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

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 d 3 y/dx 3 and is the derivative of d 2 y/dx 2

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.

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 CAUTION: Local maxima and minima can also occur when the 2 nd derivative is zero. E.g., consider y = x 4

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

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 Maximum and minimum values, contd. (slide added by JAB) Better criteria (but still not absolutely complete ones) for local maxima and minima are: LOCAL MAXIMUM at point P: The derivative (slope) is zero at P and is DECREASING around it—i.e., the slope is passing DOWN THROUGH ZERO at P. LOCAL MINIMUM at point P: The derivative (slope) is zero at P and is INCREASING around it—i.e, the slope is passing UP THROUGH ZERO at P. It’s just that the second derivative being negative is the usual way in which the slope can be decreasing, and being positive is the usual way in which the slope can be increasing.

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.

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:

STROUD Worked examples and exercises are in the text Differentiation of inverse trigonometric functions Programme 9: Differentiation applications 2 Similarly: