stroud worked examples and exercises are in the text programme 9 differentiation applications 2...

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Worked examples and exercises are in the text

PROGRAMME 9

DIFFERENTIATION APPLICATIONS 2

((edited by JAB))

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NOT IN TEST or EXAM but IMPORTANT

Differentiation of inverse trigonometric functions

Derivatives of inverse hyperbolic functions [REMOVED]

Second derivatives

Maximum and minimum values

Points of inflexion

Programme 9: Differentiation applications 2

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Derivatives of inverse hyperbolic functions

Second derivatives


Points of inflexion


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Second derivatives

Notation

(moved here from) Programme F11: Differentiation

The derivative of the derivative of y is called the second derivative of y and is written as:

So, if:

then

2

2

d dy d y

dx dx dx

4 3 2

3 2

22

2

5 4 7 2

4 15 8 7

12 30 8

y x x x x

dyx x x

dx

d yx x

dx

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Third etc. derivatives (added by JAB)

Can continue like that differentiating again and again.

E.g., the third derivative of y wrt x is notated as d3y/dx3

and is the derivative of d2 y/dx2

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Second derivatives

Local maxima, local minima and points of inflexion (partial treatment)


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Local Maximum and Local Minima

(slide added by JAB)

LOCAL MAXIMUM at point P:

Small enough deviations to the left and right of P always make y DECREASE.

LOCAL MINIMUM at point P:

Small enough deviations to the left and right of P always make y INCREASE.

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Stationary points


A stationary point is a point on the graph of a function y = f (x) where the rate of change is zero. That is where:

This can occur at a local maximum, a local minimum or a point of inflexion. Solving this equation will locate the stationary points.

0dy

dx

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Local maximum and minimum values


Having located a stationary point one can characterize it further. If, at the stationary point

2

2

2

2

0

the stationary point is a minimum

0

the stationary point is a maximum

d y

dx

d y

dx

CAUTION: Local maxima and minima can also occur when the 2nd derivative is zero. E.g., consider

y = x4

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If, at the stationary point

The stationary point may be:

a local maximum, a local minimum or a point of inflexion

The test is to look at the values of y a little to the left and a little to the right of the stationary point

2

20

d y

dx

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Maximum and minimum values, contd.

(slide added by JAB)

It’s just that the first derivative (slope) being zero and the second derivative being negative is the usual way in which the slope can be decreasing through zero, which is the usual way for there to be a local maximum,

and

the first derivative (slope) being zero and the second derivative being positive is the usual way in which the slope can be increasing through zero, which is the usual way for there to be a local minimum.

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Points of inflexion


A point of inflexion can also occur at points other than stationary points. A point of inflexion is a point where the direction of bending changes – from a right-hand bend to a left-hand bend or vice versa.

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Points of inflexion


At a point of inflexion the second derivative is zero. However, the converse is not necessarily true because the second derivative can be zero at points other than stationary points: see right hand of diagram.

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Points of inflexion


The test is the behaviour of the second derivative as we move through the point. If, at a point P on a curve:

and the sign of the second derivative changes as x increases from values to the left of P to values to the right of P, the point is a point of inflexion.

(Added by JAB:) The usual way for this to happen is for the third derivative to be non-zero.

(Added by JAB:) And all we’re saying overall is that a point of inflexion is a local maximum or minimum of the SLOPE, i.e. of the first derivative of y.

2

20

d y

dx

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Second derivatives


Points of inflexion


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(see pp.335/6 for such functions)

NB: sin-1 x is also written arcsin x, tan-1 x as arctan x etc.


If then

Then:

1siny x sin and so cos y

dxx y y

d

2

2

1

/

1

cos

1

1 sin

1

1

dy

dx dx dy

y

y

x

1

2

1sin

1

dx

dx x

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Similarly:

1

2

1cos

1

dx

dx x

12

1tan

1

dx

dx x

stroud worked examples and exercises are in the text programme 9 differentiation applications 2...

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