Page 75 - 2024-calc4e-SE proofs-4e.indd

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

Section 2.5 • The Derivative of the Trigonometric Functions 225

′
73. If y = sin x and y (n) is the nth derivative of y with respect to x, 79. Let f (x) = cos x. Show that finding f (0) is the same as
find the smallest positive integer n for which y (n) = y. cos x − 1
finding lim .
A + B A − B x→0 x
74. Use the identity sin A − sin B = 2 cos sin , ′
2 2 80. Let f (x) = sin x. Show that finding f (0) is the same as
with A = x + h and B = x, to prove that sin x
finding lim .
x→0 x
d sin(x + h) − sin x
sin x = lim = cos x 81. If y = A sin t + B cos t, where A and B are constants,
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dx h→0 h
show that y + y = 0.
′′
d
75. Use the definition of a derivative to prove cos x = −sin x.
dx Challenge Problem
76. Derivative of y = sec x Use a derivative rule to show that 82. For a differentiable function f , let f be the function defined by
∗
d ∗ f (x + h) − f (x − h)
sec x = sec x tan x f (x) = lim
dx h→0 h
77. Derivative of y = csc x Use a derivative rule to show that 2
(a) Find f (x) for f (x) = x + x.
∗
∗
(b) Find f (x) for f (x) = cos x.
d
csc x = −csc x cot x (c) Write an equation that expresses the relationship between the
dx
∗
′
functions f and f , where f denotes the derivative of f .
′
78. Derivative of y = cot x Use a derivative rule to show that Justify your answer.
d 2
cot x = −csc x
dx
R
Preparing for the AP Exam
R
AP Practice Problems
Multiple-Choice Questions
dy d 50
PAGE PAGE
219 1. If y = x sin x, then = 221 5. If y = sin x, then 50 sin x equals
dx dx
(A) x cos x + sin x (B) x cos x − sin x (A) sin x (B) − sin x (C) cos x (D) − cos x
π
(C) cos x + sin x (D) (x + 1) cos x PAGE x
220 6. If f (x) = , find f ′ .
π π
cos x 3
cos + h − cos √ √
PAGE 3 3 2 3 3
220 2. What is lim ? (A) 2 − π (B) 1 + π
h→0 h 3 3
√ √ √ √
1 3 3 3 2 3
(A) 0 (B) (C) (D) − (C) 1 − π (D) 2 + π
2 2 2 3 3
PAGE dy
7. If y = x − tan x, then equals
PAGE π 221
221 3. If f (x) = tan x, then f ′ equals dx
3
2
(A) 1 − sec x tan x (B) − tan x
√ 1
(A) 2 3 (B) 4 (C) 2 (D) (C) tan x (D) − sec x
2
2
4
PAGE x ′
PAGE 220 8. If g(x) = e cos x + 2π, then g (x) =
221 4. The position x (in meters) of an object moving along a
x
x
x
π (A) e − sin x (B) e cos x − e sin x + 3π
horizontal line at time t, 0 ≤ t ≤ , (in seconds) is given
x
x
x
x
2 (C) e cos x − e sin x (D) e cos x + e sin x
3
2
by x(t) = 6 sin t + t + 8. What is the velocity of PAGE
2 220 9. At which of the following numbers x, 0 ≤ x ≤ 2π, does the
the object when its acceleration is zero? graph of y = x + cos x have a horizontal tangent line?
π
(A) 6 m/s (B) 3 + π m/s (A) 0 only (B) only
√ 2
6 3 + π √ π 3π π
(C) m/s (D) 3 3 − m/s (C) only (D) 0 and only
2 2 2 2
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