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3.7 FSMQ Calculus (9998)

Calculus (9998)

This qualification has been developed to allow you to demonstrate your ability to use

  • differentiation
  • integration
  • differential equations

to analyse, make sense of and describe real world situations and to solve problems. You will also investigate the use of numerical methods to find gradients and evaluate integrals and compare these with analytic methods.

Before you start this qualificationYou must:This includes:
 be able to use algebraic methods to rearrange and solve linear and quadratic equationsSolution of a quadratic equation by at least one of the following methods:
  • use of a graphics calculator
  • use of formula

    (which must be memorised)

  • completing the square
Solution by factorisation will be acceptable where the quadratic factorises.
 have knowledge of basic functions and how geometric transformations can be applied to them using
  • transformations by the vector

    and by the vector

  • stretches of scale factor with the invariant line and with the invariant line
being familiar with graphs and functions of:
  • powers of ,

    eg


  • quadratics:

  • trigonometric functions:

  • exponential functions:

    ( positive or negative)

  • logarithmic functions:

Using calculators and computers

Using calculators and computers

When carrying out calculations, you may find the use of a standard scientific calculator sufficient.

You should learn to use your calculator effectively and efficiently. This will include learning to use:

  • memory facilities
  • function facilities (e.g., )

It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.

Whenever you use a calculator you should record your working as well as the result.

Understanding and using differentiation

Understanding
and
using
differentiation
You should learn to:This includes:
 understand and calculate gradient at a point, , on a function using the numerical approximation:

where is small

understanding how to improve the calculation of gradient at a point by using a smaller interval,
 understand and interpret gradients in terms of their physical significance 
 understand and use the correct units with which to measure gradients /rates of change  
 sketch graphs of gradient functions
  • curves that you do not know as functions
  • curves defined as functions
 identify the key features of gradient functions in terms of the gradient of the original function
  • zeros of gradient functions linking to local turning points
 understand how can be used to generate a gradient function 
 differentiate functions
  • using notations and
  • polynomials
  • trigonometric functions using radians
  • exponential functions
 Differentiate
  • sums and differences of functions
  • functions multiplied by a constant
  • products of functions
 
 find the second derivatives of functionsusing notations and
 identify the key features of a second derivative
  • linking positive values to increasing gradient
  • linking negative values to decreasing gradient
  • linking zero values to points of inflexion
 Applications of differentiation to gradients, maxima and minima and stationary points, increasing and decreasing functions
  • Application to determining maxima and minima
  • understanding the importance of the second derivative and its value at such points
  • understanding that zero values of the second derivative can occur at maximum and minimum points as well as points of inflexion

Understanding and using integration

Understanding
and
using
integration
You should learn to:This includes:
 estimate areas under graphs of functions using numerical methods
  • the trapezium rule
  • understanding how to improve your calculation of the area under a graph by using a smaller interval.
 understand and find areas under curves, between and using ,  
 understand integration as the reverse process of differentiation 
 understand and determine indefinite integrals of functions
  • (including and fractional)
  • ( positive or negative)
  • sums, differences and constant multiples of these using a constant of integration
  Integration by inspection and

by one use of integration by parts

 

  • eg: ,
 
  • eg: ,,
 understand the idea of constant of integration and be able to calculate this in known situations 
 be able to determine definite integrals for functions
  • those functions defined above

Understanding and using differential equations

Understanding
and
using
differential
equations
You should learn to:This includes:
 find families of solutions to first order differential equations with separable variables
  • find particular solutions when boundary conditions are given