Applications of Differentiation (CIE IGCSE Maths: Extended)

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
  • stationary means the gradient is zero (flat) at these points

 

• 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 substituting the x-coordinate of a turning point into the gradient function (aka derived function or derivative) will give an output of zero

 

How do I find the coordinates of a turning point?

• STEP 1:  Solve the equation of the gradient function (derivative / derived function) equal to zero

  ie. solve 

  This will find the x-coordinate of the turning point
• STEP 2:  To find the y-coordinate of the turning point, substitute the x-coordinate into the equation of the graph, y = ...
  • not into the gradient function

What are the different types of stationary points?

• You can see from the shape of a curve the different types of stationary points
• You need to know two different types of stationary points (turning points):
  • 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 use graphs to classify which is a maximum point and which is a minimum point?

• You can see and justify which is a maximum point and which is a minimum point from the shape of a curve...
  
  • ... either from a sketch given in the question
  • ... or a sketch drawn by yourself

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

  • ... or from 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

 

How do I use the second derivative to classify which is a maximum point and which is a minimum point?

• The second derivative, , is the derivative-of-the-derivative
  • differentiate the expression for   to get the expression for
    • this is the same as differentiating the original equation for twice
• A quick algebraic test to find out the turning point (that does not require sketching) is as follows
  • If the stationary point is at , substitute into the expression for to get a numerical value...
    • ...if this value is negative, , the stationary point is a maximum point
    • ...if this value is positive,  , the stationary point is a minimum point
    • If the value is zero,  , then unfortunately the test has failed
      • a zero means it could be any out of a max, min, or other types (stationary points-of-inflection)
      • go back to sketching the graph to classify the stationary point(s)