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![Page 1: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/1.jpg)
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Higher Outcome 2
Higher Unit 3Higher Unit 3
www.mathsrevision.comwww.mathsrevision.com
Further Differentiation Trig Functions
Further Integration
Integrating Trig Functions
Differentiation The Chain Rule
![Page 2: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/2.jpg)
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Higher Outcome 2
The Chain Rule for Differentiating
To differentiate composite functions (such as functions with brackets in them) we can
use:
dy dy du
dx du dx
Example 2 8( 3) Differentiate y x
2( 3) Let u x 8 then y u
78 dy
udu
2 du
xdx
dy dy du
dx du dx78 2 u x 2 78( 3) .2 x x 2 716 ( 3)
dyx x
dx
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Higher Outcome 2
The Chain Rule for Differentiating
Good News !
There is an easier way.
'( ) '( ( )) '( )h x g f x f x
2 8( ) ( 3)h x x Differentiate
2( ) ( 3)f x x Let 8( )g x x then
2 7'( ( )) 8( 3)g f x x 2 8( ( )) ( 3)g f x x '( ) 2f x x
2 7'( ) 16 ( 3)h x x x
You have 1 minute to come up with the rule.
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
2 7'( ) 16 ( 3)h x x x
![Page 4: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/4.jpg)
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Higher Outcome 2
The Chain Rule for Differentiating
Example
42
5'( ) ( )
3
Find when f x f x
x
2 4( ) 5( 3)f x x
2 5'( ) 20( 3)f x x
2 5
40'( )
( 3)
xf x
x
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
2 5'( ) 20( 3) 2f x x x
You are expected to do the chain rule all at once
![Page 5: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/5.jpg)
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Higher Outcome 2Example
1
2 1
Find when
dyy
dx x
1
2(2 1)dy
xdx
3-2
12 -1 2
2
dyx
dx
3
2
1
(2 1)
dy
dxx
The Chain Rule for Differentiating1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
![Page 6: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/6.jpg)
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Higher Outcome 2Example
3 2 5( ) (3 2 1) '(0) If , find the value of f x x x f
3 2 4 2'( ) 5(3 2 1) (9 - 4 )f x x x x x Let
'(0) 0 f
The Chain Rule for Differentiating
3 2 4 2'(0) 5(3 0 2 0 1) (9 0 - 4 0)f Let
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Higher Outcome 2Example1
32 1
Find the equation of the tangent to the curve where y xx
1
2 1
y
xRe-arrange: 1
2 1 y x
The slope of the tangent is given by the derivative of the equation.
Use the chain rule:
21 2 1 2
dy
xdx 2
2
2 1
x
Where x = 3:
1 2
5 25
and
dyy
dx
1 23,5 25
We need the equation of the line through with slope
The Chain Rule for Differentiating Functions
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Higher Outcome 2
1 2( 3)
5 25
y x
2 1 1 5( 3) (2 6)
25 5 25 25
y x x
1(2 1)
25
y x Is the required equation
Remember
y - b = m(x – a)
The Chain Rule for Differentiating Functions
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Higher Outcome 2Example
In a small factory the cost, C, in pounds of assembling x components in a month is given by:
21000
( ) 40 , 0
C x x xx
Calculate the minimum cost of production in any month, and the corresponding number of
components that are required to be assembled.
0 Minimum occurs where , so differentiate dC
dx
225
( ) 40
C x xx
Re-arrange 2
251600
x
x
The Chain Rule for Differentiating Functions
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Higher Outcome 2
225
1600
dC d
xdx dx x
225
1600
dx
dx x
Using chain rule 2
25 251600 2 1
x
x x 2
25 253200 1
x
x x
2
25 250 0 1 0
where or dC
xdx x x
2 225 25 i.e. where or x x
2 25No real values satisfy x 0 We must have x
5 We must have items assembled per month x
The Chain Rule for Differentiating Functions
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Higher Outcome 2
2
25 253200 1 0 5
when dC
x xdx x x
5 5 Consider the sign of when and dC
x xdx
For x < 5 we have (+ve)(+ve)(-ve) = (-ve)
Therefore x = 5 is a minimum
Is x = 5 a minimum in the (complicated) graph?
Is this a minimum
?
For x > 5 we have (+ve)(+ve)(+ve) = (+ve)
The Chain Rule for Differentiating Functions
For x = 5 we have (+ve)(+ve)(0) = 0x = 5
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Higher Outcome 2
The cost of production:2
25( ) 1600 5
, C x x x
x
225
1600 55
21600 10
160,000 £ Expensive components?
Aeroplane parts maybe ?
The Chain Rule for Differentiating Functions
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Calculus Revision
Back NextQuit
Differentiate5( 2)x
Chain rule45( 2) 1x
45( 2)x Simplify
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Calculus Revision
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Differentiate3(5 1)x
Chain Rule23(5 1) 5x
215(5 1)x Simplify
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Calculus Revision
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Differentiate2 4(5 3 2)x x
Chain Rule 2 34(5 3 2) 10 3x x x
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Calculus Revision
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Differentiate
5
2(7 1)x
Chain Rule
Simplify
3
25
2(7 1) 7x
3
235
2(7 1)x
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Calculus Revision
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Differentiate
1
2(2 5)x
Chain Rule
Simplify
3
21
2(2 5) 2x
3
2(2 5)x
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Calculus Revision
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Differentiate 3 1x
Chain Rule
Simplify
Straight line form 1
23 1x
1
21
23 1 3x
1
23
23 1x
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Calculus Revision
Back NextQuit
Differentiate
13 2( ) (8 )f x x
Chain Rule
Simplify
13 221
2( ) (8 ) ( 3 )f x x x
12 3 23
2( ) (8 )f x x x
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Calculus Revision
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Differentiate 1
2( ) 5 4f x x
Chain Rule
Simplify
1
21
2( ) 5 4 5f x x
1
25
2( ) 5 4f x x
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Calculus Revision
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Differentiate 42
5
3x
Chain Rule
Simplify
Straight line form2 45( 3)x
2 520( 3) 2x x
2 540 ( 3)x x
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Calculus Revision
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Differentiate1
2 1x
Chain Rule
Simplify
Straight line form1
2(2 1)x
3
21
2(2 1) 2x
3
2(2 1)x
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Higher Outcome 2
Trig Function Differentiation
The Derivatives of sin x & cos x
(sin ) cos (cos ) sin d d
x x x xdx dx
![Page 24: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/24.jpg)
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Higher Outcome 2Example
2( ) 2cos sin '( )
3 . Find f x x x f x
2'( ) (2cos ) sin
3
d d
f x x xdx dx
[ ( ) ( )]d df dgf x g x
dx dx dx
22 (cos ) sin
3
d dx x
dx dx. ( )
d dgc g x c
dx dx
2'( ) 2sin cos
3 f x x x
Trig Function Differentiation
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Higher Outcome 2Example
3
3
4 cos Differentiate with respect to
x xx
x
33
3
4 cos4 cos
x x
x xx
3 3(4 cos ) 4 (cos ) d d d
x x x xdx dx dx
4
4
12 ( sin )
12 sin
x x
x x
4
4
sin 12 x x
x
Trig Function Differentiation
Simplify expression -
where possible
Restore the original form of expression
![Page 26: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/26.jpg)
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Higher Outcome 2
The Chain Rule for DifferentiatingTrig Functions
Worked Example:
sin(4 )y x Differentiate
cosdy
dx cos4
dyx
du cos4 4
dyx
dx
4cos4 dy
xdx
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
![Page 27: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/27.jpg)
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Higher Outcome 2
The Chain Rule for DifferentiatingTrig Functions
Example
4cos Find when dy
y xdx
4(cos )dy
xdx
34(cos ) (-sin )dy
x xdx
34sin cos dy
x xdx
![Page 28: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/28.jpg)
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Higher Outcome 2Example
sin Find when dy
y xdx
1
2(sin )y x 1
21(sin ) cos2
dyx x
dx
cos
2 sin
dy x
dx x
The Chain Rule for DifferentiatingTrig Functions
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Calculus Revision
Back NextQuit
Differentiate2
2cos sin3
x x
22sin cos
3x x
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Calculus Revision
Back NextQuit
Differentiate 3cos x
3sin x
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Calculus Revision
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Differentiate cos 4 2sin 2x x
4sin 4 4cos 2x x
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Calculus Revision
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Differentiate 4cos x
34cos sinx x
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Calculus Revision
Back NextQuit
Differentiate sin x
Chain Rule
Simplify
Straight line form
1
2(sin )x1
21
2(sin ) cosx x
1
21
2cos (sin )x x
1
2
cos
sin
x
x
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Calculus Revision
Back NextQuit
Differentiate sin(3 2 )x
Chain Rule
Simplify
cos(3 2 ) ( 2)x
2cos(3 2 )x
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Calculus Revision
Back NextQuit
Differentiatecos5
2
x
Chain Rule
Simplify
1
2cos5xStraight line form
1
2sin 5 5x
5
2sin 5x
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Calculus Revision
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Differentiate ( ) cos(2 ) 3sin(4 )f x x x
( ) 2sin(2 ) 12cos(4 )f x x x
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Calculus Revision
Back NextQuit
Differentiate 62siny x
Chain Rule
Simplify
62cos 1dy
xdx
62cosdy
xdx
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Calculus Revision
Back NextQuit
Differentiate2 2( ) cos sinf x x x
( ) 2 cos sin 2sin cosf x x x x x Chain Rule
Simplify ( ) 4 cos sinf x x x
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Higher Outcome 2
Integrating Composite Functions
Harder integration
1( )( )
( 1)
n
n ax bax b dx c
a n
4(3 7) x dx5(3 7)
15
xc
4 1(3 7)
3(4 1)
x
c
we get
You have 1 minute to come up with the rule.
4(3 7) x dx5(3 7)
15
xc
![Page 40: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/40.jpg)
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Higher Outcome 2
Integrating Composite Functions
1( )( )
( 1)
n
n ax bax b dx c
a n
Example :4(4 2) Evaluate
dt
t
4(4 2) t dt
3(4 2)t 3(4 2)
12
tc
1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
3(4 2)
3
t
3(4 2)
4 ( 3)
t
![Page 41: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/41.jpg)
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Higher Outcome 2Example
2( 3) Evaluate u du
1( 3)u 1( 3)
1
u
12 ( 3) 1
( 3)1 ( 3)
uu du c c
u
Integrating Composite Functions1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
1( 3)
1 ( 1)
uc
You are expected to do the integration rule all at once
![Page 42: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/42.jpg)
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Higher Outcome 2Example 2 3
1(2 1) Evaluate x dx
24
1
(2 1)
8
x
68
Integrating Composite Functions
4 4(2 2 1) (1 1 1)
8 8
![Page 43: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/43.jpg)
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Higher Outcome 2Example
32
4
1(3 4)
Evaluate x dx
32
4
1(3 4)x dx
45
2
1
(3 4)5
32
x
682
5
Integrating Composite Functions
45
2
1
2(3 4)
15
x
5 5
2 22(3 4 4) 2(3 ( 1) 4)
15 15
![Page 44: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/44.jpg)
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Higher Outcome 2Example
3( ) '( ) 2 1 (1) 2f x f x x f Find given and
3( ) 2 1f x x dx
Integrating
42 1
8
xc
So we have:
42 1 1(1) 2
8f c
41
8c
1
8c
Giving:
1 1528 8
c 42 1 15( )
8 8
xf x
Integrating Functions1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
![Page 45: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/45.jpg)
Calculus Revision
Back NextQuit
Integrate2(5 3 )x dx
3(5 3 )
3 3
xc
31
9(5 3 )x c
Standard Integral
(from Chain Rule)
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Calculus Revision
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Integrate2
1
(7 3 )dx
x2(7 3 )x dx
1(7 3 )
1 3
xc
11
3(7 3 )x c
Straight line form
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Calculus Revision
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Find
1
6
0
(2 3)x dx17
0
(2 3)
7 2
x
7 7(2 3) (0 3)
14 14
7 75 3
14 14
5580.36 156.21 (4sf)5424
Use standard Integral
(from chain rule)
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Calculus Revision
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Integrate
1
2
1
0 3 1
dx
x
1
2
1
0
3 1x dx
Straight line form
1
2
1
0
13
2
3 1x
1
0
23 1
3x
2 23 1 0 1
3 3
2 24 1
3 3
4 2
3 3
2
3
![Page 49: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/49.jpg)
Calculus Revision
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Find 2
3
1
2 1x dx
24
1
2 1
4 2
x
4 44 1 2 1
4 2 4 2
4 45 3
8 8
68
Use standard Integral
(from chain rule)
![Page 50: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/50.jpg)
Calculus Revision
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Evaluate0
2
3
(2 3)x dx
Use standard Integral
(from chain rule)
03
3
(2 3)
3 2
x
3 3(2(0) 3) (2( 3) 3)
6 6
27 27
6 6
27 27
6 6
54
6 9
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Calculus Revision
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Evaluate1
01 3x dx
1
21
01 3x dx
13
2
0
3
2
1 3
3
x
13
2
0
21 3
9x
13
0
21 3
9x
3 32 21 3(1) 1 3(0)
9 9
3 32 24 1
9 9
16 2
9 9
14
9 5
91
![Page 52: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/52.jpg)
Calculus Revision
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Find p, given
1
42p
x dx 1
2
1
42p
x dx 3
2
1
2
342
p
x
3 3
2 22 2
3 3(1) 42p
2 233 3
42p 32 2 126p
3 64p 3 2 1264 2p 1
12 32 16p
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Calculus Revision
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A curve for which 26 2dy
x xdx
passes through the point (–1, 2).
Express y in terms of x.
3 26 2
3 2
x xy c 3 22y x x c
Use the point3 22 2( 1) ( 1) c 5c
3 22 5y x x
![Page 54: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/54.jpg)
Calculus Revision
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Given the acceleration a is: 1
22(4 ) , 0 4a t t
If it starts at rest, find an expression for the velocity v where dv
adt
1
22(4 )dv
tdt
3
2
31
2
2(4 )tv c
3
24(4 )3
v t c
344
3v t c Starts at rest, so
v = 0, when t = 0 34
0 43
c
340 4
3c
320
3c
32
3c
3
24 32
3 3(4 )v t
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Higher Outcome 2
Integrating Trig Functions
(sin ) cos
(cos ) sin
dx x
dxd
x xdx
cos sin
sin cos
x dx x c
x dx x c
Integration is opposite of
differentiation
Worked Example
123sin cosx x dx 3cos x 1
2 sin x c
![Page 56: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/56.jpg)
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Higher Outcome 2
Special Trigonometry Integrals are
1cos( ) sin( )
1sin( ) cos( )
ax b dx ax b ca
ax b dx ax b ca
Worked Example
sin(2 2) x dx 1cos(2 2)2
x c
Integrating Trig Functions1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.
![Page 57: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/57.jpg)
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Higher Outcome 2
Integrating Trig Functions
Example
sin5 cos( 2 ) Evaluate t t dt
Integrate
sin(5 ) s( 2 )t dt co t dt 1 1cos5 sin 25 2
t t c
Break up into two easier integrals
1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.
![Page 58: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/58.jpg)
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Higher Outcome 2Example 2
4
3cos 22
Evaluate x dx
2
4
3cos 22
x dx
2
4
3sin 2x dx
Re-arrange
2
4
3cos22
x
Integrate
3cos 2 cos 2
2 2 4
3
1 02
3
2
Integrating Trig Functionscos( ) cos cos sin sin 1. Integrate outside the bracket
2. Keep the bracket the same
3. Compensate for inside the bracket.
![Page 59: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/59.jpg)
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Higher Outcome 2
Integrating Trig Functions (Area)
Example
cosy xsiny x
1
x
y
1
0
AThe diagram shows the
graphs of y = -sin x and y = cos x
a) Find the coordinates of A
b) Hence find the shaded area cos sin The curves intersect where x x
tan 1 x
0 We want x
1tan ( 1) x C
AS
T0o180
o
270o
90o
3
2
2
3 7
4 4
and
3
4
x
![Page 60: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/60.jpg)
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Higher Outcome 2
Integrating Trig Functions (Area)
( ) The shaded area is given by top curve bottom curve dx
3
4
0
(cos ( sin ))
Area x x dx
3
4
0
(cos sin )
x x dx
3
40
sin cosx x
3 3sin cos sin0 cos0
4 4
1 11
2 2
1 2
![Page 61: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/61.jpg)
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Higher Outcome 2Example 3cos3 cos(2 ) cos3 4cos 3cos a) By writing as show that x x x x x x
3cos b) Hence find x
cos(2 ) cos2 cos sin 2 sin x x x x x x
2 2(cos sin )cos (2sin cos )sin x x x x x x
2 2 2(cos sin )cos 2sin cos x x x x x
3 2cos 3sin cos x x x
3 2cos 3 1 cos cos x x x
3cos3 4cos 3cos x x x
Integrating Trig Functions
Remember cos(x +
y) =
![Page 62: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/62.jpg)
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Higher Outcome 2
3cos3 4cos 3cos As shown above x x x
3 1cos (cos3 3cos )
4 x x x
1 1 1(cos3 3cos ) sin3 3sin
4 4 3 x x dx x x c
3 1cos sin3 9sin
12 x dx x x c
Integrating Trig Functions
![Page 63: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/63.jpg)
Calculus Revision
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Find2sin7
t dt2
7cos t c
![Page 64: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/64.jpg)
Calculus Revision
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Find 3cos x dx3sin x c
![Page 65: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/65.jpg)
Calculus Revision
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Find 2sin d 2cos c
![Page 66: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/66.jpg)
Calculus Revision
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2(6 cos )x x x dx Integrate
3 26sin
3 2
x xx c Integrate term by term
![Page 67: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/67.jpg)
Calculus Revision
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Find1
3sin cos2
x x dx1
3cos sin2
x x c Integrate term by term
![Page 68: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/68.jpg)
Calculus Revision
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Find sin 2 cos 34
x x dx
1 1
2 3cos 2 sin 3
4x x c
![Page 69: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/69.jpg)
Calculus Revision
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The curve ( )y f x ,112
passes through the point
( ) cos 2f x x Find f(x)
1
2( ) sin 2f x x c
1
2 121 sin 2 c
use the given point ,112
1
2 61 sin c
1 1
2 21 c 3
4c 1 3
2 4( ) sin 2f x x
1
6 2sin
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Calculus Revision
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If
( ) sin(3 )f x x passes through the point 9, 1
( )y f x express y in terms of x.
1
3( ) cos(3 )f x x c Use the point 9
, 1
1
31 cos 3
9c
1
31 cos
3c
1 1
3 21 c
7
6c 1 7
3 6cos(3 )y x
![Page 71: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/71.jpg)
Calculus Revision
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A curve for which 3sin(2 )dy
xdx
5
12, 3passes through the point
Find y in terms of x. 3
2cos(2 )y x c
3 5
2 123 cos(2 ) c Use the point 5
12, 3
3 5
2 63 cos( ) c 3 3
2 23 c
3 33
4c
4 3 3 3
4 4c
3
4c
3 3
2 4cos(2 )y x
![Page 72: Higher Outcome 2 Higher Unit 3 Further Differentiation Trig Functions Further Integration Integrating Trig.](https://reader035.fdocuments.net/reader035/viewer/2022062303/551a94c7550346b52d8b5fd0/html5/thumbnails/72.jpg)
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Higher Outcome 2
Are you on Target !
• Update you log book
• Make sure you complete and correct
ALL of the Calculus questions in the
past paper booklet.