c2 differentiation jan 22
TRANSCRIPT
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Differentiation
You need to know the difference between
Increasing and Decreasing Functions
An increasing function is one with a positive gradient.
A decreasing function is one with a negative gradient.
9A
x
x
y
y
This function is increasing for all values
of x
This function is decreasing for all values
of x
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Differentiation
You need to know the difference between
Increasing and Decreasing Functions
An increasing function is one with a positive gradient.
A decreasing function is one with a negative gradient.
Some functions are increasing in one interval and decreasing in
another.
9A
x
y
This function is decreasing for x > 0, and increasing for x <
0
At x = 0, the gradient is 0. This is known as a
stationary point.
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Differentiation
You need to know the difference between
Increasing and Decreasing Functions
An increasing function is one with a positive gradient.
A decreasing function is one with a negative gradient.
Some functions are increasing in one interval and decreasing in another.
You need to be able to work out ranges of values where a function is
increasing or decreasing..
9A
Example Question
Show that the function ;3( ) 24 3f x x x
is an increasing function.
3( ) 24 3f x x x 2'( ) 3f x x 24
Differentiate to get the gradient
function
Since x2 has to be positive, 3x2 + 24 will be as well
So the gradient will always be positive, hence an increasing
function
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Differentiation
You need to know the difference between
Increasing and Decreasing Functions
An increasing function is one with a positive gradient.
A decreasing function is one with a negative gradient.
Some functions are increasing in one interval and decreasing in another.
You need to be able to work out ranges of values where a function is
increasing or decreasing..
9A
Example Question
Find the range of values where:3 2( ) 3 9f x x x x
is an decreasing function.
3 2( ) 3 9f x x x x 2'( ) 3f x x 6x 9
23 6 9 0x x 23( 2 3) 0x x
3( 3)( 1) 0x x
1x 3x OR
3 1x
Differentiate for the gradient function
We want the gradient to be
below 0Factorise
Factorise again
Normally x = -3 and 1
BUT, we want values that will
make the function negative…
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Differentiation
You need to know the difference between
Increasing and Decreasing Functions
9A
Example Question
Find the range of values where:3 2( ) 3 9f x x x x
is an decreasing function.
3 2( ) 3 9f x x x x 2'( ) 3f x x 6x 9
23 6 9 0x x 23( 2 3) 0x x
3( 3)( 1) 0x x
1x 3x OR
3 1x
Differentiate for the gradient function
We want the gradient to be
below 0Factorise
Factorise again
Normally x = -3 and 1
BUT, we want values that will
make the function negative…
x
y
-3 1
Decreasing Function range
f(x)
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
A point where f(x) stops increasing and starts decreasing is called a maximum point
A point where f(x) stops decreasing and starts increasing is called a minimum point
A point of inflexion is where the gradient is locally a maximum or minimum (the
gradient does not have to change from positive to negative, for example)
These are all known as turning points, and occur where f’(x) = 0 (for now at least!)
9B
y
x
Local maximum
Local minimum
Point of inflexion
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
9B
y
x
Local maximum
Local minimum
Point of inflexion
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
9B
Example Question
Find the coordinates of the turning point on the curve y = x4 – 32x, and state whether it
is a minimum or maximum.
4 32y x x
34 32dy
xdx
34 32 0x 34 32x
2x 4 32y x x
4(2) 32(2)y 48y
Differentiate
Set equal to 0
Add 32
Divide by 4, then cube root
Sub 2 into the original equation
Work out the y-coordinate
The stationary point is at (2, -48)
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
4) To determine whether the point is a minimum or a maximum, you need to work out f’’(x)
(differentiate again!)
9B
Example Question
Find the coordinates of the turning point on the curve y = x4 – 32x, and state whether it
is a minimum or maximum.
4 32y x x
34 32dy
xdx
The stationary point is at (2, -48)
22
212
d yx
dx
212x
212(2)
48
Differentiate again
Sub in the x coordinate
Positive = Minimum
Negative = Maximum
So the stationary point is a MINIMUM
in this case!
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
4) To determine whether the point is a minimum or a maximum, you need to work out f’’(x)
(differentiate again!)
9B
Example Question
Find the stationary points on the curve: y = 2x3 – 15x2 + 24x + 6, and state
whether they are minima, maxima or points of inflexion
3 22 15 24 6y x x x 2'( ) 6f x x 30x 24
26 30 24 0x x 26( 5 4) 0x x
6( 4)( 1) 0x x
4x 1x OR
Substituting into the original formula will give the following coordinates as stationary
points:
(1, 17) and (4, -10)
Differentiate
Set equal to 0
Factorise
Factorise again
Write the solutions
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
4) To determine whether the point is a minimum or a maximum, you need to work out f’’(x)
(differentiate again!)
9B
Example Question
Find the stationary points on the curve: y = 2x3 – 15x2 + 24x + 6, and state
whether they are minima, maxima or points of inflexion
3 22 15 24 6y x x x 2'( ) 6f x x 30x 24
Stationary points at: (1, 17) and (4, -
10)Differentiate
again''( ) 12 30f x x
''( ) 12 30f x x
''(1) 12(1) 30f
''( ) 12 30f x x
''(4) 12(4) 30f
''(1) 18f ''(4) 18f
Sub in x = 1
Sub in x = 4
So (1,17) is a Maximum
So (4,-10) is a
Minimum
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Differentiation
You need to be able to calculate the co-ordinates of Stationary
points, and determine their nature
To find the coordinates of these points, you need to:
1) Differentiate f(x) to get the Gradient Function
2) Solve f’(x) by setting it equal to 0 (as this represents the gradient being 0)
3) Substitute the value(s) of x into the original equation to find the corresponding y-coordinate
4) To determine whether the point is a minimum or a maximum, you need to work out f’’(x)
(differentiate again!)
9B
Example Question
Find the maximum possible value for y in the formula y = 6x – x2. State the range of the
function.
26y x x
6 2dy
xdx
6 2 0x
3x
26y x x 26(3) (3)y
9y
9y
Differentiate
Set equal to 0
Solve
Sub x into the original equation
Solve
9 is the maximum, so the range is less than but
including 9
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Differentiation
You need to be able to recognise practical problems that can be solved
by using the idea of maxima and minima
Whenever you see a question asking about the maximum value or minimum value of a quantity, you will most likely need to use
differentiation at some point.
Most questions will involve creating a formula, for example for Volume or Area, and then calculating the maximum value
of it.
A practical application would be ‘If I have a certain amount of material to make a box, how can I make the one with the largest
volume? (maximum)’
9C
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A large tank (shown) is to be made from 54m2 of sheet metal. It has no top.
Show that the Volume of the tank will be given by:
3218
3V x x x
xy
2V x y Formula for the Volume
22 3SA x xy
1) Try to make formulae using the information you have
Formula for the Surface Area (no
top)
254 2 3x xy 2) Find a way to remove a constant, in
this case ‘y’. We can rewrite the Surface Area formula in terms of y.
254 2 3x xy 254 2 3x xy 254 2
3
xy
x
3) Substitute the SA formula into the Volume formula, to replace y.
22 54 2
3
xV x
x
2V x y
2 454 2
3
x xV
x
2 454 2
3 3
x xV
x x
3218
3V x x
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A large tank (shown) is to be made from 54m2 of sheet metal. It has no top.
Show that the Volume of the tank will be given by:
3218
3V x x x
xy
b) Calculate the values of x that will give the largest volume possible, and
what this Volume is.
3218
3V x x
218 2dV
xdx
218 2 0x 218 2x
3x
254 2 3x xy
3218
3V x x
3218(3) (3)
3V 336V m
Differentiate
Set equal to 0
Rearrange
Solve
Sub the x value in
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A wire of length 2m is bent into the shape shown, made up of a Rectangle and a Semi-
circle.
x
y
y a) Find an expression for y in terms of x.
1) Find the length of the semi-circle, as this makes up part of the length.
2 2y x 2
x πx
2
2 22
xx y
12 4
x xy
Rearrange to get y alone
Divide by 2
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A wire of length 2m is bent into the shape shown, made up of a Rectangle and a Semi-
circle.
x
y
y a) Find an expression for y in terms of x.
1) Work out the areas of the Rectangle and Semi-circle
separately.
b) Show that the Area is:
(8 4 )8
xA x x
xy2
22
x
Rectangle
Semi Circle
2 2r
2
24
x
2
8
x
12 4
x xy
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A wire of length 2m is bent into the shape shown, made up of a Rectangle and a Semi-
circle.
x
y
y a) Find an expression for y in terms of x.
1) Work out the areas of the Rectangle and Semi-circle
separately.
b) Show that the Area is:
(8 4 )8
xA x x
xyRectangl
eSemi Circle
2
8
x
12 4
x xy
A xy
2
8
x
A 12 4
x xx
2
8
x
A2 2
2 4
x xx
2
8
x
A2 2
2 8
x xx
(8 4 )8
xA x x
Replace y
Expand
Factorise
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Differentiation
You need to be able to recognise practical problems that can be
solved by using the idea of maxima and minima
9C
Example Question
A wire of length 2m is bent into the shape shown, made up of a Rectangle and a Semi-
circle.
x
y
y a) Find an expression for y in terms of x.
1) Use the formula we have for the Area
b) Show that the Area is:
(8 4 )8
xA x x
12 4
x xy
c) Find the maximum
possible Area
(8 4 )8
xA x x
2 2
2 8
x xA x
14
dA xx
dx
21 0
8
xx
8 8 2 0x x 4 4 0x x
4 4x x
4 4x
0.56 x20.28A m
Expand
Differentiate
Set equal to 0
Multiply by 8
Divide by 2
Factorise
Divide by (4 + π)