Www.mathsrevision.com Higher Unit 1 Finding the gradient for a polynomial Differentiating Brackets ( Type 1 ) Differentiating Harder

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Higher Unit 1Higher Unit 1

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Finding the gradient for a polynomial

Differentiating Brackets ( Type 1 ) Differentiating Harder Terms (Type 2)Differentiating with Leibniz NotationEquation of a Tangent Line ( Type 3 )

Increasing / Decreasing functionsMax / Min and inflexion Points

Curve Sketching

Max & Min Values on closed IntervalsOptimization

Higher Outcome 3

Mind Map of Chapter

Using differentiation (Application)

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On a straight line the gradient remains constant, however with curves the gradient changes continually, and the gradient at any point is in fact the same as the gradient of the tangent at that point.

The sides of the half-pipe are very steep(S) but it is not very steep near the base(B).

B

S

Gradients & CurvesHigher Outcome 3

Demo

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A

Gradient of tangent = gradient of curve at A

B

Gradient of tangent = gradient of curve at B


Demo

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For the function y = f(x) we do this by taking the point (x, f(x))

and another “very close point” ((x+h), f(x+h)).Then we find the gradient between the two.

(x, f(x))

((x+h), f(x+h))

True gradient

Approx gradient

To find the gradient at any point on a curve we need

to modify the gradient formula

2 1

2 1

--

y ym

x x

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The gradient is not exactly the same but is

quite close to the actual value We can improve the approximation by making the value of h

smallerThis means the two points are closer together.

(x, f(x))

((x+h), f(x+h))

True gradient

Approx gradient


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We can improve upon this approximation by making the value of h even smaller.

(x, f(x))

((x+h), f(x+h))True gradientApprox gradient

So the points are even closer together.


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Higher Outcome 3

Derivative

We have seen that on curves the gradient changes continually and is dependant on the position on

the curve. ie the x-value of the given point.

The process of finding the gradient is called

DIFFERENTIATING

or

FINDING THE DERIVATIVE (Gradient)

Differentiating

Finding the GRADIENT

Finding the rate of change

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If the formula/equation of the curve is given by f(x)Then the derivative is called f '(x) - “f dash x”

There is a simple way

of finding f '(x) from f(x).

f(x) f '(x)

2x2 4x 4x2 8x 5x10 50x9

6x7 42x6

x3 3x2

x5 5x4

x99 99x98

DerivativeHigher Outcome 3

Have guessed the

rule yet !

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If f(x) = axn

n-1n

Rule for Differentiating

It can be given by this simple flow diagram ...

multiply by the power

reduce the power by 1

then f '(x) =

NB: the following terms & expressions mean the same

GRADIENT, DERIVATIVE, RATE OF CHANGE, f '(x)

Derivative

Higher Outcome 3

ax

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Rule for Differentiating

To be able to differentiate

it is VERY IMPORTANT that you are

comfortable using indices rules

Derivative

Higher Outcome 3

0

1

11

nn

mm n m n m n

n

nm n n mm

x xx

xx x x x

x

x x x

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Special Points

(I) f(x) = ax (Straight line function)

If f(x) = ax = ax1

then f '(x) = 1 X ax0

= a X 1 = a

Index Laws

x0 = 1

So if g(x) = 12x then g '(x) = 12

Also using y = mx + c

The line y = 12x has gradient 12,

and derivative = gradient !!

Higher Outcome 3

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If f(x) = a = a X 1 = ax0

then f '(x) = 0 X ax-1 = 0

Index Laws

x0 = 1

So if g(x) = -2 then g '(x) = 0

Also using formula y = c , (see outcome 1 !)

The line y = -2 is horizontal so has gradient 0 !

Special PointsHigher Outcome 3

Name :

3( ) 4f x x 2

2( )

3f x

x( ) 3f x x 3 2

1( )

2f x

x

3

2( )

3

xf x

x

( ) ( 2)(3 1)f x x x 2 3

( )x

f xx

41

( ) 5f x xx

Differentiation techniquesGradien

t

=Rate of change=

Differentiation

Differentiation

2'( ) 12f x x

1

2( ) 3f x x1

23

'( )2

f x x

3'( )

2f x

x

22( )

3

xf x

34

'( )3

xf x

3

4'( )

3f x

x

2

3

( )2

xf x

3 5

1'( )

3f x

x

3( ) 23

xf x x

41'( ) 6

3f x x

4

1 6'( )

3f x

x

1 1

2 4( ) 5f x x x

3 3

2 41 5

'( )2 4

f x x x

3 34

1 5'( )

2 4f x

x x

2( ) 3 5 2f x x x

'( ) 6 5f x x

2 3( )

xf x

x x

1 1

2 2( ) 2 3f x x x

3

1 3'( )

2f x

x x

Calculus Revision

Differentiate 24 3 7x x

8 3x

Calculus Revision

Differentiate 3 22 7 4 4y x x x

26 14 4y x x

Calculus Revision

Differentiate 216( ) 240

3A x x x

32( ) 240

3A x x

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Example 1 A curve has equation f(x) = 3x4

Its gradient is f '(x) = 12x3

f '(2) = 12 X 23 = 12 X 8 = 96

Example 2 A curve has equation f(x) = 3x2

Find the formula for its gradient and find the gradient when x = 2

Its gradient is f '(x) = 6x

At the point where x = -4 the gradient isf '(-4) = 6 X -4 =-24


Find the formula for its gradient and find the gradient when x = -4

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g '(x) = 20x3 - 20x4 g '(2) = 20 X 23 - 20 X 24

= 160 - 320

= -160


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Example 4

h(x) = 5x2 - 3x + 19

so h '(x) = 10x - 3

and h '(-4) = 10 X (-4) - 3

= -40 - 3 = -43

Example 5

k(x) = 5x4 - 2x3 + 19x - 8, find k '(10) .

k '(x) = 20x3 - 6x2 + 19

So k '(10) = 20 X 1000 - 6 X 100 + 19

= 19419


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Example 6 : Find the points on the curve

f(x) = x3 - 3x2 + 2x + 7 where the gradient is 2.

NB: gradient = derivative = f '(x)

We need f '(x) = 2

ie 3x2 - 6x + 2 = 2

or 3x2 - 6x = 0

ie 3x(x - 2) = 0

ie 3x = 0 or x - 2 = 0

so x = 0 or x = 2

Now using original formula

f(0) = 7

f(2) = 8 -12 + 4 + 7

= 7

Points are (0,7) & (2,7)


Calculus Revision

Differentiate

3 1

2 22 5x x

1 3

2 23 12 2

2 5x x

1 3

2 25

23x x

Calculus Revision

Differentiate3

1 1

x x

1 3x x Straight line form

Differentiate2 4( 3)x x

2 43x x

Calculus Revision

Differentiate 2 8003 r

r

2 13 800r r Straight line form

Differentiate26 ( 1)800r r

Calculus Revision

Differentiate 2 43200( )A x x

x

Chain Rule

Simplify

Straight line form2 1( ) 43200A x x x

2( ) 2 43200A x x x

2

43200( ) 2A x x

x

Calculus Revision

Differentiate2

3 1xx

1 1

2 23 2 1x x

Straight line form

Differentiate

1 3

2 21

2

32

2x x

1 3

2 23

2x x

Calculus Revision

Differentiate 2

2( )f x x

x

Differentiate

Straight line form1

22( ) 2f x x x

1321

2( ) 4f x x x

Calculus Revision

Differentiate16

, 0y x xx

Differentiate

Straight line form1

216y x x

3

21 8dy

xdx

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Brackets

Basic Rule: Break brackets before you differentiate !

Example h(x) = 2x(x + 3)(x -3)

= 2x(x2 - 9)

= 2x3 - 18x

So h'(x) = 6x2 -18

Higher Outcome 3

Calculus Revision

Differentiate (3 5)( 2)x x

23 6 5 10x x x Multiply out

Differentiate 6 1x

23 10x x

Calculus Revision

Differentiate2( 2 )x x x

multiply out3 22x x

differentiate23 4x x

Calculus Revision

Differentiate 2x x x

1

22x x xStraight line form

multiply out

Differentiate

5 3

2 2x x3 1

2 25 3

2 2x x

Calculus Revision

Differentiate3

(8 )4

A a a

multiply out

Differentiate

236

4A a a

36

2A a

Calculus Revision

Differentiate 2 4x x

multiply out

Simplify

4 8 2x x x

Differentiate

6 8x x

Straight line form1

26 8x x 1

23 1x

Calculus Revision

Differentiate 23 3 16( )

2A x x

x

Straight line form

Multiply out 23 3 3 3 16( )

2 2A x x

x

Differentiate

2 13 3( ) 24 3

2A x x x

2( ) 3 3 24 3A x x x

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Recall 17 7 7

Fractions

Reversing the above we get the following “rule” !

This can be used as follows …..

a + bc

a bc c

Higher Outcome 3

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Example

f(x) = 3x3 - x + 2 x2

= 3x - x-1 + 2x-2

f '(x) = 3 + x-2 - 4x-3

Fractions

= 3x3 - x + 2 x2 x2 x2

= 3 + 1 - 4 x2 x3

Higher Outcome 3

Calculus Revision

Differentiate

3 26 3 9x x x

x

3 26 3 9x x x

x x x x

2 16 3 9x x x

Split up

Straight line form

22 6 9x x Differentiate

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If y is expressed in terms of x then the derivative is written as dy/dx .

Leibniz Notation

Leibniz Notation is an alternative way of expressing derivatives to f'(x) , g'(x) , etc.

eg y = 3x2 - 7x so dy/dx = 6x - 7 .

Example 19 Find dQ/dR

NB: Q = 9R2 - 15R-3

So dQ/dR = 18R + 45R-4 = 18R + 45 R4

Q = 9R2 - 15 R3

Higher Outcome 3

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Example 20

A curve has equation y = 5x3 - 4x2 + 7 .

Find the gradient where x = -2 ( differentiate ! )

gradient = dy/dx = 15x2 - 8x

if x = -2 then

gradient = 15 X (-2)2 - 8 X (-2)

= 60 - (-16) = 76

Leibniz NotationHigher Outcome 3

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Newton’s 2ndLaw of Motion

s = ut + 1/2at2 where s = distance & t = time.

Finding ds/dt means “diff in dist” “diff in time”

ie speed or velocity

so ds/dt = u + at

but ds/dt = v so we get v = u + at

and this is Newton’s 1st Law of Motion

Real Life ExamplePhysics

Higher Outcome 3

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y = mx +c

y = f(x)

Equation of Tangents

tangent

NB: at A(a, b) gradient of line = gradient of curve

gradient of line = m (from y = mx + c )

gradient of curve at (a, b) = f (a)

it follows that m = f (a)

Higher Outcome 3

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Higher Outcome 3


Example 21

Find the equation of the tangent line to the curve

y = x3 - 2x + 1 at the point where x = -1.Point: if x = -1 then y = (-1)3 - (2 X -1) + 1

= -1 - (-2) + 1= 2 point is (-1,2)

Gradient: dy/dx = 3x2 - 2

when x = -1 dy/dx = 3 X (-1)2 - 2

= 3 - 2 = 1 m = 1

Straight line so we need a point plus the gradient then we can use the formula y - b = m(x -

a) .

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we get y - 2 = 1( x + 1)

or y - 2 = x + 1

or y = x + 3

point is (-1,2)

m = 1


Higher Outcome 3

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Example 22

Find the equation of the tangent to the curve y = 4 x2

at the point where x = -2. (x 0)

Also find where the tangent cuts the X-axis and Y-axis.Point:when x = -2 then y = 4

(-2)2 = 4/4 =

1

point is (-2, 1)

Gradient: y = 4x-2 so dy/dx = -8x-3 = -8 x3

when x = -2 then dy/dx = -8 (-2)3

= -8/-8 = 1m = 1

Equation of TangentsHigher Outcome 3

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Now using y - b = m(x - a)

we get y - 1 = 1( x + 2)

or y - 1 = x + 2

or y = x + 3

Axes Tangent cuts Y-axis when x = 0

so y = 0 + 3 = 3

at point (0, 3)

Tangent cuts X-axis when y = 0

so 0 = x + 3 or x = -3 at point (-3, 0)


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Example 23 - (other way round)

Find the point on the curve y = x2 - 6x + 5 where the gradient of the tangent is 14.

gradient of tangent = gradient of curve

dy/dx = 2x - 6

so 2x - 6 = 14

2x = 20 x = 10

Put x = 10 into y = x2 - 6x + 5

Giving y = 100 - 60 + 5 = 45 Point is (10,45)


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Increasing & Decreasing Functions and Stationary Points

Consider the following graph of y = f(x) …..

X

y = f(x)

a b c d e f+

+

+

++

-

-

0

0

0

Higher Outcome 3

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In the graph of y = f(x)

The function is increasing if the gradient is positive

i.e. f (x) > 0 when x < b or d < x < f or x > f . The function is decreasing if the gradient is negativeand f (x) < 0 when b < x < d .

The function is stationary if the gradient is zeroand f (x) = 0 when x = b or x = d or x = f .These are called STATIONARY POINTS.

At x = a, x = c and x = e

the curve is simply crossing the X-axis.


Higher Outcome 3

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Example 24

For the function f(x) = 4x2 - 24x + 19 determine the intervals when the function is decreasing and

increasing.f (x) = 8x - 24

f(x) decreasing when f (x) < 0 so 8x - 24 < 0 8x < 24

x < 3

f(x) increasing when f (x) > 0 so 8x - 24 > 0

8x > 24x > 3

Check: f (2) = 8 X 2 – 24 = -8

Check: f (4) = 8 X 4 – 24 = 8


Higher Outcome 3

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Example 25

For the curve y = 6x – 5/x2 Determine if it is increasing or decreasing when x = 10.

= 6x - 5x-2

so dy/dx = 6 + 10x-3

when x = 10 dy/dx = 6 + 10/1000

= 6.01

Since dy/dx > 0 then the function is increasing.


y = 6x - 5 x2

= 6 + 10 x3

Higher Outcome 3

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Example 26Show that the function g(x) = 1/3x3 -3x2 + 9x -10

is never decreasing.

g (x) = x2 - 6x + 9

= (x - 3)(x - 3)= (x - 3)2

Since (x - 3)2 0 for all values of x

then g (x) can never be negative

so the function is never decreasing.

Squaring a negative or a positive value produces a positive value, while 02 = 0. So you will never

obtain a negative by squaring any real number.


Higher Outcome 3

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Example 27

Determine the intervals when the function

f(x) = 2x3 + 3x2 - 36x + 41

is (a) Stationary (b) Increasing (c) Decreasing.

f (x) = 6x2 + 6x - 36

= 6(x2 + x - 6)

= 6(x + 3)(x - 2)

Function is stationary when f (x) = 0

ie 6(x + 3)(x - 2) = 0 ie x = -3 or x = 2


Higher Outcome 3

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determine when f (x) is positive & negative.

x -3 2

f’(x) +

Function increasing when f (x) > 0

ie x < -3 or x > 2

Function decreasing when f (x) < 0

ie -3 < x < 2


Higher Outcome 3

-0 +0

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Stationary Points and Their Nature

Consider this graph of y = f(x) again

X

y = f(x)

a b c+

+

+

+

+-

-

0

0

0

Higher Outcome 3

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When x = a we have a maximum turning point (max TP)When x = b we have a minimum turning point (min TP)When x = c we have a point of inflexion (PI)

Each type of stationary point is determined by the gradient ( f(x) ) at either side of the stationary

value.


Higher Outcome 3

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Maximum Turning point

x af(x) + 0 -

Minimum Turning Point

x bf(x) - 0

+


Higher Outcome 3

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of inflexion

x c

f(x) + 0 +

Other possible type of inflexion

x d

f(x) - 0 -


Higher Outcome 3

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Example 28Find the co-ordinates of the stationary point on the curve y = 4x3 + 1 and determine its nature.

SP occurs when dy/dx = 0so 12x2 = 0

x2 = 0

x = 0

Using y = 4x3 + 1

if x = 0 then y = 1

SP is at (0,1)


Higher Outcome 3

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x 0

dy/dx +

So (0,1) is a rising point of inflexion.


dy/dx = 12x2

Higher Outcome 3

+

0

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Example 29

Find the co-ordinates of the stationary points on the curve y = 3x4 - 16x3 + 24 and determine their nature.

SP occurs when dy/dx = 0

So 12x3 - 48x2 = 0

12x2(x - 4) = 0

12x2 = 0 or (x - 4) = 0

x = 0 or x = 4

Using y = 3x4 - 16x3 + 24

if x = 0 then y = 24

if x = 4 then y = -232

SPs at (0,24) & (4,-232)


Higher Outcome 3

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Nature Table

x 0 4

dy/dx - 0 - 0 +

So (0,24) is a Point of inflexion

and (4,-232) is a minimum Turning Point


dy/dx=12x3 - 48x2

Higher Outcome 3

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Example 30Find the co-ordinates of the stationary points on the curve y = 1/2x4 - 4x2 + 2 and determine their nature.

SP occurs when dy/dx = 0

So 2x3 - 8x = 0

2x(x2 - 4) = 0

2x(x + 2)(x - 2) = 0

x = 0 or x = -2 or x = 2

Using y = 1/2x4 - 4x2 + 2if x = 0 then y = 2

if x = -2 then y = -6

SP’s at(-2,-6), (0,2) & (2,-6)

if x = 2 then y = -6


Higher Outcome 3

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Nature Table

x 0

dy/dx

-2 2

- 0 + 0 - 0 +

So (-2,-6) and (2,-6) are Minimum Turning Points

and (0,2) is a Maximum Turning Points


Higher Outcome 3

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Curve Sketching

Note: A sketch is a rough drawing which includes important details. It is not an accurate scale

drawing.Process

(a) Find where the curve cuts the co-ordinate axes. for Y-axis put x = 0

for X-axis put y = 0 then solve.(b) Find the stationary points & determine their

nature as done in previous section.

(c) Check what happens as x +/- .This comes automatically if (a) & (b) are correct.

Higher Outcome 3

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Dominant Terms

Suppose that f(x) = -2x3 + 6x2 + 56x - 99

As x +/- (ie for large positive/negative values) The formula is approximately the same as f(x) = -2x3

As x + then y -

As x - then y

+

Graph roughly

Curve SketchingHigher Outcome 3

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Example 31

Sketch the graph of y = -3x2 + 12x + 15(a) Axes If x = 0 then y = 15

If y = 0 then -3x2 + 12x + 15 = 0

( -3)

x2 - 4x - 5 = 0

(x + 1)(x - 5) = 0

x = -1 or x = 5

Graph cuts axes at (0,15) , (-1,0) and (5,0)


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(b) Stationary Points occur where dy/dx = 0

so -6x + 12 = 0

6x = 12

x = 2

If x = 2

then y = -12 + 24 + 15 = 27

Nature Table

x 2

dy/dx + 0 -So (2,27)

is a Maximum Turning Point

Stationary Point is (2,27)


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using y = -3x2

as x + then y -

as x - then y -

Sketching

X

Y

y = -3x2 + 12x + 15

Curve Sketching

Cuts x-axis at -1 and 5

Summarising

Cuts y-axis at 15-1 5

Max TP (2,27)(2,27)

15

Higher Outcome 3

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Example 32Sketch the graph of y = -2x2 (x - 4)

(a) Axes If x = 0 then y = 0 X (-4) = 0If y = 0 then -2x2 (x - 4) = 0

x = 0 or x = 4

Graph cuts axes at (0,0) and (4,0) .

-2x2 = 0 or (x - 4) = 0

(b) SPs

y = -2x2 (x - 4) = -2x3 + 8x2

SPs occur where dy/dx = 0

so -6x2 + 16x = 0


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-2x(3x - 8) = 0

-2x = 0 or (3x - 8) = 0

x = 0 or x = 8/3

If x = 0 then y = 0 (see part (a) ) If x = 8/3 then y = -2 X (8/3)2 X (8/3 -4) =512/27

naturex 0 8/3

dy/dx -


-

+

0

0

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(c) Large valuesusing y = -2x3 as x + then y -

as x - then y +Sketch

Xy = -2x2 (x – 4)

Curve Sketching

Cuts x – axis at 0 and 40 4

Max TP’s at (8/3, 512/27) (8/3, 512/27)

Y

Higher Outcome 3

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Example 33

Sketch the graph of y = 8 + 2x2 - x4

(a) Axes If x = 0 then y = 8 (0,8)If y = 0 then 8 + 2x2 - x4 = 0

Graph cuts axes at (0,8) , (-2,0) and (2,0)

Let u = x2 so u2 = x4

Equation is now 8 + 2u - u2 = 0

(4 - u)(2 + u) = 0

(4 - x2)(2 + x2) = 0

or (2 + x) (2 - x)(2 + x2) = 0So x = -2 or x = 2 but x2 -2


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(b) SPs SPs occur where dy/dx = 0

So 4x - 4x3 = 0 4x(1 - x2) = 0

4x(1 - x)(1 + x) = 0 x = 0 or x =1 or x = -1

Using y = 8 + 2x2 - x4

when x = 0 then y = 8

when x = -1 then y = 8 + 2 - 1 = 9 (-1,9) when x = 1 then y = 8 + 2 - 1 = 9

(1,9)


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x -1 0 1

dy/dx +

So (0,8) is a min TP while (-1,9) & (1,9) are max TPs .


+- -0 00

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Cuts y – axis at 8Cuts x – axis at -2 and 2

(c) Large valuesUsing y = -

x4

as x + then y -as x - then y -

Sketch is

X

Y

-2 28

(-1,9) (1,9)

y = 8 + 2x2 - x4

Max TP’s at (-1,9) (1,9)

Curve Sketching

Summarising

Higher Outcome 3

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Max & Min on Closed Intervals

In the previous section on curve sketching we dealt with the entire graph.

In this section we shall concentrate on the important details to be found in a small section of

graph.Suppose we consider any graph between the points

where x = a and x = b (i.e. a x b)

then the following graphs illustrate where we would expect to find the maximum & minimum values.

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y =f(x)

Xa b

(a, f(a))

(b, f(b)) max = f(b) end point

min = f(a) end point


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x

y =f(x)(b, f(b))

(a, f(a))

max = f(c ) max TP

min = f(a) end point

a b

(c, f(c))

c NB: a < c < b


Higher Outcome 3

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y =f(x)

xa b

c

(a, f(a))

(b, f(b))

(c, f(c))

max = f(b) end point

min = f(c) min TP

NB: a < c < b


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From the previous three diagrams we should be able to see that the maximum and minimum values of f(x) on the closed interval a x b can be found either at the end points or at a stationary point between the two end points

Example 34

Find the max & min values of y = 2x3 - 9x2 in the interval where -1 x 2.

End points If x = -1 then y = -2 - 9 = -11

If x = 2 then y = 16 - 36 = -20


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Stationary pointsdy/dx = 6x2 - 18x = 6x(x - 3)

SPs occur where dy/dx = 0

6x(x - 3) = 0

6x = 0 or x - 3 = 0

x = 0 or x = 3

in interval not in interval

If x = 0 then y = 0 - 0 = 0

Hence for -1 x 2 , max = 0 & min = -20


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Using function notation we can say that

Domain = {xR: -1 x 2 }

Range = {yR: -20 y 0 }


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Optimization

Note: Optimum basically means the best possible.

In commerce or industry production costs and profits can often be given by a mathematical

formula.

Optimum profit is as high as possible so we would look for a max value or max TP.

Optimum production cost is as low as possible so we would look for a min value or min TP.

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OptimizationHigher Outcome 3

Problem

Practical exercise on optimizing volume.

Graph

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Example 35Higher Outcome 3

OptimizationQ. What is the maximum

volume

We can have for the given dimensions

A rectangular sheet of foil measuring 16cm X 10 cm has four small squares each x cm cut from each

corner. 16cm

10cm

x cm

NB: x > 0 but 2x < 10 or x < 5ie 0 < x < 5

This gives us a particular interval to consider !

x cm

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(16 - 2x) cm

(10 - 2x) cmx cm

The volume is now determined by the value of x so we can write

V(x) = x(16 - 2x)(10 - 2x)

= x(160 - 52x + 4x2)

= 4x3 - 52x2 +160x

We now try to maximize V(x) between 0 and 5

Optimization

By folding up the four flaps we get a small cuboid

Higher Outcome 3

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5End Points

V(0) = 0 X 16 X 10 = 0

V(5) = 5 X 6 X 0 = 0

SPs V '(x) = 12x2 - 104x + 160

= 4(3x2 - 26x + 40)

= 4(3x - 20)(x - 2)


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ie 4(3x - 20)(x - 2) = 0

3x - 20 = 0 or x - 2 = 0

ie x = 20/3 or x = 2

not in intervalin interval

When x = 2 then

V(2) = 2 X 12 X 6 = 144

We now check gradient near x = 2


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x 2

V '(x) +

Hence max TP when x = 2

So max possible volume = 144cm3

Nature


- 0

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Example 36

When a company launches a new product its share of the market after x months is calculated by the formula

So after 5 months the share is

S(5) = 2/5 – 4/25 = 6/25

Find the maximum share of the market

that the company can achieve.

(x 2)2

2 4( )S x

x x


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End points S(2) = 1 – 1 = 0

There is no upper limit but as x S(x) 0.

SPs occur where S (x) = 0

3 2

8 2'( ) 0S x

x x

1 22

2 4( ) 2 4 S x x x

x x

1 2 2 32 3

2 8'( ) 2 4 2 8S x x x x x

x x


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8x2 = 2x3

8x2 - 2x3 = 0

2x2(4 – x) = 0

x = 0 or x = 4

Out with interval In interval

We now check the gradients either side of 4

3 2

8 2'( ) 0S x

x xrearrange


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x 4

S (x)

S (3.9 ) = 0.00337…

S (4.1) = -0.0029…

Hence max TP at x = 4

And max share of market = S(4) = 2/4 – 4/16

= 1/2 – 1/4

= 1/4

Optimization

Nature

Higher Outcome 3

+ -0

Differentiationof Polynomials

f(x) = axn

then f’x) = anxn-1

Derivative

= gradient

= rate of change

Graphs

f’(x)=0

54

2( )

3f x

x

5

42( )

3

xf x

1

4

4

552'( )

3 6

xf x

x

1

2( ) 2 1f x x x 3 1

2 2( ) 2f x x x 1 1

2 21

'( ) 32

f x x x

1

21

'( ) 32

f x xx

f’(x)=0

Stationary Pts

Max. / Mini Pts

Inflection Pt

Nature Table-1 2 5+ 0 -

xf’(x)

MaxGradient at a point

Equation of tangent line

Straight Line

Theory

Leibniz Notation

'( )dy

f xdx

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Are you on Target !

• Update you log book

• Make sure you complete and correct

ALL of the Differentiation 1

questions in the past paper booklet.

Outcome 3

Documents

Www.mathsrevision.com Higher Unit 1 Finding the gradient for a polynomial Differentiating Brackets ( Type 1 ) Differentiating Harder