VCE Specialist Mathematics Units 3 and 4 – All Topics

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Description

1. Functions and Graphs
2. algebra
3. Calculus
4. Vectors
5. Mechanics
6. Probability and Statistics




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Outcome 1

On the completion of each unit, the student should be able to define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures.

Key knowledge
• functions and relations, the form of their sketch graphs and their key features, including asymptotic behaviour
• complex numbers, cartesian and polar forms, operations and properties and representation in the complex plane
• geometric interpretation of vectors in the plane and of complex numbers in the complex plane
• specification of curves in the complex plane using complex relations
• techniques for finding derivatives of explicit and implicit functions, and the meaning of first and second derivatives of a function
• techniques for finding anti-derivatives of functions, the relationship between the graph of a function and the graph of its anti-derivative functions, and graphical interpretation of definite integrals
• analytical, graphical and numerical techniques for setting up and solving equations involving functions and relations
• simple modelling contexts for setting up differential equations and associated solution techniques, including
numerical approaches and representation of direction(slope) fields
• the definition and properties of vectors, vector operations, the geometric representation of vectors and the geometric interpretation of linear dependence and independence
• standard contexts for the application of vectors to the motion of a particle and to geometric problems
• techniques for solving kinematics problems in one and two dimensions
• Newton’s laws of motion and related concepts
• the distribution of sample means
• linear combinations of independent random variables
• hypothesis testing for a sample mean.

Key skills
• sketch graphs and describe behaviour of specified functions and relations with and without the assistance of technology, clearly identifying their key features and using the concepts of first and second derivatives
• perform operations on complex numbers expressed in cartesian form or polar form and interpret them geometrically
• represent curves on an argand diagram using complex relations
• apply implicit differentiation, by hand in simple cases
• use analytic techniques to find derivatives and anti-derivatives by pattern recognition, and apply anti-derivatives to evaluate definite integrals
• set up and evaluate definite integrals to calculate arc lengths, areas and volumes
• set up and solve differential equations of specified forms
• represent and interpret differential equations by direction(slope) fields
• perform operations on vectors and interpret them geometrically
• apply vectors to the motion of a particle and to geometric problems
• solve kinematics problems using a variety of techniques
• set up and solve problems involving Newton’s laws of motion
• apply a range of analytical, graphical and numerical processes to obtain solutions (exact or approximate) to equations
• set up and solve problems involving the distribution of sample means
• construct approximate confidence intervals for sample means
• undertake a hypothesis test for a mean of a sample from a normal distribution or a large sample.

Outcome 2

On the completion of each unit, the student should be able to apply mathematical processes, with an emphasis on general cases, in non-routine contexts, and analyse and discuss these applications of mathematics.

Key knowledge
• the key mathematical content from one or more areas of study relating to a given application context
• specific and general formulations of concepts used to derive results for analysis within a given application context
• the role of examples, counter-examples and general cases in developing mathematical analysis
• the role of proof in establishing a general result
• the use of inferences from analysis to draw valid conclusions related to a given application context.

Key skills
• specify the relevance of key mathematical content from one or more areas of study to the investigation of various questions related to a given context
• give mathematical formulations of specific and general cases used to derive results for analysis within a given application context
• develop functions as possible models for data presented in graphical form and apply a variety of techniques to decide which function provides an appropriate model
• use a variety of techniques to verify results
• establish proofs for general case results
• make inferences from analysis and use these to draw valid conclusions related to a given application context
• communicate conclusions using both mathematical expression and everyday language, in particular in relation to a given application context.

Outcome 3

On completion of each unit, the student should be able to select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.

Key knowledge
• the exact and approximate specification of mathematical information such as numerical data, graphical forms and general or specific forms of solutions of equations produced by technology
• domain and range requirements for specification of graphs of functions and relations, when using technology
• the role of parameters in specifying general forms of functions and equations
• the relation between numerical, graphical and symbolic forms of information about functions and equations and the corresponding features of those functions and equations
• similarities and differences between formal mathematical expressions and their representation by technology
• the selection of an appropriate functionality of technology in a variety of mathematical contexts.

Key skills
• distinguish between exact and approximate presentations of mathematical results produced by technology, and interpret these results to a specified degree of accuracy
• use technology to carry out numerical, graphical and symbolic computation as applicable
• produce results using a technology that identifies examples or counter-examples for propositions
• produce tables of values, symbolic expressions, families of graphs and collections of other results using technology, which supports general analysis in problem-solving, investigative and modelling contexts
• use appropriate domain and range specifications to illustrate key features of graphs of functions and relations
• identify the relation between numerical, graphical and symbolic forms of information about functions and equations and the corresponding features of those functions and equations
• specify the similarities and differences between formal mathematical expressions and their representation by technology, in particular, equivalent forms of symbolic expressions
• select an appropriate functionality of technology in a variety of mathematical contexts, and provide a rationale for these selections
• apply suitable constraints and conditions, as applicable, to carry out required computations
• relate the results from a particular technology application to the nature of a particular mathematical task (investigative, problem solving or modelling) and verify these results
• specify the process used to develop a solution to a problem using technology, and communicate the key stages of mathematical reasoning (formulation, solution, interpretation) used in this process.

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