# Year 13 Calculus

TOPIC ASSESSED

AS90638 COMPLEX NUMBERS

(5 CREDITS EXTERNAL)

SIMULTANEOUS EQ'S

(2 CREDITS INTERNAL)

AS90637 3.3 TRIGONOMETRY

(4 CREDITS INTERNAL)

AS90635 DIFFERENTIATION

(6 CREDITS EXTERNAL)

AS90636 INTEGRATION

(4 CREDITS EXTERNAL)

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:

-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

AS90644

ACHIEVE

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.

MERIT

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.

EXCELLENCE

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.

AS90637

ACHIEVE

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

MERIT

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.

EXCELLENCE

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

AS90635

ACHIEVE

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

AS90636

ACHIEVE

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

MERIT

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

EXCELLENCE

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