In Unit 2 students focus on the study of simple transcendental functions and the calculus of simple algebraic functions. The areas of study are ‘Functions and graphs’, ‘Algebra’, ‘Calculus’, and ‘Probability and statistics’. At the end of Unit 2, students are expected to have covered the material outlined in each area of study. Material from the ‘Functions and graphs’, ‘Algebra’, ‘Calculus’, and ‘Probability and statistics’ areas of study should be organised so that there is a clear progression of skills and knowledge from Unit 1 to Unit 2 in each area of study. In undertaking this unit, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, graphs, differentiation and anti-differentiation with and without the use of technology. They should have facility with relevant mental and by-hand approaches to estimation and computation. The use of numerical, graphical, geometric, symbolic and statistical functionality of technology for teaching and learning mathematics, for working mathematically, and in related assessment, is to be incorporated throughout the unit as applicable.

OUTCOME

For this unit the student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass all of the areas of study for the unit.

Outcome 1

On completion of this 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. To achieve this outcome the student will draw on knowledge and skills outlined in all the areas of study.

Key skills

sketch by hand graphs of the sine, cosine and exponential functions, and simple transformations of these to the form Af (bx) + c, and sketch by hand graphs of loga(x) and the tangent function
draw graphs of circular, exponential and simple logarithmic functions over a given domain and identify and discuss key features and properties of these graphs
describe the effect of transformations of the plane on the graphs of the sine, cosine, tangent and exponential functions, and apply matrices to transformations, by hand in simple cases
solve simple equations over a specified interval related to circular, exponential and simple logarithmic functions using graphical, numerical and analytical approaches
use Newton’s method to find a numerical approximation to a root of a cubic polynomial function
recognise characteristics of data which suggest that a circular or exponential function is an appropriate model for the data
evaluate limiting values of a function
use a variety of approaches (numerical, graphical, first principles and by rule) to find the value of the derivative of a function at a given point
use first principles to find by hand the derivative of simple polynomial functions up to degree 3
find by hand the derivative function and an anti-derivative function for a simple power function, or a polynomial function of low degree
use derivatives to assist in the sketching of graphs of simple polynomial functions and to solve simple maximum and minimum optimisation problems
find a family of anti-derivative functions for a given power or polynomial function, and determine a specific antiderivative given a boundary condition
apply counting techniques to solve probability problems and calculate probabilities for compound events, by hand in simple cases.

Outcome 2

On completion of this unit the student should be able to apply mathematical processes in non-routine contexts, including situations requiring problem-solving, modelling or investigative techniques or approaches, and analyse and discuss these applications of mathematics. To achieve this outcome the student will draw on knowledge and skills outlined in one or more areas of study.

Key skills

specify the relevance of key mathematical content from one or more areas of study to the investigation of various questions in a given context
develop mathematical formulations of specific and general cases used to derive results for analysis within a given context for investigation
use a variety of techniques to verify results
make inferences from analysis and use these to draw valid conclusions related to a given context for investigation
communicate conclusions using both mathematical expression and everyday language, in particular, the interpretation of mathematics with respect to the context for investigation.

Outcome 3

On completion of this unit the student should be able to select and 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. To achieve this outcome the student will draw on knowledge and skills outlined in all the areas of study.

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 which identify examples or counter-examples for propositions
produce tables of values, families of graphs and collections of other results using technology, which support 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.