STROUD
Worked examples and exercises are in the text
PROGRAMME 9
DIFFERENTIATION APPLICATIONS 2
((edited by JAB))
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Maximum and minimum values
Points of inflexion
Programme 9: Differentiation applications 2
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
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
STROUD
Worked examples and exercises are in the text
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Local maxima, local minima and points of inflexion (partial treatment)
Programme 9: Differentiation applications 2
STROUD
Worked examples and exercises are in the text
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.
STROUD
Worked examples and exercises are in the text
Stationary points
Programme 9: Differentiation applications 2
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
STROUD
Worked examples and exercises are in the text
Local maximum and minimum values
Programme 9: Differentiation applications 2
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
STROUD
Worked examples and exercises are in the text
Maximum and minimum values
Programme 9: Differentiation applications 2
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
STROUD
Worked examples and exercises are in the text
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.
STROUD
Worked examples and exercises are in the text
Points of inflexion
Programme 9: Differentiation applications 2
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.
STROUD
Worked examples and exercises are in the text
Points of inflexion
Programme 9: Differentiation applications 2
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.
STROUD
Worked examples and exercises are in the text
Points of inflexion
Programme 9: Differentiation applications 2
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
STROUD
Worked examples and exercises are in the text
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Maximum and minimum values
Points of inflexion
Programme 9: Differentiation applications 2
STROUD
Worked examples and exercises are in the text
Differentiation of inverse trigonometric functions
(see pp.335/6 for such functions)
NB: sin-1 x is also written arcsin x, tan-1 x as arctan x etc.
Programme 9: Differentiation applications 2
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
STROUD
Worked examples and exercises are in the text
Differentiation of inverse trigonometric functions
Programme 9: Differentiation applications 2
Similarly:
1
2
1cos
1
dx
dx x
12
1tan
1
dx
dx x