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PROGRAMME 8

PROGRAMME 8. DIFFERENTIATION APPLICATIONS . Programme 8: Differentiation applications . Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion. Programme 8: Differentiation applications . Equation of a straight line

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PROGRAMME 8

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  1. PROGRAMME 8 DIFFERENTIATION APPLICATIONS

  2. Programme 8: Differentiation applications Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion

  3. Programme 8: Differentiation applications Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion

  4. Programme 8: Differentiation applications Equation of a straight line (1) The basic equation of a straight line is: where:

  5. Programme 8: Differentiation applications Equation of a straight line How about the equation of the line? Found it.

  6. Programme 8: Differentiation applications Equation of a straight line (2) Given the gradient m of a straight line and one point (x1, y1) through which it passes, the equation can be used in the form: Example:

  7. Programme 8: Differentiation applications Equation of a straight line Exercise 1 Found the equation of the straight line of: Line passing through (2, -3), gradient -2. Line passing through (5, 3), gradient 2.

  8. Programme 8: Differentiation applications Answers: 1 2.

  9. Programme 8: Differentiation applications Equation of a straight line (3) If the gradient of a straight line is m and the gradient of a second straight line is m1 where the two lines are mutually perpendicular then:

  10. Programme 8: Differentiation applications Exercise A point P has coordinates (4,3) and the gradient m of straight line through P is 2. Then there is a line perpendicularly through P. Found the equation of the line. Answer

  11. Programme 8: Differentiation applications Are these two straight line perpendicular each other?

  12. Programme 8: Differentiation applications Exercise 2

  13. Programme 8: Differentiation applications Answers

  14. Programme 8: Differentiation applications Further Example 1. 2.

  15. Programme 8: Differentiation applications Answer (1)

  16. Programme 8: Differentiation applications Answer (2)

  17. Programme 8: Differentiation applications Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion

  18. Programme 8: Differentiation applications Tangents and normals to a curve at a given point Tangent The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point: The equation of the tangent can then be found from the equation:

  19. Programme 8: Differentiation applications Tangents and normals to a curve at a given point Example

  20. Programme 8: Differentiation applications Tangents and normals to a curve at a given point Normal The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point: The equation of the normal (perpendicular to the tangent) can then be found from the equation:

  21. Programme 8: Differentiation applications Tangents and normals to a curve at a given point Example Found the normal of the last exercise!

  22. Programme 8: Differentiation applications 1 Tangents and normals to a curve at a given point Exercise

  23. Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion

  24. Programme 9: Differentiation applications Maximum and minimum values 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.

  25. Programme 9: Differentiation applications Maximum and minimum values Having located a stationary point it is necessary to identify it. If, at the stationary point

  26. Programme 9: Differentiation applications 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

  27. Programme 9: Differentiation applications Example

  28. Programme 8: Differentiation applications Equation of a straight line Tangents and normals to a curve at a given point Maximum and minimum values Points of inflexion

  29. Programme 9: Differentiation applications 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.

  30. Programme 9: Differentiation applications 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 points of inflexion.

  31. Programme 9: Differentiation applications 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.

  32. Thank you

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