IB Maths. Turning points. First derivative test

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IB Maths. Turning points. First derivative test

  1. 1. By the end of the lesson you will be able to: • Use the derivative of a function to find      maximum and minimum points. • Use the second derivative to test the nature of a  stationary point and/or point of inflexion.
  2. 2. increasing increasing decreasing stationary stationary http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html
  3. 3. •   •   •   •   A B P Q At  A   ⇒  f '(x) > 0 ⇒ f is increasing           At  P   ⇒  f '(x) < 0 ⇒ f is decreasing           At  B and Q   ⇒  f '(x) = 0 ⇒ B and Q are  stationary points.          f '  = 0 f '  < 0 f '  > 0 f '  > 0 f ' = 0
  4. 4. If the derivative is positive then the function is increasing. If the derivative is negative then the function is decreasing.  
  5. 5. f '( a ) = 0   ⇒ (a, f(a))   is a stationary point •   •   •   •   A B P Q A point on a curve at which the gradient is zero is called a stationary point. At a stationary point, the tangent to the curve is horizontal.
  6. 6. •   •   •   Local Maximum point  f '   > 0  f '   = 0  f '   < 0 P To the left of P At point P To the right of P  f '   > 0  f '   < 0 f '   = 0 P  is a local maximum point
  7. 7. Local Minimum point  f '   > 0  f '   = 0  f '   < 0 To the left of P At point P To the right of P  f '   > 0 f '   < 0  f '   = 0 P  is a local minimum point P •   •  •   Maximum and minimum points are also called  turning points.
  8. 8. Point of inflexion •   •   •   f ' =0 f ' > 0 f ' > 0P •   •   •   f ' = 0 f ' < 0 f ' < 0 P f ' ( a) = 0 but  f ' has the same sign to the right  and left of a,  a is called a horizontal point of  inflexion.(because the tangent is horizontal at P) Point of inflection.ggb
  9. 9. •   non­horizontal point of inflexion ( tangent is not  horizontal) f ' < 0 f ' < 0 tangent •  
  10. 10. Find the coordinates of the stationary points on the  curve y= x3 +3x2 +1 and determine their nature.
  11. 11. y= x3 +3x2 +1 
  12. 12. Find the coordinates of the stationary points on the  curve y= x4  ­ 4 x3   and determine their nature.
  13. 13. y= x4  ­ 4 x3  
  14. 14. Attachments Point of inflection.ggb

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