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Programme 9: Differentiation Applications 2

This programme covers differentiation of inverse trigonometric functions, derivatives of inverse hyperbolic functions, second derivatives, maximum and minimum values, and points of inflexion.

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Programme 9: Differentiation Applications 2

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  1. PROGRAMME 9 DIFFERENTIATION APPLICATIONS 2 ((edited by JAB))

  2. Programme 9: Differentiation applications 2 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

  3. Programme 9: Differentiation applications 2 Differentiation of inverse trigonometric functions Derivatives of inverse hyperbolic functions Second derivatives Maximum and minimum values Points of inflexion

  4. (moved here from) Programme F11: Differentiation Second derivatives Notation The derivative of the derivative of y is called the second derivative ofy and is written as: So, if: then

  5. 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

  6. Programme 9: Differentiation applications 2 Differentiation of inverse trigonometric functions Derivatives of inverse hyperbolic functions Second derivatives Local maxima, local minima and points of inflexion (partial treatment)

  7. (slide added by JAB) Local Maximum and Local Minima 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.

  8. Programme 9: Differentiation applications 2 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.

  9. Programme 9: Differentiation applications 2 Local maximum and minimum values 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 2nd derivative is zero. E.g., consider y = x4

  10. Programme 9: Differentiation applications 2 Maximum and minimum values 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

  11. (slide added by JAB) Maximum and minimum values, contd. 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.

  12. Programme 9: Differentiation applications 2 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.

  13. Programme 9: Differentiation applications 2 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.

  14. Programme 9: Differentiation applications 2 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.

  15. Programme 9: Differentiation applications 2 Differentiation of inverse trigonometric functions Derivatives of inverse hyperbolic functions Second derivatives Maximum and minimum values Points of inflexion

  16. Programme 9: Differentiation applications 2 Differentiation of inverse trigonometric functions (see pp.335/6 for such functions) NB: sin-1x is also written arcsin x, tan-1x as arctan x etc. If then Then:

  17. Programme 9: Differentiation applications 2 Differentiation of inverse trigonometric functions Similarly:

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