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# VCE Mathematical Methods Units 3&4

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The full course description can be found at the VCAA website
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## AREAS OF STUDY

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##### 1. Functions and graphs

In this area of study students cover transformations of the plane and the behaviour of some elementary functions of a single real variable, including key features of their graphs such as axis intercepts, stationary points, points of inflection, domain (including maximal, implied or natural domain), co-domain and range, asymptotic behaviour and symmetry. The behaviour of these functions and their graphs is to be linked to applications in practical situations.
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This area of study will include:

• graphs and identification of key features of graphs of the following functions:
• power functions, $y = x^n, \; n \in Q$
• exponential functions, $y = a^x, \; a \in R^+$,
• in particular $y = e^x,\; y = log_e(x), \; y = log_{10}(x)$
• circular functions, $y = \sin(x), \; y = \cos(x), \; y = \tan(x)$
• graphs of polynomial functions
• transformation from $y = ƒ(x)$ to $y = Aƒ(n(x + b)) + c$,
• where, $A,\; n,\; b, \; c \in R$,
• and $ƒ(x)$ is one of the functions specified above and the inverse transformation
• the relation between the graph of an original function and the graph of a corresponding transformed function
• including families of transformed functions for a single transformation parameter
• graphs of sum, difference, product and composite functions of ƒ and g
• where ƒ and g are functions of the types specified above
• not including composite functions that result in reciprocal or quotient functions
• modelling practical situations using polynomial, power, circular, exponential and logarithmic functions,
• and simple transformation and combinations of these functions,
• including simple piecewise (hybrid) functions,

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##### 2. Algebra

In this area of study students cover the algebra of functions, including composition of functions, simple functional relations, inverse functions and the solution of equations. They also study the identification of appropriate solution processes for solving equations, and systems of simultaneous equations, presented in various forms. Students also cover recognition of equations and systems of equations that are solvable using inverse operations or factorisation, and the use of graphical and numerical approaches for problems involving equations where exact value solutions are not required or which are not solvable by other methods. This content is to be incorporated as applicable to the other areas of study
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This area of study will include:

• review of algebra of polynomials, equating coefficients and solution of polynomial equations with real coefficients of degree n having up to n real solutions;
• use of simple functional relations such as $f(x + k) = f(x)$, $f(x^n ) = nf(x)$,$f(x) + f(-x) = 0$, $f(xy) = f(x)f(y)$, to characterise properties of functions including periodicity and symmetry, and to specify algebraic equivalence, including the exponent and logarithm laws
• functions and their inverses, including conditions for the existence of an inverse function, and use of inverse functions to solve equations involving exponential, logarithmic, circular and power functions
• composition of functions, where f composition g is defined by $f(g(x))$, given rg ⊆ df
• the notation f ° g may be used, but is not required
• solution of equations of the form $f(x) = g(x)$ over a specified interval, where f and g are functions of the type specified in the ‘Functions and graphs’ area of study, by graphical, numerical and algebraic methods, as applicable
• solution of literal equations and general solution of equations involving a single parameter
• solution of simple systems of simultaneous linear equations,
• including consideration of cases where no solution or an infinite number of possible solutions exist
• geometric interpretation only required for two equations in two variables

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##### 3. Calculus

In this area of study students cover graphical treatment of limits, continuity and differentiability of functions of a single real variable, and differentiation, anti-differentiation and integration of these functions. This material is to be linked to applications in practical situations.
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This area of study will include:

• review of average and instantaneous rates of change, tangents to the graph of a given function and the derivative function
• deducing the graph of the derivative function from the graph of a given function and deducing the graph of an anti-derivative function from the graph of a given function
• derivatives of xn for n ∈ Q, ex, loge(x), sin(x), cos(x), tan(x)
• derivatives of f(x) ± g(x), f(x) × g(x), f(x)/g(x) and f(g(x)) where f and g are polynomial functions, exponential, circular, logarithmic or power functions and transformations or simple combinations of these functions
• application of differentiation to graph sketching and identification of key features of graphs,
• identification of intervals over which a function is constant, stationary, strictly increasing or strictly decreasing,
• identification of the maximum rate of increase or decrease in a given application context
• consideration of the second derivative is not required,
• identification of local maximum/minimum values over an interval and application to solving problems,
• identification of interval endpoint maximum and minimum values
• anti-derivatives of polynomial functions and functions of the form f(ax + b) where f is xn , for n ∈ Q, ex, sin(x), cos(x) and linear combinations of these
• informal consideration of the definite integral as a limiting value of a sum involving quantities such as area under a curve,
• including examples such as distance travelled in a straight line and cumulative effects of growth such as inflation
• anti-differentiation by recognition that F'(x) = f(x) implies $\int f(x)dx = F(x) + c$
• informal treatment of the fundamental theorem of calculus, $\int_a^b f(x)dx = F(b) - F(a)$
• properties of anti-derivatives and definite integrals
• application of integration to problems involving
• finding a function from a known rate of change given a boundary condition,
• calculation of the area of a region under a curve and
• simple cases of areas between curves,
• distance travelled in a straight line,
• average value of a function
• and other situations.

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##### 4. Probability

In this area of study students cover 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 measures of spread; and statistical inference for sample proportions. The focus is on understanding the notion of a random variable, related parameters, properties and application and interpretation in context for a given probability distribution.
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This area of study will include:

• random variables, including: including the concept of a random variable as a real function defined on a sample space and examples of discrete and continuous random variables
• discrete random variables:
• specification of probability distributions for discrete random variables using graphs, tables and probability mass functions
• calculation and interpretation and use of mean (μ), variance (σ2) and standard deviation (σ) of a discrete random variable and their use
• bernoulli trials and the binomial distribution, Bi(n, p), as an example of a probability distribution for a discrete random variable
• effect of variation in the value/s of defining parameters on the graph of a given probability mass function for a discrete random variable
• calculation of probabilities for specific values of a random variable and intervals defined in terms of a random variable, including conditional probability
• continuous random variables:
• construction of probability density functions from non-negative functions of a real variable
• specification of probability distributions for continuous random variables using probability density functions
• calculation and interpretation of mean (μ), median, variance (σ2) and standard deviation (σ) of a continuous random variable and their use
• standard normal distribution, N(0, 1), and transformed normal distributions, N(μ,σ2), as examples of a probability distribution for a continuous random variable
• effect of variation in the value/s of defining parameters on the graph of a given probability density function for a continuous random variable
• calculation of probabilities for intervals defined in terms of a random variable, including conditional probability
• the cumulative distribution function may be used but is not required
• Statistical inference, including definition and distribution of sample proportions, simulations and confidence intervals:
• distinction between a population parameter and a sample statistic and the use of the sample statistic to estimate the population parameter
• concept of the sample proportion $\hat{P} = \frac{X}{n}$ as a random variable whose value varies between samples, where X is a binomial random variable which is associated with the number of items that have a particular characteristic and n is the sample size
• approximate normality of the distribution of $\hat{P}$ for large samples and, for such a situation, the mean p, (the population proportion) and standard deviation, $\sqrt{\frac{p(1-p)}{n}}$
• simulation of random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of $\hat{P}$
• determination of, from a large sample, an approximate confidence interval for a population proportion where z is the appropriate quantile for the standard normal distribution $\left( \hat{p} - z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \; \hat{p} + z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right)$
• in particular the 95% confidence interval as an example of such an interval where z ≈ 1.96
• the term standard error may be used but is not required.

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