f (x)

Sine & Cosine Differentiation Exercises-1
Sa
Sine & Cosine Differentiation — Exercise Set I.
sk
DEO
PAT-
ET
RIÆ
atc h
sis
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Un
e w ane
n
2002 Doug MacLean
Find the derivative f (x) for the following functions, and express the subsets of [0, 2π ] on which f (x) > 0
using Interval Notation. Try to produce a rough sketch of the graph of f .
(I.1)
(I.2)
(I.3)
(I.4)
(I.5)
(I.6)
(I.7)
(I.8)
(I.9)
(I.10)
f (x) = 2 sin x cos x
f (x) =
sin 4x
2 + cos 4x
f (x) = sin3 x cos2 x
f (x) = sin x + sin x cos x
Solution
Solution
Solution
Solution
f (x) = 2 cos x + cos 2x
Solution
f (x) = 2 sin x + sin 2x
Solution
f (x) = sin2 x + cos x
Solution
f (x) = sin x + cos2 x
Solution
f (x) = sin 2x cos 3x
Solution
f (x) = sin 2x sin 3x
Solution
Sine & Cosine Differentiation Exercises-2
Sa
Sine & Cosine Differentiation — Exercise Set II.
sk
DEO
PAT-
ET
RIÆ
atc h
sis
iversitas
Un
e w ane
n
2002 Doug MacLean
Find the derivative f (x) for the following functions, and express the subsets of [0, 2π ] on which f (x) > 0
using Interval Notation. Try to produce a rough sketch of the graph of f .
(II.1)
(II.2)
(II.3)
(II.4)
(II.5)
(II.6)
(II.7)
(II.8)
(II.9)
(II.10)
f (x) = 3 sin(5x)
f (x) = 7 cos(3x − 2)
f (x) = sin
√ x
Solution
Solution
Solution
f (x) = cos 2(2x − 1)2
Solution
f (x) = cos3 x − sin3 x
Solution
f (x) = x 3 cos(2x) − x 2 sin(3x)
f (x) = sin2 x cos x
1
f (x) = x sin
x
1
f (x) = x 2 sin
x
1
3
f (x) = x sin
x
Solution
Solution
Solution
Solution
Solution
Sine & Cosine Differentiation Exercises-3
Sa
Sine & Cosine Differentiation — Exercise Set III.
sk
DEO
PAT-
ET
RIÆ
atc h
sis
iversitas
Un
e w ane
n
2002 Doug MacLean
Find the derivative f (x) for the following functions, and express the subsets of [0, 2π ] on which f (x) > 0
using Interval Notation.
(III.1)
(III.2)
(III.3)
(III.4)
(III.5)
(III.6)
(III.7)
(III.8)
(III.9)
(III.10)
f (x) = x sin x
Solution
f (x) = sin2 x
Solution
f (x) = sin3 x
Solution
f (x) = sin4 x
Solution
f (x) = sin5 x
Solution
f (x) =
1 − cos x
1 + cos x
Solution
f (x) =
1 + cos x
1 − cos x
Solution
f (x) =
1 − sin x
1 + sin x
Solution
f (x) =
1 + sin x
1 − sin x
Solution
f (x) =
sin x
1 − sin x
Solution
Sine & Cosine Differentiation Exercises-4
Sa
Solution Set I
sk
f (x) = 2 sin x cos x
Solution:
f (x) = 2(sin x) (cos x) + 2(sin x)(cos x) = 2(cos x)(cos x) + 2(sin x)(− sin x) =
3π 5π
7π
π
2
2
∪
,
∪
, 2π . The graph of y = f (x) is
2(cos x − sin x) = 2 cos 2x > 0 on the set 0,
4
4
4
4
shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
0
-1
-2
PAT-
ET
RIÆ
atc h
e w ane
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2002 Doug MacLean
Back to Questions
(I.1)
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-5
Solution:
Sa
(I.2)
sin 4x
f (x) =
2 + cos 4x
sk
Back to Questions
(sin 4x) (2 + cos 4x) − (sin 4x)(2 + cos 4x)
cos 4x)(4x) (2 + cos 4x) − (sin 4x)(− sin 4x)(4x)
=
=
(2 + cos 4x)2
(2 + cos 4x)2
cos 4x(4)(2 + cos 4x) − (sin 4x)(− sin 4x)(4)
2 cos 4x + cos2 4x + sin2 4x
2 cos 4x + 1
=
4
=
4
> 0 if
(2 + cos 4x)2
(2 + cos 4x)2
(2 + cos 4x)2
π 2π
5π 7π
4π 5π
11π
1
π
∪
,
∪
,
∪
,
∪
, 2π . The graph
cos 4x > − , so f (x) > 0 on the set 0,
2
6
3 3
6
6
3
3
6
of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
PAT-
ET
RIÆ
atc h
e w ane
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2002 Doug MacLean
f (x) =
0
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-6
Sa
(I.3)
sk
f (x) = sin3 x cos2 x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin3 x) cos2 x + sin3 x(cos2 x) = 3 sin2 x(sin x) cos2 x + sin3 x(2(cos x)(cos x) ) =
3 sin2 x(cos x) cos2 x + sin3 x(2(cos x)(− sin x)) = 3 sin2 x cos3 x − 2 sin4 x cos x =
sin2 x cos x(3 cos2 x − 2 sin2 x) = sin2 x cos x(5 cos2 x − 2).
0
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-7
Sa
(I.4)
sk
f (x) = sin x + sin x cos x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
-1
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin x) + (sin x) cos x + sin x(cos x) = cos x + cos x
cos x + sin x(− sin x) = cos x + cos2 x − sin2 x =
√
2 − 4(2)(−1)
−1
±
−1 ± 3
1
1
9
−1
±
cos x + 2 cos2 x − 1 = 2 cos2 x + cos x − 1 = 0 if cos x =
=
=
= −1, ,
2(2)
4
4
2
π
5π
i.e. x = π or x = 3 or x = 3 .
5π
π
∪
, 2π .
The set on which f (x) > 0 is 0,
3
3
0
DEO
sis
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-8
Sa
(I.5)
sk
f (x) = 2 cos x + cos 2x
Solution:
Back to Questions
5π
π
f (x) => 0 on the set 0,
∪
, 2π . The graph of y = f (x) is shown in red , and the graph of
3
3
y = f (x) is shown in blue .
y
2
1
0
-1
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-9
Sa
(I.6)
sk
f (x) = 2 sin x + sin 2x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
4
3
2
1
-1
-2
-3
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = 2 cos x + cos 2x(2x) = 2 cos x + cos
2x = 2 cos x + 2(2 cos2 x − 1) =
2x(2) = 2 cos x + 2 cos
√
−1 ± 9
−1 ± 3
1
−1 ± 12 − 4(2)(−1)
2(2 cos2 x + cos x − 1) = 0 if cos x =
=
=
= −1, .
2(2)
4
4
2
5π
π
∪
, 2π .
f (x) => 0 on the set 0,
3
3
0
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-10
Sa
(I.7)
sk
f (x) = sin2 x + cos x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
-1
-2
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin2 ) x + (cos x) = 2 sin x(sin x) − sin x = 2 sin x cos x − sin x = sin x(2 cos x − 1) = 0 if sin x = 0
5π
π
1
π
5π
or cos x = 2 , i.e., if x = 0, π , 2π , x = 3 or x = 3 so f (x) > 0 on the set 0,
∪
, 2π .
3
3
0
DEO
sis
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-11
Sa
(I.8)
sk
f (x) = sin x + cos2 x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
-1
-2
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin x) + (cos2 x) = cos x + 2(cos x)(cos x) = cos x + 2(cos x)(− sin x) = cos x(1 − 2 sin x) = 0 if
1
π
3π
π
5π
cos x = 0 or sin x = , i.e., if x = , x =
, x = , or x =
so f (x) > 0 on the set
2
2
2
6
6
5π 7π
3π
π
∪
,
∪
, 2π . .
0,
6
6
6
2
0
DEO
sis
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-12
Sa
(I.9)
sk
f (x) = sin 2x cos 3x
Solution:
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
3
2
1
-1
-2
-3
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin 2x) cos 3x + sin 2x(cos 3x) = cos 2x(2x) cos 3x + sin 2x(− sin 3x)(3x) =
cos 2x(2) cos 3x + sin 2x(− sin 3x)(3) = 2 cos 2x cos 3x − 3 sin 2x sin 3x.
0
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-13
Solution:
Sa
(I.10)
sk
f (x) = sin 2x sin 3x
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
3
2
1
-1
-2
-3
π/2
PAT-
ET
RIÆ
atc h
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n
2002 Doug MacLean
f (x) = (sin 2x) sin 3x + sin 2x(sin 3x) = (cos 2x)(2x) sin 3x + sin 2x(cos 3x)(3x) =
cos 2x(2) sin 3x + sin 2x(cos 3x)(3) = 2 cos 2x sin 3x + 3 sin 2x(cos 3x).
0
DEO
sis
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-14
Sa
Solution Set II
sk
DEO
PAT-
ET
RIÆ
atc h
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Un
e w ane
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2002 Doug MacLean
(II.1)
f (x) = 3 sin(5x)
Solution:
Back to Questions
f (x) = 3 cos(5x)(5x) = 15 cos(5x) > 0. Now cos x > 0 if 0 < x <
π
2
or
3π
< x < 2π , and cos 5x , depicted
2
2π
in red below, has period
, so f (x) > 0 on the set.
5
3π 5π
7π 9π
11π 5π
15π 17π
19π
π
∪
,
∪
,
∪
,
∪
,
∪
, 2π . The graph of y = f (x) is
0,
10
10 10
10 10
10 13
10
10
10
shown in blue .
y
3
2
1
0
-1
-2
-3
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-15
Sa
(II.2)
sk
f (x) = 7 cos(3x − 2)
Solution:
Back to Questions
1 21 f (x)
is shown in red , and the graph of y = f (x) is shown in blue .
y
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = 7(− sin(3x − 2))(3x − 2) = −21 sin(3x − 2) > 0 − sin(3x − 2) , depicted in red below, has
2π
2 4π
2 π 2 2π
2
2 5π
2
period
, so f (x) > 0 on the set. 0,
∪
+ , +
∪
+ π, +
∪
+
, 2π . The
3
3
3
3 3
3
3
3
3
3
3
graph of y =
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-16
Solution:
Sa
(II.3)
√ f (x) = sin x
sk
Back to Questions
√
√ √ π
cos ( x)
√
> 0 on the set 0,
.
x =
f (x) = cos x
2 x
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
0
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-17
Sa
(II.4)
sk
f (x) = cos 2(2x − 1)2
Solution:
Back to Questions
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = − sin 2(2x − 1)2 2(2x − 1)2 = −8(2x − 1) sin 2(2x − 1)2 . It is not easy to explicitly describe the set
1
on which f (x) > 0, but we can find the roots of f (x): f (x) = 0 if x = , or if 2(2x − 1)2 = kπ for any
2
π
1
positive integer k, or, equivalently, x = + k . The graph of y = f (x) is shown in red , and the graph of
2
8
y = f (x) is shown in blue . The figure below shows the complications we will encounter with this function:
0
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-18
Sa
(II.5)
sk
f (x) = cos3 x − sin3 x
Solution:
Back to Questions
y
2
1
-1
-2
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = 3(cos x)2 (− sin x) − 3(sin x)2 (cos x) = −3 sin x cos2 x − 3 sin2 x cos x =
3
3π
7π
π 3π
−3 sin x cos x(cos x + sin x) = − sin 2x(cos x + sin x) > 0 on the set
,
∪ π,
∪
2π . The
2
2 4
4
4
graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
0
DEO
sis
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-19
Sa
(II.6)
sk
f (x) = x 3 cos(2x) − x 2 sin(3x)
Solution:
Back to Questions
y
250
200
150
100
50
-50
-100
π/2
PAT-
ET
RIÆ
atc h
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n
2002 Doug MacLean
f (x) = (x 3 ) cos(2x) + x 3 (cos(2x)) − [(x 2 ) sin(3x) + x 2 (sin(3x)) ] =
3x 2 cos(2x) + x 3 (− sin(2x)(2x) ) − [2x sin(3x) + x 2 (cos(3x)(3x) )] =
3x 2 cos(2x) − x 3 sin(2x)(2) − [2x sin(3x) + x 2 cos(3x)(3)] =
3x 2 cos(2x) − 2x 3 sin(2x) − 2x sin(3x) − 3x 2 cos(3x). The graph of y = f (x) is shown in blue . It is not
easy to find the roots of either f (x) or f (x).
0
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-20
Sa
(II.7)
sk
f (x) = sin2 x cos x
Solution:
Back to Questions
sin x(3 cos2 x − 1) = 0 and sin x = 0 if x = 0, π , 2π and 3 cos2 x − 1 = 0 is cos x = ± 13 on the set .
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = (sin2 x) cos x + sin2 x(cos x) =
(2 sin x cos x) cos x + sin2 x(− sin x) =
2 sin x cos2 x − sin3 x =
sin x(2 cos2 x − sin2 x) =
0
DEO
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Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-21
(II.8)
1
f (x) = x sin
x
Solution:
Sa
sk
Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
0
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
1
1
+ x sin
=
f (x) = (x) sin
x
x 1
1
1
+ x cos
=
sin
x x x 1
1
−1
sin
+ x cos
=
2
x x x
1
1
1
sin
− cos
.
x
x
x
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-22
1
f (x) = x 2 sin
x
Solution:
Sa
(II.9)
sk
2 Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
5
4
3
2
1
0
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
1
1
2
+ x sin
=
f (x) = (x ) sin
x
x 1 1
1
+ x 2 cos
=
2x sin
x x x 1
−1
1
2x sin
+ x 2 cos
=
x2
x x
1
1
2x sin
− cos
.
x
x
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-23
Solution:
1
f (x) = x 3 sin
x
Sa
(II.10)
sk
3 Back to Questions
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
2
1
0
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
1
1
3
+ x sin
=
f (x) = (x ) sin
x x
1 1
1
+ x 3 cos
=
3x 2 sin
x x x −1
1
1
3x 2 sin
+ x 3 cos
=
x2
x
x 1
1
3x 2 sin
− x cos
.
x
x
DEO
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Un
π/2
x
Sine & Cosine Differentiation Exercises-24
Sa
Solution Set III
sk
DEO
PAT-
ET
RIÆ
atc h
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Un
e w ane
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2002 Doug MacLean
(III.1)
Solution:
f (x) = x sin x
Back to Questions
f (x) = (x) sin x + x(sin x) = sin x + x cos x
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-25
Solution:
Sa
(III.2)
sk
f (x) = sin2 x
Back to Questions
f (x) = 2 sin x cos x = sin 2x > 0 on the set
π
0,
2
3π
∪ π,
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
0
DEO
sis
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-26
f (x) = sin3 x
Solution:
Sa
(III.3)
sk
Back to Questions
f (x) = 3 sin x cos x > 0 on the set
π
0,
2
3π
∪
, 2π
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
2
0
DEO
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-27
f (x) = sin4 x
Solution:
Sa
(III.4)
sk
3
Back to Questions
2
π
0,
2
3π
∪ π,
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) = 4 sin x cos x = 2 sin x sin 2x > 0 on the set
0
DEO
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Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-28
Sa
(III.5)
sk
f (x) = sin5 x
Solution:
Back to Questions
4
f (x) = 5 sin x cos x > 0 on the set
π
0,
2
3π
∪
, 2π
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
y
1
0
-1
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
DEO
sis
iversitas
Un
π/2
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-29
Solution:
Sa
(III.6)
1 − cos x
f (x) =
1 + cos x
sk
Back to Questions
(1 − cos x) (1 + cos x) − (1 − cos x)(1 + cos x)
=
(1 + cos x)2
sin x(1 + cos x) − (1 − cos x)(− sin x)
=
(1 + cos x)2
1 + cos x + 1 − cos x
sin x
=
(1 + cos x)2
2 sin x
> 0 on the interval (0, π )
(1 + cos x)2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) =
y
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-30
Solution:
Sa
(III.7)
1 + cos x
f (x) =
1 − cos x
sk
Back to Questions
(1 + cos x) (1 − cos x) − (1 + cos x)(1 − cos x)
=
(1 − cos x)2
− sin x(1 − cos x) − (1 + cos x)(−(− sin x))
=
(1 − cos x)2
−(1 − cos x) − (1 + cos x)
sin x
=
(1 − cos x)2
−2 sin x
> 0 on the set (π , 2π ) ∪
(1 − cos x)2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) =
y
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-31
Solution:
Sa
(III.8)
1 − sin x
f (x) =
1 + sin x
sk
Back to Questions
(1 − sin x) (1 + sin x) − (1 − sin x)(1 + sin x)
=
(1 + sin x)2
(− cos x)(1 + sin x) − (1 − sin x)(cos x)
=
(1 + sin x)2
−(1 + sin x) − (1 − sin x)
cos x
=
(1 + sin x)2
−2 cos x
π 3π
,
> 0 on the set
(1 + sin x)2
2 2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) =
y
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-32
Solution:
Sa
(III.9)
1 + sin x
f (x) =
1 − sin x
sk
Back to Questions
(1 + sin x) (1 − sin x) − (1 + sin x)(1 − sin x)
=
(1 − sin x)2
(cos x)(1 − sin x) − (1 + sin x)(−(cos x))
=
(1 − sin x)2
(1 − sin x) + (1 + sin x)
cos x
=
(1 − sin x)2
3π
2 cos x
π
∪
, 2π
> 0 on the set 0,
(1 − sin x)2
2
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) =
y
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
DEO
sis
iversitas
Un
π
3π/2
2π
x
Sine & Cosine Differentiation Exercises-33
Solution:
Back to Questions
Sa
(III.10)
sin x
f (x) =
1 − sin x
sk
(sin x) (1 − sin x) − (sin x)(1 − sin x)
=
(1 − sin x)2
(cos x)(1 − sin x) − (sin x)(− cos x)
=
(1 − sin x)2
1 − sin x + sin x
cos x
=
(1 − sin x)2
cos x
3π
π
∪
, 2π
> 0 on the set 0,
(1 − sin x)2
2
2
The graph of y = f (x) is shown in red , and the graph of y = f (x) is shown in blue .
π/2
PAT-
ET
RIÆ
atc h
e w ane
n
2002 Doug MacLean
f (x) =
y
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
DEO
sis
iversitas
Un
π
3π/2
2π
x

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