Section 5.4 Extra Practice
STUDENT BOOK PAGES 250–257
dy
1. Determine dx for each of the following.
a. y ⫽ sin 4x
b. y ⫽ 5 sin (⫺x)
c. y ⫽ cos 2x
d. y ⫽ ⫺cos 9x
e. y ⫽ cos (x 5 ⫺ x 3 ⫺ 1)
f. y ⫽ sin 4x ⫺ cos 3x
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2. Determine f ⬘(x) for each of the following functions.
a. f (x) ⫽ sin (e x )
b. f (x) ⫽ cos (4x )
c. f (x) ⫽ 4 sin (5x ⫺ 19)
1
d. f (x) ⫽ cos
x
e. f (x) ⫽ sin (4x ⫺ 2) ⫹ cos ␲
3. Determine f ⬘(x) for each of the following functions.
sin 2x
a. f (x) ⫽
x
b. f (x) ⫽ sin (cos 2x)
c. f (x) ⫽ ⫺3x sin x
sin 2x
d. f (x) ⫽
cos 2x
4. For each of the following functions, determine the
slope of the tangent to the curve at the point with the
given x-coordinate.
␲
a. f (x) ⫽ cos x, x ⫽
6
7␲
b. f (x) ⫽ x ⫹ cos x, x ⫽
6
␲
c. f (x) ⫽ sin 5x, x ⫽
2
␲
d. f (x) ⫽ cos 3x ⫹ sin x, x ⫽
4
1
␲
2
e. f (x) ⫽ sin x, x ⫽
3
4
2␲
f. f (x) ⫽ ⫺sin x cos x, x ⫽
3
5. For each of the following functions, determine an
equation for the tangent to the curve at the point
with the given x-coordinate.
␲
a. f (x) ⫽ ⫺sin x, x ⫽
4
b. f (x) ⫽ 1 ⫹ cos (⫺3x), x ⫽ 2␲
␲
␲
c. f (x) ⫽ sin ax ⫹ b , x ⫽
3
2
␲
d. f (x) ⫽ ⫺2cos ax ⫺ b , x ⫽ ␲
6
6. a. In the interval 0 ⱕ x ⱕ 2␲, find the x-coordinates
of the points where the tangent to the graph of
f (x) ⫽ cos x ⫺ sin x is horizontal.
b. Graph the function f (x) on the same interval.
7. A particle moves along a line so that, at time t
measured in seconds, its position in metres is given by
the function s(t) ⫽ 4 cos 2t.
a. Differentiate the function with respect to t.
b. When is the first time that the particle changes
direction?
c. When does the particle reach maximum velocity
for the interval 0 ⬍ t ⬍ 2␲?
d. What is the maximum velocity of the particle?
8. Differentiate each function.
a. f (x) ⫽ cos (e x )
b. f (x) ⫽ sin (32x )
c. f (x) ⫽ cos (sin (e x ))
Section 5.4 Extra Practice
395