Mathematics is the study of order, relation and pattern
From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand realworld phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work. Mathematics Methods Level 4 provides the study of algebra, functions, differential and integral calculus, probability and statistics. These are necessary prerequisites for the study of Mathematics Specialised Level 4 and as a foundation for tertiary studies in disciplines in which mathematics and statistics have important roles, including engineering, the sciences, commerce and economics, health and social sciences.
Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand realworld phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.
Mathematics Methods Level 4 provides the study of algebra, functions, differential and integral calculus, probability and statistics. These are necessary prerequisites for the study of Mathematics Specialised Level 4 and as a foundation for tertiary studies in disciplines in which mathematics and statistics have important roles, including engineering, the sciences, commerce and economics, health and social sciences.
Mathematics Methods Level 4 aims to develop learners’:
On successful completion of this course, learners will be able to:
It is highly recommended that learners attempting this course will have successfully completed either of the courses Mathematics Methods – Foundation Level 3 or the AC Mathematics 10A subject with some additional studies in introductory calculus.
Assumed knowledge and skills are contained in both of these courses and will be drawn upon in the development of key knowledge and skills in Mathematics Methods.
Learners must have access to calculator algebraic system (CAS) graphics calculators and become proficient in their use. These calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems. Refer to 'What can I take to my exam?' for the current TASC Calculator Policy that applies to Level 3 and 4 courses.
The use of computer software is also recommended as an aid to students’ learning and mathematical development. A range of packages such as, but not limited to: Wolfram Mathematica, Microsoft Excel, Autograph, Efofex Stat, Graph and Draw are appropriate for this purpose.
This course has a complexity level of 4.
In general, courses at this level provide theoretical and practical knowledge and skills for specialised and/or skilled work and/or further learning, requiring:
This level 4 course has a size value of 15.
This course is made up of five (5) areas of study. While each of these is compulsory, the order of delivery is not prescribed.
These areas of study relate to Assessment Criteria 4–8. Assessment Criteria 1–3 apply to all five areas of study.
Whilst the areas of study may be addressed separately and in any order, much of the content is interrelated and a more integrated approach is recommended and encouraged. It is also recommended that, where possible, concepts be developed within a context of practical applications. Such an approach provides learners with mathematical experiences that are much richer than a collection of skills. Learners thereby have the opportunity to observe and make connections between related aspects of the course and the real world and to develop further some important abstract ideas.
FUNCTION STUDY
Learners will develop their understanding of the behaviour of a number of different functions.
This area of study will include:
CIRCULAR (TRIGONOMETRIC) FUNCTIONS
Learners will develop their understanding of a range of circular (trigonometric) functions.
This area of study will include:
CALCULUS
Learners will study the graphical treatment of limits and differentiability of functions of a single real variable, and the differentiation and integration of these functions.
This area of study will include:
Differential Calculus:
Integral Calculus:
PROBABILITY AND STATISTICS
Learners will study scenarios involving discrete and continuous random variables, their representation using tables, probability functions specified by rule and defining parameters (as appropriate); the calculation and interpretation of central measures and measure and spread; and statistical inference for sample proportions.
This area of study will include:
Criterionbased assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate endpoint of study. Although assessment – as part of the learning program – is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect endpoint achievement.
The standard of achievement each learner attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.
A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.
A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.
Providers offering this course must participate in quality assurance processes specified by TASC to ensure provider validity and comparability of standards across all awards. To learn more, see TASC's quality assurance processes and assessment information.
Internal assessment of all criteria will be made by the provider. Providers will report the learner’s rating for each criterion to TASC. TASC will supervise the external assessment of designated criteria which will be indicated by an asterisk (*). The ratings obtained from the external assessments will be used in addition to internal ratings from the provider to determine the final award.
The following processes will be facilitated by TASC to ensure there is:
Process – TASC gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.
Additionally, the Office of TASC may select to undertake scheduled audits of this course and its work requirements (Provider standards 1, 2, 3 and 4).
The external assessment for this course will comprise:
For further information, see the current external assessment specifications and guidelines for this course available in the Supporting Documents below.
The assessment for Mathematics Methods Level 4 will be based on the degree to which the learner can:
* = denotes criteria that are both internally and externally assessed
The learner:
Rating A  Rating B  Rating C 

presents work that conveys a logical line of reasoning that has been followed between question and answer  presents work that conveys a line of reasoning that has been followed between question and answer  presents work that shows some of the mathematical processes that have been followed between question and answer 
uses mathematical conventions and symbols correctly  uses mathematical conventions and symbols correctly  uses mathematical conventions and symbols. There may be some errors or omissions in doing so. 
presents work with the final answer clearly identified, and articulated in terms of the question as required  presents work with the final answer clearly identified  presents work with the final answer apparent 
uses correct units and includes them in an answer for routine and nonroutine problems  uses correct units and includes them in an answer for routine problems  uses correct units and includes them in an answer for routine problems 
presents detailed tables, graphs and diagrams that convey accurate meaning and precise information  presents detailed tables, graphs and diagrams that convey clear meaning  presents tables, graphs and diagrams that include some suitable annotations 
adds a detailed diagram to illustrate and explain a solution  adds a diagram to illustrate a solution  adds a diagram to a solution as directed 
ensures an appropriate degree of accuracy is maintained and communicated throughout a problem.  determines and works to an appropriate degree of accuracy.  works to an appropriate degree of accuracy as directed. 
The learner:
Rating A  Rating B  Rating C 

selects and applies an appropriate strategy, where several may exist, to solve routine and nonroutine problems  selects and applies an appropriate strategy to solve routine and simple nonroutine problems  identifies an appropriate strategy to solve routine problems 
interprets solutions to routine and nonroutine problems  interprets solutions to routine and simple nonroutine problems  describes solutions to routine problems 
explains the reasonableness of results and solutions to routine problems and nonroutine problems  describes the reasonableness of results and solutions to routine problems  describes the appropriateness of the results of calculations 
identifies and describes limitations of simple models  identifies and describes limitations of simple models  identifies limitations of simple models 
uses available technological aids in familiar and unfamiliar contexts  chooses to use available technological aids when appropriate to solve routine problems  uses available technological aids to solve routine problems. 
explores calculator techniques in familiar and unfamiliar contexts  explores calculator techniques in familiar contexts  
constructs and solves problems derived from routine and nonroutine scenarios.  constructs and solves problems derived from routine scenarios. 
The learner:
Rating A  Rating B  Rating C 

uses planning tools and strategies to achieve and manage activities within proposed times  uses planning tools to achieve objectives within proposed times  uses planning tools, with prompting, to achieve objectives within proposed times 
divides a task into appropriate subtasks  divides a task into subtasks  divides a task into subtasks as directed 
selects strategies and formulae to successfully complete routine and nonroutine problems  selects from a range of strategies and formulae to successfully complete routine problems  uses given strategies and formulae to successfully complete routine problems 
plans timelines and monitors and analyses progress towards meeting goals, making adjustments as required  plans timelines and monitors progress towards meeting goals  monitors progress towards meeting goals 
addresses all of the required elements of a task with a high degree of accuracy  addresses the elements of required tasks  addresses most elements of required tasks 
plans future actions, effectively adjusting goals and plans where necessary.  plans future actions, adjusting goals and plans where necessary.  uses prescribed strategies to adjust goals and plans where necessary. 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

can graph functions which may involve multiple transformations, sophisticated algebra (e.g. use of log laws), and functions presented in an unfamiliar form  can graph functions which may involve multiple transformations when presented in standard form  can graph functions which involve a single transformation 
can recognise and determine the equation of a graph of a function which has undergone more complex transformations  can recognise and determine the equation of a graph of a function which has undergone multiple transformations  can recognise and determine the equation of a graph of a function which has undergone a single transformation 
can (using appropriate notation) restrict the domain of a function in order to allow the existence of an inverse function  can use the graph of a function to determine whether or not an inverse function exists, giving reasons  understands the difference between a relation and a function 
can algebraically find the inverse of complex functions (for example: exponential and logarithmic functions)  can algebraically find the inverse of routine functions  
can produce the graph of an inverse from the graph of a function  distinguishes between: onetoone and manyto one and manytomany relations  
explains the relationship between the domain and the range of a function and the domain and range of its inverse  can identify the domain and the range given the graph of a function or a relation  
solves more complex logarithmic equations, including cases which involve the application of multiple log laws and cases where an algebraic substitution may be required  uses the definition of logarithm and the log laws to solve logarithmic equations  can use the definition of logarithm to change between index and logarithmic statements 
recognises when an answer is not feasible  can apply log laws to simplify expressions  
can establish the existence of a composite function by considering the domain and range of the component functions.  determines more complex composite functions.  determines simple composite functions. 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

can use a suitable method to evaluate a second ratio from a given one. E.g. Given that `sin x = 12/13` and `pi < x < (3pi)/2`, find `cos x` and `tan x`  can use the CAST diagram to simplify expressions of the type: `sin (pi/2 + theta)`, etc...  can convert between degrees and radians 
can use the CAST diagram to evaluate a trig ratio of an angle of greater than `pi/2` expressing the answer as an exact value. E.g. Evaluate `cos ((7pi)/6)`  can recall exact values of trigonometric ratios for `pi/6`, `pi/4`, `pi/2`, `0`, `pi`, `(3pi)/2`, `2pi` and also recall exact values when these angles are expressed in degrees  
can use the Pythagorean identity to evaluate a second ratio from a given one within the first quadrant. E.g. Given that `sin x = 12/13` and `0 < x < pi/2` find `tan x`  
can graph trigonometric functions which involve multiple transformations which may be presented in an unfamiliar form. E.g. `y = a sin (bx + c) + d`. Can identify endpoints and `y`intercept as exact values.  can graph trigonometric functions which involve multiple transformations but which are presented in the form: `y = a sin (bx + c) + d`. Graphs of this type include functions where the horizontal dilation factor is a multiple of `pi`. E.g. `y = 2 sinpi x + 1`  can graph trigonometric functions which involve a single transformation. E.g. `y = 2 sin x`, `y = sin 2x`, `y = sin(x + pi/6)`, `y = sin x + 1` 
can give the equation of a graph that has undergone multiple transformations  can give the equation of a graph that has undergone a single transformation  
can find specific solutions over a given domain to trig equations, presented in unfamiliar forms or which might involve a horizontal dilation factor as a multiple of `pi` .  can find specific solutions over a given domain to trig equations which are presented in standard form. E.g. `cos 2(x + pi/3) = 1/2`  can use the inverse trig functions to identify the principal angle that has a given trig value. E.g. Find `x` if `sin x = 1/2` 
can produce a graphical model of such an equation.  can find specific solutions over a given domain to simple trig equations which relate directly to the standard formula. E.g. `sin x = 1/2` where `0 < x < 2pi` . 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

can evaluate limits that require initial simplification (e.g. factorisation)  can explain the concept of a limit and evaluate a simple limit  
can explain the definition of a derivative and how it relates to the gradient of a tangent to a curve  can use the definition of derivative to differentiate a linear expression  
presents detailed working when using the definition of derivative and can use it on more complex examples  
applies differentiation rules to differentiate complex expressions. Such cases include examples which involve multiple use of the composite rule and examples where several rules are used.  applies differentiation rules to differentiate products, quotients, rational functions and simple composite expressions  applies differentiation rules to differentiate expressions of the type: `kx^n`, `ke^x`, `k ln x`, `k sin x`, `k cos x`, `k tan x` and functions that are the sum of each of these types. E.g. Differentiate `y = 5x^4  3e^x` 
finds the equation(s) of the tangent and/or normal to a curve given a point on the curve or the gradient in cases where significant algebraic manipulation may be involved. Examples include use of log laws and evaluation of trigonometric exact values.  finds the equation(s) of the tangent and/or normal to a curve given a point on the curve or the gradient for routine nonpolynomial cases  finds the equation(s) of the tangent and/or normal to a curve given a point on the curve or the gradient for polynomial cases 
can choose the appropriate rate of change (average or instantaneous)  articulates the difference between the average rate of change and the instantaneous rate of change and can calculate both in the context of a practical example  uses the derivative to calculate an instantaneous rate of change in a practical example 
can deduce the graph of a derivative from the graph of a more complex (possibly discontinuous) function  can deduce the graph of a derivative from the graph of a polynomial or other simple functions  
finds and justifies stationary points of routine and nonroutine functions and interprets the results.  finds and justifies stationary points of routine functions and interprets the results.  finds the stationary points of routine functions. 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

integrates functions which may require algebraic manipulation before applying a rule. Such examples include integration by recognition.  integrates more complex functions which involve the direct application of a rule including `sin (ax+b)` and `1/(ax + b)`  determines the indefinite integral of a polynomial function in expanded form, e.g. the function of `(ax + b)^n`, where `n` is a positive integer, and `e^(ax)` 
calculates the area between the graph of more complex functions and the `x`axis by evaluating the definite integral. This could include hyperbolic, trigonometric or logarithmic functions.  calculates the area between the graph of a polynomial or a simple exponential function and the `x`axis by evaluating the definite integral (in cases where the graph could cut the `x`axis within the range of the integration)  calculates the area between the graph of a polynomial function and the `x`axis by evaluating the definite integral (in cases where the graph does not cut the `x`axis within the range of the integration) 
evaluates the area between the graphs of two or more complex functions with two or more points of intersection that are easily determined  evaluates the area between the graphs of two polynomial functions with points of intersection that are easily determined  evaluates the area between the graphs of two polynomial functions that do not intersect within the range of the integration 
integrates more complex functions and makes similar applications of integration to calculate net changes  determines the distance travelled by an object moving in a straight line with one or two changes in direction by integrating a polynomial velocity function in time  determines the distance travelled by an object moving in a straight line in one direction by integrating a simple polynomial velocity function in time 
can determine the equation of a function given an expression representing its gradient and a point on the curve of the function, in cases where sophisticated algebra such as logarithm laws or trigonometric exact values are required.  can determine the equation of a more complex function given its gradient and a point on the curve of the function.  can determine the equation of a polynomial function given its gradient and a point on the curve of the function. Such examples may involve the tangent at the contact point. 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

identifies and defines randomness, discrete and continuous variables  
can solve problems involving finding unknown probabilities for a discrete random variable where the mean and variance is given  calculates the standard deviation and/or variance for a discrete random variable. Can determine the 95% confidence interval for a given random variable.  calculates the expected value (mean) for a discrete random variable 
uses the formula `P = ((n),(r))p^r(1  p)^(n  r)` to determine binomial probabilities for cumulative outcomes. Calculates (using technology) a binomial probability for single and multiple outcomes where sophisticated modelling may be required.  uses the formula `P = ((n),(r))p^r(1  p)^(n  r)` to determine binomial probabilities for cumulative outcomes. Calculates (using technology) a binomial probability for single and multiple outcomes where sophisticated modelling may be required.  calculates (using technology) a binomial probability for a single outcome in routine cases 
can predict changes to the graph of a binomial distribution given changing values of `x`, `n` and `p`  can draw the graph of a binomial distribution given values of `x`, `n` and `p`  can draw the graph of a probability distribution for a discrete random variable 
can use technology to determine mean and standard deviation in a normal distribution. Can use the ‘689599% approximation’ rule.  
can determine the mean and standard deviation of normally distributed data given proportion information. E.g. 10% of a population of animals has a mass < 1.2 kg, 20% has a mass > 3.8 kg. Find the mean and standard deviation.  can, given the population size, use technology to determine quantities in a normal distribution. Can, given a proportion, use inverse normal calculations to determine appropriate percentiles.  can use technology to determine probabilities in a normal distribution 
can construct confidence intervals for a population proportion and can articulate their findings. This may include the 90% and the 99% confidence intervals.  can construct the 95% confidence interval for a population proportion and can articulate their findings.  can calculate the mean and the standard (margin of) error of the sample proportion. 
Mathematics Methods Level 4 (with the award of):
EXCEPTIONAL ACHIEVEMENT
HIGH ACHIEVEMENT
COMMENDABLE ACHIEVEMENT
SATISFACTORY ACHIEVEMENT
PRELIMINARY ACHIEVEMENT
The final award will be determined by the Office of Tasmanian Assessment, Standards and Certification from 13 ratings (8 from the internal assessment, 5 from the external assessment).
The minimum requirements for an award in Mathematics Methods Level 4 are as follows:
EXCEPTIONAL ACHIEVEMENT (EA)
11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings and 1 ‘B’ rating from external assessment)
HIGH ACHIEVEMENT (HA)
5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings and 1 ‘C’ rating from external assessment)
COMMENDABLE ACHIEVEMENT (CA)
7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings and 2 ‘C’ ratings from external assessment)
SATISFACTORY ACHIEVEMENT (SA)
11 ‘C’ ratings (3 ‘C’ ratings from external assessment)
PRELIMINARY ACHIEVEMENT (PA)
6 ‘C’ ratings
A learner who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.
The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment.
In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.
Requests for amendments to an accredited course will be forwarded by Curriculum Services to the Office of TASC for formal consideration.
Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.
A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.
The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.
For the content areas of Mathematics Methods, the proficiency strands – Understanding; Fluency; Problem Solving; and Reasoning – build on learners’ learning in F10 Australian Curriculum: Mathematics. Each of these proficiencies is essential, and all are mutually reinforcing. They are still very much applicable and should be inherent in the five areas of study.
MATHEMATICAL METHODS
Unit 1 – Topic 1: Functions and Graphs
Lines and linear relationships:
Review of quadratic relationships:
Inverse proportion:
Powers and polynomials:
Functions:
Unit 1 – Topic 2: Trigonometric Functions
Cosine and sine rules:
Circular measure and radian measure:
Trigonometric functions:
Unit 1 – Topic 3: Counting and Probability
Combinations:
Unit 2 – Topic 1: Exponential Functions
Indices and the index laws:
Exponential functions:
Unit 2 – Topic 3: Introduction to Differential Calculus
Rates of change:
The concept of the derivative:
Computation of derivatives:
Properties of derivatives:
Applications of derivatives:
Antiderivatives:
Unit 3 – Topic 1: Further Differentiation and Applications
Exponential functions:
Trigonometric functions:
Differentiation rules:
The second derivative and applications of differentiation:
Unit 3 – Topic 2: Integrals
Antidifferentiation:
Definite integrals:
Fundamental theorem:
Applications of integration:
Unit 3 – Topic 3: Discrete Random Variables
General discrete random variables:
Bernoulli distributions:
Binomial distributions:
Unit 4 – Topic 1: The Logarithmic Function
Logarithmic functions:
Calculus of logarithmic functions:
Unit 4 – Topic 2: Continuous Random Variables and the Normal Distribution
General continuous random variables:
Normal distributions:
Unit 4 – Topic 3: Interval Estimates for Proportions
Random sampling:
Sample proportions:
Confidence intervals for proportions:
The accreditation period for this course has been renewed from 1 January 2022 until 31 December 2025.
During the accreditation period required amendments can be considered via established processes.
Should outcomes of the Years 912 Review process find this course unsuitable for inclusion in the Tasmanian senior secondary curriculum, its accreditation may be cancelled. Any such cancellation would not occur during an academic year.
Version 1 – Accredited on 17 August 2016 for use from 1 January 2017. This course replaces Mathematics Methods (MTM315114) that expired on 31 December 2016.
Version 1.1 – Renewal of accreditation on 13 August 2017 for use in 2018.
Accreditation renewed on 22 November 2018 for the period 1 January 2019 until 31 December 2021.
Version 1.2  Renewal of Accreditation on 14 July 2021 for the period 31 December 2021 until 31 December 2024, without amendments.
GLOSSARY
Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(x) = 1/a^x`, `(a^x)^y = a^(xy)`, `a^0 = 1`, where `x`, `y` and `a` are real.
Additivity property of definite integrals
The additivity property of definite integrals refers to ‘addition of intervals of integration’: `int_a^b f(x) dx + int_b^c f(x) dx = int_a^c f(x) dx` for any numbers `a`, `b` and `c`, and any function `f(x)`.
Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(x) = 1/a^x`, `(a^x)^y = a^(xy)`, `a^0 = 1`, for any real numbers `x`, `y`, and `a`, with `a > 0`.
Algebraic properties of logarithms
The algebraic properties of logarithms are the rules: `log_a (xy) = log_a x + log_a y`, `log_a (1/x) = log_a x`, and `log_a 1 = 0`, for any positive real numbers `x`, `y` and `a`.
Antidifferentiation
An antiderivative, primitive or indefinite integral of a function `f(x)` is a function `F(x)` whose derivative is `(x)`, i.e. `F'(x) = f(x)`.
The process of solving for antiderivatives is called antidifferentiation.
Antiderivatives are not unique. If `F(x)` is an antiderivative of `f(x)`, then so too is the function `F(x) + c` where `c` is any number. We write `int f(x) dx = F(x) + c` to denote the set of all antiderivatives of `f(x)`. The number `c` is called the constant of integration. For example, since `d/dx (x^3) = 3x^2`, we can write `int 3x^2 dx = x^3 + c`.
Asymptote
A straight line is an asymptote of the function `y = f(x)` if graph of `y = f(x)` gets arbitrarily close to the straight line. An asymptote can be horizontal, vertical or oblique. For example, the line with equation `x = pi/2` is a vertical asymptote to the graph of `y = tan x`, and the line with equation `y = 0` is a horizontal asymptote to the graph of `y = 1/x`.
Bernoulli random variable
A Bernoulli random variable has two possible values, namely `0` and `1` . The parameter associated with such a random variable is the probability `p` of obtaining a `1`.
Bernoulli trial
A Bernoulli trial is a chance experiment with possible outcomes, typically labelled ‘success’ and failure’.
Binomial distribution
The expansion `(x + y)^n = x^n + ((n),(1))x^(n1)y + ... + ((n),(r))x^(nr)y^r + ... + y^n` is known as the binomial theorem. The numbers `((n),(r)) = (n!)/(r!(n  r)!) = (n xx (n  1) xx ... xx (n  r + 1))/(r xx (r  1) xx ... xx 2 xx 1)` are called binomial coefficients.
Central Limit Theorem
There are various forms of the Central Limit Theorem, a result of fundamental importance in statistics. For the purposes of this course, it can be expressed as follows: "If `bar X` is the mean of `n` independent values of random variable `X` which has a finite mean `mu` and a finite standard deviation `sigma`, then as `n > oo`, the distribution of `(bar X  mu)/(sigma/sqrt n)` approaches the standard normal distribution.”
In the special case where `X` is a Bernoulli random variable with parameter `p`, `bar X` is the sample proportion `hat p`, `mu = p` and `sigma = sqrt (p(1  p)`. In this case, the central limit theorem is a statement that as `n > oo` the distribution of `(hat p  p)/sqrt ((hat p(1  hat p))/n` approaches the standard normal distribution.
Chain rule
The chain rule relates the derivative of the composite of two functions to the functions and their derivatives. If `h(x) = f@g(x)`, then `(f@g)'(x) = f'(g(x))g'(x)`, and in Leibniz notation: `dz/dx = dz/dy dy/dx`.
Circular measure
A rotation, typically measured in radians or degrees.
Composition of functions
If `y = g(x)` and `z = f(y)` for functions `f` and `g`, then `z` is a composite function of `x`. We write `z = f@g(x) = f(g(x))`. For example, `z = sqrt (x^2 + 3)` expresses `z` as a composite of the functions `f(y) = sqrt y` and `g(x) = x^2 + 3`.
Concave up and concave down
A graph of `y = f(x)` is concave up at a point `P` if points on the graph near `P` lie above the tangent at `P`. The graph is concave down at `P` if points on the graph near `P` lie below the tangent at `P`.
Effect of linear change
The effects of linear changes of scale and origin on the mean and variance of a random variable are summarised as follows:
If `X` is a random variable and `Y = aX + b`, where `a` and `b` are constants, then `E(Y) = aE(X) + b` and `Var(Y) = a^2 Var(X)`.
Euler’s number
Euler’s number `e` is an irrational number whose decimal expansion begins `e = 2.7182818284590452353602874713527...`.
It is the base of the natural logarithms, and can be defined in various ways including: `e = 1 + 1/(1!) + 1/(2!) + 1/(3!) + ...` and `e = lim_(n > oo) (1 + 1/n)^n`.
Expected value
The expected value `E(X)` of a random variable `X` is a measure of the central tendency of its distribution.
If `X` is discrete, `E(X) = sum_i p_i x_i`, where the `x_i` are the possible values of `X` and `p_i = P(X = x_i)`.
If `X` is continuous, `E(x) = int_oo^oo xp(x) dx`, where `p(x)` is the probability density function of `X`.
Function
A function `f` is a rule such that for each `x`value there is only one corresponding `y`value. This means that if `(a, b)` and `(a, c)` are ordered pairs, then `b = c`.
Gradient (Slope)
The gradient of the straight line passing through points `(x_1, y_1)` and `(x_2, y_2)` is the ratio `(y_2  y_1)/(x_2  x_1)`. Slope is a synonym for gradient.
Graph of a function
The graph of a function `f` is the set of all points `(x, y)` in Cartesian plane where `x` is in the domain of `f` and `y = f(x)`.
Index laws
The index laws are the rules: `a^x a^y = a^(x + y)`, `a^(x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, and `(ab)^x = a^x b^x`, where `a`, `b`, `x` and `y` are real numbers.
Level of confidence
The level of confidence associated with a confidence interval for an unknown population parameter is the probability that a random confidence interval will contain the parameter.
Linearity property of the derivative
The linearity property of the derivative is summarised by the equations: `d/dx(ky) = k dy/dx` for any constant `k` and `d/dx (y_1 + y_2) = dy_1/dx + dy_2/dx`.
Local and global maximum and minimum
A stationary point on the graph `y = f(x)` of a differentiable function is a point where `f'(x) = 0`.
We say that `f(x_0)` is a local maximum of the function `f(x)` if `f(x) <= f(x_0)`for all values of `x` near `x_0`.
We say that `f(x_0)` is a global maximum of the function `f(x)` if `(x) <= f(x_0)` for all values of `x` in the domain of `f`.
We say that `f(x_0)` is a local minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` near `x_0`.
We say that `f(x_0)` is a global minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` in the domain of `f`.
Margin of error
The margin of error of a confidence interval of the form `f  E < p < f + E` is `E`, the halfwidth of the confidence interval. It is the maximum difference between `f` and `p` if `p` is actually in the confidence interval.
Mean of a random variable
The mean of a random variable is another name for its expected value.
The variance `Var(X)` of a random variable `X` is a measure of the ‘spread’ of its distribution.
If `X` is discrete, `(X) = sum_i p_i(x_i  mu)^2`, where `mu = E(X)` is the expected value.
If `X` is continuous, `Var(X) = int_oo^oo (x  mu)^2 p(x)dx`.
Nonroutine problems
Problems solved using procedures not regularly encountered in learning activities.
Pascal’s triangle
Pascal’s triangle is a triangular arrangement of binomial coefficients. The `n^"th"` row consists of the binomial coefficients `((n),(r))`, for `0 <= r <= n`, each interior entry is the sum of the two entries above it, and sum of the entries in the `n^"th"` row is `2^n`.
Period of a function
The period of a function `f(x)` is the smallest positive number `p` with the property that `f(x + p) = f(x)` for all `x`. The functions `sin x` and `cos x` both have period `2pi`, and `tan x` has period `pi`.
Point and interval estimates
In statistics estimation is the use of information derived from a sample to produce an estimate of an unknown probability or population parameter. If the estimate is a single number, this number is called a point estimate. An interval estimate is an interval derived from the sample that, in some sense, is likely to contain the parameter.
A simple example of a point estimate of the probability `p` of an event is the relative frequency `f` of the event in a large number of Bernoulli trials. An example of an interval estimate for `p` is a confidence interval centred on the relative frequency `f`.
Point of inflection
A point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
Probability density function
The probability density function of a continuous random variable is a function that describes the relative likelihood that the random variable takes a particular value. Formally, if `p(x)` is the probability density of the continuous random variable `X`, then the probability that `X` takes a value in some interval `[a, b]` is given by `int_a^b p(x) dx`.
Probability distribution
The probability distribution of a discrete random variable is the set of probabilities for each of its possible values.
Product rule
The product rule relates the derivative of the product of two functions to the functions and their derivatives.
If`h(x) = f(x) g(x)`, then `h'(x) = f(x) g'(x) + f'(x) g(x)`, and in Leibniz notation: `d/dx(uv) = u (dv)/dx + (du)/dx v`.
Quadratic formula
If `ax^2 + bx + c = 0` with `a != 0`, then `x = (b + sqrt(b^2  4ac))/(2a)`. This formula for the roots is called the quadratic formula.
Quantile
A quantile `t_alpha` for a continuous random variable `X` is defined by `P(X > t_alpha) = alpha`, where `0 < alpha < 1`.
The median `m` of `X` is the quantile corresponding to `alpha = 0.5 : P(X > m) = 0.5`.
Quotient rule
The quotient rule relates the derivative of the quotient of two functions to the functions and their derivatives
If `h(x) = f(x)/g(x)`, then `h'(x) = (g(x) f'(x)  f(x) g'(x))/g(x)^2`.
Radian measure
The radian measure `theta` of an angle in a sector of a circle is defined by `theta = l/r`, where `r` is the radius and `l` is the arc length. Thus, an angle whose degree measure is `180` has radian measures `pi`.
Random variable
A random variable is a numerical quantity, the value of which depends on the outcome of a chance experiment. For example, the proportion of heads observed in 100 tosses of a coin.
A discrete random variable is one which can only take a countable number of value, usually whole numbers.
A continuous random variable is one whose set of possible values are all of the real numbers in some interval.
Relative frequency
If an event `E` occurs `r` times in `n` trials of a chance experiment, the relative frequency of `E` is `r/n`.
Routine problems
Problems solved using procedures regularly encountered in learning activities.
Secant
A secant of the graph of a function is a straight line passing through two points on the graph. The line segment between the two points is called a chord.
Second derivative test
According to the second derivative test, if `f'(x) = 0`, then `f(x)` is a local maximum of `f` if `f''(x) < 0`, and `f(x)` is a local minimum if `f''(x) > 0`.
Sine and cosine functions
In the unit circle definition of cosine and sine, `cos theta` and `sin theta` are the `x` and `y` coordinates of the point on the unit circle corresponding to the angle `theta` measured as a rotation from the ray `OX`. If `theta` is measured in the counterclockwise direction, then it is said to be positive; otherwise it is said to be negative.
Standard deviation of a random variable
The standard deviation of a random variable is the square root of its variance.
Tangent line
The tangent line (or simply the tangent) to a curve at a given point `P` can be described intuitively as the straight line that "just touches" the curve at that point. At the point where the tangent touches the curve, the curve has “the same direction” as the tangent line. In this sense it is the best straightline approximation to the curve at the point.
The fundamental theorem of calculus
The fundamental theorem of calculus relates differentiation and definite integrals. It has two forms: `d/dx (int_a^x f(t) dt) = f(x)` and `int_a^b f'(x) dx = f(b)  f(a)`.
The linearity property of antidifferentiation
The linearity property of antidifferentiation is summarised by the equations: `int kf(x) dx = k int f(x) dx` for any constant `k`, and `int (f_1(x) + f_2(x)) dx = int f_1(x) dx + int f_2(x) dx` for any two functions `f_1(x)` and `f_2(x)`.
Similar equations describe the linearity property of definite integrals: `int_a^b kf(x)dx = kint_a^b f(x)dx` for any constant `k`, and `int_a^b(f_1(x) + f_2(x))dx = int_a^b f_1(x)dx + int_a^b f_2(x)dx` for any two functions `f_1(x)` and `f_2(x)`.
Uniform continuous random variable
A uniform continuous random variable `X` is one whose probability density function `p(x)` has constant value on the range of possible values of `X`. If the range of possible values is the interval `[a, b]`, then `p(x) = 1/(b  a)` if `a <= x <= b`.
Vertical line test
A relation between two real variables `x` and `y` is a function and `y = f(x)` for some function `f`, if and only if each vertical line, i.e. each line parallel to the `y`axis, intersects the graph of the relation in at most one point. This test to determine whether a relation is, in fact, a function is known as the vertical line test.
LINE OF SIGHT – Mathematics Methods Level 4
Learning Outcomes  Separating out content from skills in 4 of LOs  Criteria and Elements  Content  
understand the concepts and techniques in algebra, graphs, function study, differential and integral calculus, probability and statistics 
function study (log, poly, exp, hyper) 
concepts and techniques problem solving interpret and evaluate select and use appropriate tools 
C4 E19 C4 E19 C4 E2, 3, 8 C2 E5, 6 
Function Study 
solve problems using algebra, graphs, function study, differential and integral calculus, probability and statistics 
function study (circular) 
concepts and techniques problem solving interpret and evaluate select and use appropriate tools 
C5 E16 C5 E2, 4, 5, 6 C5 E26 C2 E5, 6 
Circular Functions 
interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems 
differential calculus 
concepts and techniques problem solving interpret and evaluate select and use appropriate tools 
C6 E18 C6 E48 C6 E58 C2 E5, 6 
Calculus  Differential Calculus 
select and use appropriate tools, including computer technology, when solving mathematical problems 
intergral calculus 
concepts and techniques problem solving interpret and evaluate select and use appropriate tools 
C7 E15 C7 E25 C7 E25 C2 E5, 6 
Calculus  Integral Calculus 
binomial, normal, statistical inference 
concepts and techniques problem solving interpret and evaluate select and use appropriate tools 
C8 E17 C8 E2, 3, 4, 6, 7 C8 E3, 5, 6, 7 C8 E37 C2 E5, 6 
Probability and Statistics 

communicate their arguments and strategies when solving problems 
C1 E17 
All content areas 

apply reasoning skills in the context of algebra, graphs, function study, differential and integral calculus, probability and statistics 
C2 E17 
All content areas 

plan activities and monitor and evaluate progress; use strategies to organise and complete activities and meet deadlines in the context of mathematics 
C3 E16 
All content areas 