9) C2 Differentiation

8/12/2019 9) C2 Differentiation

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Introduction

We have seen Differentiation in C1

In C2 we will be looking at solving moretypes of problem

We are also going to be applying theprocess to worded practical problems


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Differentiation

You need to know thedifference between Increasing

and Decreasing Functions

An increasing function is one with a

positive gradient.

A decreasing function is one with anegative gradient.

9A

x

x

y

y

This functionis increasingfor all values

of x

This functionis decreasingfor all values

of x


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Differentiation




positive gradient.


Some functions are increasing in one

interval and decreasing in another.

9A

x

y

This function isdecreasing for x > 0,

and increasing for x < 0

At x = 0, the gradient is0. This is known as a

stationary point.


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Differentiation




positive gradient.




You need to be able to work outranges 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 toget the gradient

function

Since x2has to be positive, 3x2+ 24will be as well

So the gradient will always bepositive, hence an increasing function


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Differentiation




positive gradient.




You need to be able to work outranges 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 923 6 9 0x x

23( 2 3) 0x x

3( 3)( 1) 0x x

1x 3x OR

3 1x

Differentiate for thegradient function

We want the gradientto be below 0

Factorise

Factorise again

Normally x = -3 and1

BUT, we want valuesthat will make the

function negative


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Differentiation



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 923 6 9 0x x

23( 2 3) 0x x

3( 3)( 1) 0x x

1x 3x OR

3 1x

Differentiate for thegradient function

We want the gradientto be below 0

Factorise

Factorise again

Normally x = -3 and1

BUT, we want valuesthat 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 calculatethe co-ordinates of Stationary

points, and determine their nature

A point where f(x) stops increasing and

starts decreasing is called a maximumpoint

A point where f(x) stops decreasing andstarts increasing is called a minimum point

A point of inflexion is where the gradientis locally a maximum or minimum (the

gradient does not have to change frompositive to negative, for example)

These are all known as turning points, and

occur where f(x) = 0 (for now at least!)9B

y

x

Localmaximum

Local

minimum

Point ofinflexion


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Differentiation



To find the coordinates of these points,

you need to:

1) Differentiate f(x) to get the GradientFunction

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 theoriginal equation to find thecorresponding y-coordinate

9B

y

x

Localmaximum

Local

minimum

Point ofinflexion


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Differentiation



To find the coordinates of these points,

you need to:


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 theoriginal equation to find thecorresponding y-coordinate

9B

Example Question

Find the coordinates of the turning point on thecurve y = x432x, and state whether it is a

minimum or maximum.

4

32y x x 34 32

dyx

dx

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 originalequation

Work out the y-coordinate

The stationary point is at(2, -48)


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Differentiation



To find the coordinates of these points, you

need to:


2) Solve f(x) by setting it equal to 0 (as thisrepresents the gradient being 0)

3) Substitute the value(s) of x into the originalequation to find the corresponding y-coordinate

4) To determine whether the point is a minimumor a maximum, you need to work out f(x)

(differentiate again!)

9B

Example Question

Find the coordinates of the turning point on thecurve y = x432x, and state whether it is a

minimum or maximum.

4

32y x x 34 32

dyx

dx

The stationarypoint is at (2, -48)

22

2 12

d yx

dx

212x

212(2)

48

Differentiate again

Sub in the xcoordinate

Positive = Minimum

Negative = Maximum

So the stationary

point is a MINIMUMin this case!


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Differentiation




need to:






9B

Example Question

Find the stationary points on the curve:y = 2x315x2+ 24x + 6, and state whether they

are minima, maxima or points of inflexion

3 2

2 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 givethe following coordinates as stationary points:

(1, 17) and (4, -10)

Differentiate

Set equal to 0

Factorise

Factorise again

Write thesolutions


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Differentiation




need to:






9B

Example Question

Find the stationary points on the curve:y = 2x315x2+ 24x + 6, and state whether they

are minima, maxima or points of inflexion

3 2

2 15 24 6y x x x 2'( ) 6f x x 30x 24

Stationary points at:(1, 17) and (4, -10)

Differentiateagain

''( ) 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 MaximumSo (4,-10) is

a Minimum


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Differentiation



To find the coordinates of these points, youneed to:






9B

Example Question

Find the maximum possible value for y in theformula y = 6x x2. State the range of the

function.

2

6y 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 theoriginal 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 recognisepractical problems that can be solved by

using the idea of maxima and minima

Whenever you see a question asking aboutthe maximum value or minimum value of aquantity, you will most likely need to use

differentiation at some point.

Most questions will involve creating aformula, for example for Volume or Area,and then calculating the maximum value of

it.

A practical application would be If I havea certain amount of material to make abox, how can I make the one with the

largest volume? (maximum)

9C


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Differentiation



9C

Example Question

A large tank (shown) is to be made from 54m2ofsheet metal. It has no top.

Show that the Volume of the tank will be givenby:

32183

V x x xx

y

2

V x y Formula for the Volume

22 3SA x xy

1) Try to make formulae using theinformation you have

Formula for theSurface Area (no

top)

254 2 3x xy

2) Find a way to remove a constant, in thiscase y. We can rewrite the Surface Areaformula in terms of y.

254 2 3x xy 254 2 3x xy

254 2

3

xy

x

3) Substitute the SA formula into theVolume formula, to replace y.

22 54 2

3

xV x

x

2V x y

2 454 2

3

x x

V x

2 4

54 23 3

x xVx x

32183

V x x


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Differentiation



9C

Example Question

A large tank (shown) is to be made from 54m2ofsheet metal. It has no top.

Show that the Volume of the tank will be givenby:

32

18 3V x x xx

y

b) Calculate the values of x that will givethe largest volume possible, and what this

Volume is.

32183

V x x

218 2dV

xdx

218 2 0x

218 2x

3x

254 2 3x xy

32183

V x x

32

18(3) (3)3V 336V m

Differentiate

Set equal to 0

Rearrange

Solve

Sub the x value in


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Differentiation



9C

Example Question

A wire of length 2m is bent into the shapeshown, made up of a Rectangle and a Semi-circle.

x

y

y a) Find an expressionfor 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

xx2

2 22

xx y

12 4

x x y

Rearrangeto get y

alone

Divide by 2


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Differentiation



9C

Example Question


x

y


1) Work out the areas of theRectangle and Semi-circle separately.

b) Show that the Area is:

(8 4 )8

xA x x

xy

2

2

2

x

Rectangle Semi Circle

2 2r

2

24

x

2

8

x

1

2 4

x xy


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Differentiation



9C

Example Question


x

y


1) Work out the areas of theRectangle and Semi-circle separately.


(8 4 )8

xA x x

xy

Rectangle Semi Circle2

8

x

1

2 4

x xy

A xy2

8

x

A 12 4

x xx

2

8

x

A

2 2

2 4

x x

x

2

8

x

A 2 2

2 8

x xx

(8 4 )8

xA x x

Replacey

ExpandFactorise


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Differentiation



9C

Example Question


x

y


1) Use the formula we have for theArea


(8 4 )8

xA x x

1

2 4

x xy

c) Find the maximum

possible Area

(8 4 )8

xA x x

2 2

2 8

x xA x

14

dA xxdx

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 + )


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Summary

We have expanded our knowledge ofDifferentiation to include stationarypoints

We have looked at using maxima andminima to solve more practical problems

Documents

9) C2 Differentiation