Year 13 Calculus

Calculus: What is it?


















 I can...
  • Manipulate equations based on:
− conversion between polar and rectangular forms of real and 
complex numbers

− simplification of sums, differences, products and quotients of
surds or complex numbers expressed in rectangular form

− simplification of products or quotients of complex numbers
expressed in polar form

− use of De Moivre’s theorem in the simplification of
expressions such as (5cis Pi/2)^10
  • Manipulate different types of equations:
- Quadratic
-Cubic (rational roots only)
-Exponential such as 2^(3x+1)=5
-Logarithmic such as ln(x+5)=1.34 (any base)
  • Use the remainder and factor theorem
  • Complete a square
I can...
  • Solve irrational equations such as x+2=2sqr(x)
  • Solve cubic equations with one integer root and two complex roots
  • Solve equations of the form z^n = a, z^n=r cis(theta)

I can...
  • Solve a problem that requires an extended chain of reasoning
  • Prove a theorem
  • Solve a complicated equation


I can...

  • Solve equations involving:
    • solving systems of three linear equations in three variables, where there is a unique solution (this may involve re-arrangement of equations and/or interpreting solutions).
    • solving a non-linear equation using the Newton-Raphson method with a given starting value, or the bisection method with a given starting interval (Newton-Raphson method includes derivatives of polynomials only)
    • optimising an objective function for a situation requiring techniques of linear programming, where the constraints and the objective function for the problem are given.

I can...

  • Solve problems involving:
    • optimising an objective function for a linear programming problem, where you are expected to form your own constraints and objective function, and round the solution in relation to the context
    • using a suitable method to find an approximate solution to a non-linear equation (graphical, table, graphics calculator etc)
    • finding appropriate solutions to a non-linear equation using either the Newton-Raphson method or the bisection method to improve the approximation to a stated precision or for a specified number of iterations. Derivatives of functions other than polynomials will be given
    • forming and solving a 3x3 system of linear equations.
I can...
  • Analyse or interpret the outcome, or the process used to solve equations or linear programming problems by:
    • discussing consistency or non-independence of 3x3 systems of linear equations, including geometric representatios
    • determine the effect of varying the constrains or objective function of a linear programming problem
    • considering the possibility of multiple solutions to a linear programming problems
    • giving advantages and disadvantages of the Newton-Raphson method or the bisection method for the problem
    • giving geometric description of the Newton-Raphson method or the bisection method.

 I can...

  •  Solve straightforward problems that involve trigonometric funcstions like:
  • AsinB(x + C) + D
  • AcosB(x + C) + D
  • AtanB(x + C) + D,       where C or D may be zero. 
  •  Solve problems that require knowledge of amplitude, period and frequency.
  •  Solve equations such as:
  • AsinB(x + C) = K 
  •  AsinBx = K 
  •  AcosB(x + C) = K 
  •  AcosBx = K 
  •  AtanB(x + C) = K 
  •  AtanBx = K
I can...
  • form an equation for the model and use the model to solve problems such us: 
    • y = AsinB(x + C) + D 
    •  y = AcosB(x + C) + D 
    •  y = AtanB(x + C) + D  

  • Work out the constants A, B, C from a worded problem
  • Manipulate trigonometric functions including
      • reciprocal relationships 
      • Pythagorean identities
      • compound angle formulae 
      • double angle formulae 
      • sum and product formulae , and combinations of these. 
  • Solve equations providing a general solution or the solution within a specific domain.

 I can...
  • Solve problems that will require a chain of reasoning or may involve
    • a proof 
    • developing a formula from a given starting points
    • rewriting a trigonometric expression in terms of a single trigonometric function
    • identifying and rectifying a flaw in reasoning
    • evaluation of the model (limitations, improvements, long-term accuracy)
    • solving more complex equations 
    • solving 3-D trigonometric problems.
  • Candidates will be required to choose and apply appropriate trig relationships

 I can...
  • Differentiate functions such us:
      • polynomial ax^n
      • exponential (base e only)
      • logarithmic (base e only)
      • trigonometric, including reciprocals
      • (x^2+5x)^7
      • (2x-3)^(1/3)
      • 7e^(2x)
      • ln(2x+7)
      • sin(5x)
  • Solve problems involving:

-optimization of a given function
-related rates of change, involving two directly related rates
-finding equations of tangents
-locating maxima and minima of polynomial functions

I can...
  • Demonstrate knowledge of the following techniques:
      • differentiation from first principles of polynomial functions of degree less 3
      • differentiate products, quotients, implicit and parametric functions
  • Identify feature of a given graph including:
      • limits
      • differentiability
      • discontinuity
      • gradients
      • concavity
  • Sketch graphs of polynomials of degree more than 3 and identify features such as:
      • turning points
      • points of inflection
      • concavity.
  • Solve diffeerntiation problems involving:
      • interpretation of features of graph
      • optimisation
      • related rates of change, which may involve more than two directly related rates.
 I can...
  • Solve problems involving a combination of different techniques such as:
      • establishing a model
      • proving a theorem

 I can...
  • Integrate functions such as:
      • polynomials a{x ^ n , including negative powers of n and n=-1
      • exponentials ae^(bx+c), base e only
      • trig functions
      • rational functions (ax+b)/x
  • Solve problems involving the following techniques:
      • rates of change, eg, kinematics
      • differential equations of the forms y'=f(x)
      • separating variables that are easy to separate
      • finding areas under graphs
      • finding volumes of solids of revolution around x
      • finding areas using Simpson's Rule or the Trapezium Rule
  • Solve problems given the diagram for the area and volume

 I can...
  • Integrate functions that involve techniques such as:
    • products of trig functions
    • simple algebraic substitution
    • rational functions of the type f'(x)/f(x)
    • rational functions of the type (ax+b)/(cx+d)
  • Solve problems involving:
    • areas between graphs of polynomials
    • areas under graphs of combined 
    • volumes of solids of revolution formed by around x or y axis
    • rates of change problems including 
    • differential equations where required to write a differential including growth and decay, inflation
    • Newton's Law of Cooling and similar similar situations eg y'=ky
 I can...
  • Solve more complex integration problems involving techniques such as:
    • areas between graphs of functions, other than polynomials
    • volumes of solids of revolution formed by rotating around an axis parallel to either x or y 
    • differential equations involving more difficult manipulation