4 ln 5
1
1
parallel to the line 6x − 3y = 2?
Step 3
(D) Apply the math rule
(C)
(A) (B) 2 III. For some c,0 < c < 7, f (c) =−2. or procedure. Using Form (1),
2 e + 1 e
(A) I only (B) I and II only (C) 4 + 4 ln 5 G � (5) (D) = lim 1 G(t) (B) (1, 2) (4000 − 3t 2 ) − (4000 − 3 · 5 2 )
(C) (2, 0)
(A) (1, −3)− G(5)
(D) (2, 4) 4),
= lim
+ 4ln5
t − 5
t − 5
t→5
Become a savvy test-taker. 2. Find y � for e x sin y − x 2 cos y = x. (D) I, II, and III Free-Response Question = lim 5 − 3(t 2 − 25) = lim − 3(t − 5)(t + 5)
5
t→5
(C) II and III only
1 + 2x cos y − e x sin y
t − 5
t − 5
t→5
t→5
Every chapter ends with AP ® Review (A) y � = x 2 sin y − e x cos y 4. The resistance R, in ohms, of a wire of radius x cm is given
= lim[− 3(t + 5)] =− 30
t→5
Problems that include multiple-choice 1 − 2 cos y − e x sin y by the formula R(x) = 0.0048 . The radius x is given
x 2
Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25 (B) y � = x 2 sin y + e x cos y Step 4 Clearly communicate your answer. (a) G � (5) =− 30
by x = 0.991 + 10 −5 T , where T is the temperature in Kelvin.
(b) A derivative is a rate of change. In this problem, G � (t) equals
problems for exam prep all year as well (C) y � = 1 + 2x cos y − e x sin y Determine how R is changing with respect to T
the rate of change of G with respect t, that is, the rate of change
232
x 2 sin y + e x cos y
Chapter 2 • The Derivative and Its Properties
as short-answer open-ended questions 1 + 2x cos y − e x sin y when T = 320 K. of the amount of oil, in gallons, with respect to the time in hours.
Because G � (5) =− 30, we say the amount of oil in the tank is
(D) y � = decreasing at the rate of 30 gallons per hour when t = 5 hours past
x 2 sin y + e x cos y midnight, or at 5AM.
to build your skills at answering free- 10. Find an equation of the line tangent to the graph
6. If y = ln x + xe x + 6, what is the instantaneous rate of
change of y with respect to x at x = 5? x + 3
of f (x) = at x = 1.
response questions. 1 + 5e x 2 + 2
(A) 5 + 6e 5
(B)
R
R
5 (A) 5x + 9y = 17 (B) 9y − 5x = 7 AP Review Problems: Chapter 2 Preparing for the AP Exam
1
(C) 5 + 5e 5 (D) 6e 5 + (C) 5x + 3y = 9 (D) 5x + 9y = 7
5
tan x − 1 Multiple-Choice Questions π
More test prep, more comprehensive practice. 1. If f (x) = sec x, then f � 4 = √ 4. The graph of the function f is shown below.
7. An equation of the line tangent to the graph of
=
x − π
11. lim π
4
x→ 4
f (x) = 3xe x + 5 at x = 0 is
Which statement about the function is true?
√
(B) −1
(A) 0
2
1
Starting with Chapter 2 , every chapter ends with an (A) 2 (B) 2 (C) 1 (D) 2 y 2 y 5 f (x)
(B) y =− x + 5
(A) y = 3x + 5
(C) 2
3
(D) Does not exist.
1
x + 5
(D) y =−3x + 5
AP ® Cumulative Review Practice Problem set. These 2. If a function f is differentiable at c, then f � (c) is given by
(C) y =
Free-Response Questions
3
12. An object moves on a line according to I. lim f (x) − f (c) 22 2 x
8. An object moves along a horizontal line so that its position the position function s = 2t 3 − 15t 2 + 24t + 3, x→c x − c
at time t is x(t) = t 4 − 6t 3 − 2t − 1. At what time t is the
sets reinforce concepts across the course. They prepare meters. II. lim f (x + h) − f (x) (A) f is differentiable everywhere.
where t is measured in minutes and s in
acceleration of the object zero?
h
you for the AP ® exam by keeping the most important x→c f (c + h) − f (c) (B) 0 ≤ f � (x) ≤ 1, for all real numbers.
(a) When is the velocity of the object 0?
(B) at 1 only
(A) at 0 only
h→0
(C) at 3 only (D) at 0 and 3 only (b) Find the object’s acceleration when t = 3. III. lim h (C) f is continuous everywhere.
(D) f is an even function.
13. Find the value
concepts and techniques fresh in your mind. of the limit below and specify (A) I only (B) III only
9. If f (x) = e x (sin x + cos x), then f � (x) = the function f for which this is the derivative. (C) I and II only (D) I and III only 5. The table displays select values of a differentiable
(A) 2e x (cos x + sin x) (B) e x cos x [4 − 2(x + h)] 2 − (4 − 2x) 2 function f . What is an approximate value of f � (2)?
lim 3 , then dy
(C) 2e x cos x (D) e x (cos 2 x − sin 2 x) h→0 h 3. If y = x 2 − 5 dx = x 1.996 1.998 2.002 2.004
6x 6x f (x) 3.168 3.181 3.207 3.220
(A) (x 2 − 5) 2 (B) − (x 2 − 5) 2
AP Cumulative Review Problems: Chapters 1–2 Preparing for the AP® Exam (C) 6x (D) 2x (A) 6.5 (B) 0.154 (C) 0.013 (D) 1.5
R
x 2 − 5 (x 2 − 5) 2
Multiple-Choice Questions
x − 4
1. lim = 5. Suppose the function f is continuous at all real
x→4 4 − x
numbers and f (−2) = 1 and f (5) =−3. Suppose the
(A) −4 (B) −1 (C) 0 (D) The limit does not exist. function g is also continuous at all real numbers
3x + sin x and g(x) = f −1 (x) for all x. The Intermediate Value Need more help?
2. lim = Theorem guarantees that
x→0 2x
(A) g(c) = 2 for at least one c between −3 and 1.
(A) 0 (B) 1 (C) 2 (D) The limit does not exist. Every AP ® Review Problem is accompanied
(B) g(c) = 0 for at least one c between −2 and 5.
3. Let h be defined by (C) f (c) = 0 for at least one c between −3 and 1.
(D) f (c) = 2 for at least one c between −2 and 5. by a short video clip in that
f (x) · g(x) if x ≤ 1
h(x) =
k + x if x > 1
where f and g are both continuous at all real numbers. 6. The line x = c is a vertical asymptote to the graph of guides you through the solution. Let an
If lim f (x) = 2 and lim g(x) =−2, then for what the function f . Which of the following statements
x→1 x→1 cannot be true? experienced AP ® Calculus teacher help you
number k is h continuous?
(A) lim f (x) =∞ (B) lim f (x) = c
x→c
(A) −5 (B) −4 (C) −2 (D) 2 (C) f (c) is not defined. (D) f is continuous at x = c. when you need it most.
x→∞
4. Which function has the horizontal asymptotes y = 1 7. The position function of an object moving along a
and y =−1?
1 1
2 straight line is s(t) = 15 t 3 − t 2 + 5t −1 . What is the
2
(A) f (x) = tan −1 x (B) f (x) = e −x + 1
π object’s acceleration at t = 5?
1 − x 2 2x 2 − 1
(C) f (x) = (D) f (x) = (A) − 27 (B) − 1 (C) 1 (D) 27
1 + x 2 2x 2 + x 25 5 5 25
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For Review Purposes Only, all other uses prohibited
Do Not Copy or Post in Any Form.
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