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Calculus 3.1 Differentiate functions and use derivatives to solve problems

Achievement criteria

The level of achievement that you reach is decided by the questions that you answer correctly.

Achievement

• You need to differentiate functions and use differentiation to solve problems which could include the following types:
  
  • power
  • exponential (base e only)
  • logarithmic (base e only)
  • trigonometric (including reciprocal functions).
• Differentiation of functions may include the use of the chain rule and product and quotient rules for expanded polynomials:
  
  • chain rule with polynomials in expanded form such as
    • (x² + 5x)⁷
    • 
    • 7e^2x
    • In(2x + 7)
    • sin 5x
  • product and quotient rules for combinations of straightforward functions, at least one of which is in expanded polynomial form, such as:
    • x².sinx
    • (2x³ – 4).e^x
    • 
• Problems may include:
  
  • optimisation of a given function
  • rates of change which may involve kinematics
  • finding equations of normals tangents
  • locating maxima and minima of polynomial functions
• Calculators may be used but you must be able to demonstrate the skill of differentiation.

Achievement with Merit

• You need to reach Achievement.
• You need to demonstrate knowledge of concepts and techniques of differentiation which may be selected from the following types:
  
  • differentiation from first principles of polynomial functions of degree ≤ 3
  • sketching the graph of a derived function from a given graph
  • differentiation of combinations of functions including:
    
    • products such as (3x² – 7)³(4x + 8) or x²sin x
    • quotients, such as
    • implicit differentiation such as x² + 3y² = 15
    • parametric differentiation for first derivative only
  • identifying features of given graphs involving a selection from:
    
    • limits
    • differentiability
    • discontinuity
    • gradients
    • concavity
    • turning points
    • points of inflection
  • sketching graphs to demonstrate knowledge of the above features.
• You need to solve problems which may involve:
  
  • interpretation of features of graph
  • optimisation
  • modelling of a situation
  • related rates of change, involving two directly related rates.
  
  

  Achievement with Excellence

  • You need to reach Achievement with Merit.
  • You need to solve complex differentiation problems which may involve:
    
    • establishing a model
    • a proof
    • testing the nature of turning points and verifying points of inflection
    • related rates of change involving more than two related rates, for example dh/dt = dh//dv.dv/dt
    • the use of higher derivatives including parametric and implicit differentiation techniques.
  
  

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