Remind me of the rule for differentiating powers

• If?y = kxⁿ?then?dy/dx = knx^n-1

What is a turning point?

• The easiest way to think of a turning point is that it is a point at which a curve?changes?from?moving?upwards?to?moving?downwards, or vice versa
• Turning points are also called?stationary?points
• Ensure you are familiar with?Differentiation – Basics?before moving on

• At a?turning?point?the?gradient?of the curve is?zero
  • If a?tangent?is drawn at a turning point it will be a?horizontal?line
  • Horizontal lines have a gradient of zero

   

• This means at a turning point the?derivative?or?gradient?function?equals zero

How do I know if a curve has turning points?

• You can see from the shape of a curve whether it has turning points or not
• At IGCSE, two types of turning point are considered:
  • Maximum?points?– this is where the graph reaches a “peak”
  • Minimum?points?– this is where the graph reaches a “trough”

   

• These are sometimes called?local?maximum/minimum points as other parts of the graph may still reach higher/lower values

How do I find the coordinates of a turning point?

• STEP 1? Solve the equation of the gradient function (derivative) equal to zeroie. solve?dy/dx = 0

  This will find the?x-coordinate of the turning point

• STEP 2? To find the?y-coordinate substitute the?x-coordinate into the equation of the graph 

  ie. substitute?x?into “y = ...”

How do I know which point is a maximum and which is a minimum?

• The easiest way to do this is to?recognise?the?shape?of the?curve
  • ... either from a given?sketch?of the?curve
  • ... a?sketch?of the?curve?you can quickly draw yourself 

    (You may even be asked to do this as part of a question)

  • ... the?equation?of the?curve

   

• For?parabolas?(quadratics) it should be obvious ...
  • ... a?positive?parabola (positive?x²?term) has a?minimum?point
  • ... a?negative?parabola (negative?x²?term) has a?maximum?point

   

• Cubic?graphs are also easily recognisable ...
  • ... a?positive?cubic has a?maximum?point on the?left,?minimum?on the?right
  • ... a?negative?cubic has a?minimum?on the?left,?maximum?on the?right