Edexcel IGCSE Maths 复习笔记 3.11.2 Differentiation - Turning Points

Edexcel IGCSE Maths 复习笔记 3.11.2 Differentiation - Turning Points

Remind me of the rule for differentiating powers

  • If y = kxn then dy/dx = knxn-1

 

3.11.2-Rule-for-IGCSE-Diff 

 

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

 

3.11.2-Turn-Pts-Notes-fig1 

 

  • 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

 

3.11.2-Turn-Pts-Notes-fig2

 

 

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

     

 

3.11.2-Turn-Pts-Notes-fig3

  • 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 = ...

 

3.11.2-Turn-Pts-Notes-fig4 

 

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

     

 

3.11.2-Turn-Pts-Notes-fig5

 

 

  • Cubic graphs are also easily recognisable ...
    • ... a positive cubic has a maximum point on the leftminimum on the right
    • ... a negative cubic has a minimum on the leftmaximum on the right

     

 

3.11.2-Turn-Pts-Notes-fig6 

Exam Tip

Read questions carefully – sometimes only the x-coordinate of a turning point is required.Differentiating accurately is crucial in leading to equations you can work with and solve.

Worked Example

3.11.2-Turn-Pts-Example-fig1-qu

Worked Example

3.11.2-Turn-Pts-Example-fig2-sol

转载自savemyexam

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