# Year 13 Calculus

** 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**

** 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**