26/02/2015
What you should know for VCE Mathematical Methods Units 3&4
AREAS OF STUDY For Mathematical methods Unit 3 &4
1. Functions and graphs
This area of study covers the behaviour of functions of a single real variable, including key featuresof their graphs such as axis intercepts, stationary points and points of inflection, domain (including maximal domain) and range, asymptotic behaviour and symmetry. The behaviour of these functions is to be linked to applications in practical situations.
This area of study will include:
• graphs and identification of key features of graphs of the following functions:
– power functions, y = xn;
– exponential functions, y = ax;
– logarithmic functions, y = loge(x) and y = log10(x), the relationship a = ek where k = loge(a);
– circular functions, y = sin(x), y = cos(x) and y = tan(x);
– modulus function, y = |x| where |x| = x when x ≥ 0 and |x| = -x when x < 0;
• transformation from y = ƒ(x) to y = Aƒ(n(x + b)) + c, where, A, n, b and c ∈ R, and ƒ is one of the functions specified above and the relation between the graph of the original function and the graph
of the transformed function (including families of transformed functions for a single transformation parameter) such as:
y = 0.4/(x+3)2 , y = –10sin(n(x – c)) –5, c ∈ R, y= A(x +7)5 – 2, A ∈ R;
• graphs of polynomial functions;
• graphs of sum, difference, product and composite functions of ƒ and g where ƒ and g are functions
of the types specified above such as:
y = sin (x) + 2x y = |cos (2x)| y = x2ekx, k ∈ R
y = a /(x2 +1) a∈ R y = (x2 – 2)n, n ∈ N;
• graphical and numerical solution of equations;
• graphs of inverse functions;
• recognition of the general form of possible models for data presented in graphical or tabular form, using polynomial, power, circular, exponential and logarithmic functions;
• applications of simple combinations of the above functions (including simple hybrid functions),and interpretation of features of the graphs of these functions in modelling practical situations;
for example, y = ax + b + msin(nx) as a possible pattern for economic growth cycles or y = axne-kx + b as a model for the amount of medication remaining in the blood stream after a dose
of the medication.
2. Algebra
This area of study covers the algebra of functions, including composition of functions, simple functional equations, inverse functions and the solution of equations. This area of study includes the identification of appropriate solution processes for solving equations, and systems of simultaneous equations, presented in various forms. It covers 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 should support work in the other areas of study.
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 and including n real solutions;
• the relationship of ƒ(x ± y), ƒ(xy) and ƒ to values of ƒ(x) and ƒ(y) for different functions ƒ;
• logarithm laws and exponent laws, recognition of equivalent forms using compound and double angle formulas for sine, cosine and tangent;
• solution of systems of simultaneous linear equations, including consideration of cases where no solution or an infinite number of possible solutions exist; for example, to find a cubic polynomial
function ƒ that satisfies the conditions ƒ′(3) = 0, ƒ(3) = 4 and ƒ(10) = –1 (familiarity with matrix representation of systems of simultaneous linear equations with up to five equations in five unknowns will be assumed);
• composition of functions, where ƒ composition g is defined by ƒ(g(x)), given rg ⊆ dƒ, such as loge(x2+ 1), e2x– 4ex– 5, |sin(x)|;
• 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;
• solution of equations of the form aƒ(n(x + b)) + c = k and recognition of the inverse function for ƒ (over a suitable principal value domain where necessary) where a, b, c, n and k ∈ R;
• graphical and numerical approaches to solving equations where exact methods may not apply or be required, such as equations of the form aƒ(n(x + b)) + c = g(x) , where f and g are power, exponential, logarithmic or circular functions; for example, finding approximate values for the coordinates of the points of intersection of the graphs of y = 3sin(2x) and y = e-x + 1, with specification of values to a required accuracy;
• solution of literal equations such as ax3+ b = c or emx + n = k;
• general solutions of equations such as cos(x) + cos(3x) = 1/2 , x ∈ R and the specification of exact solutions or numerical solutions, as appropriate, within a restricted domain;
• solution of general equations which arise from finding the points of intersection of graphs of functions, such as a straight line with a given parabola, or y =1/x with ƒ(x) = ax2 + bx + c.
3. Calculus
This area of study covers graphical treatment of limits, continuity and differentiability (including local linearity) 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.
This area of study will include:
• deducing the graph of the derivative function from the graph of a function and the relation between the graph of an anti-derivative function and the graph of the original function;
• derivatives of xn, for n ∈ Q, ex, loge(x), sin(x) and cos(x) and tan(x) (formal derivation is not required);
• properties of derivatives, (aƒ(x) ± bg(x))′ = aƒ′(x) ± bg′(x) where a, b ∈ R;
• derivatives of ƒ(x) ± g(x), ƒ(x) × g(x), and ƒ(g(x)) where ƒ and g are polynomial functions, exponential, circular, logarithmic or power functions (or combinations of these functions) such
as:
x5+ (1-x2)1/2 x sin(2x) ecos(x)
• application of differentiation to curve sketching and identification of key features of curves, 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) and tangents and normals to curves;
• identification of local maximum/minimum values over an interval and application to solving problems, identification of interval endpoint maximum and minimum values;
• average and instantaneous rates of change, including formulation of expressions for rates of change and related rates of change and solution and interpretation of problems involving rates of change and simple cases of related rates of change;
• the relationship ƒ(x + h) ≈ ƒ(x) + hƒ′(x) for a small value of h and its geometric interpretation;
• anti-derivatives of polynomial functions and of ƒ(ax + b) where ƒ is xn for n ∈ Q, ex, sin(x), or cos(x) and linear combinations of these;
• definition of the definite integral as the limiting value of a sum where the interval [a, b] is partitioned into n subintervals, with the i th subinterval of length δxi and containing xi*, and δx =max{δxi: i = 1, 2, … n} and evaluation of numerical approximations based on this definition;
• examples of the definite integral as a limiting value of a sum involving quantities such as area under a curve, distance travelled in a straight line and cumulative effects of growth such as inflation;
• anti-differentiation by recognition that F´ (x) = ƒ(x) implies ;
• informal treatment of the fundamental theorem of calculus, ;
• properties of anti-derivatives and definite integrals:
• application of integration to problems involving calculation of the area of a region under a curve and simple cases of areas between curves, such as distance travelled in a straight line; average value of a function; other situations modelled by the use of the definite integral as a limiting value of a sum over an interval; and finding a function from a known rate of change.
4. Probability
This area of study includes the study of discrete and continuous random variables, their representation using tables, probability functions or probability density functions (specified by rule and defining parameters as appropriate); and the calculation and interpretation of central measures and measures of spread. 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.
This area of study will include:
• random variables, including:
– the concept of discrete and continuous random variables;
– calculation and interpretation of the expected value, variance and standard deviation of a random variable (for discrete and continuous random variables, including consideration of the
connection between these);
– calculation and interpretation of central measures (mode, median, mean);
– property that, for many random variables, approximately 95 per cent of their probability distribution is within two standard deviations of the mean;
– bernoulli trials and two state markov chains, including the length of run in a sequence, steady values for a markov chain (familiarity with the use of transition matrices to compute values of
a markov chain will be assumed);
• discrete random variables:
– specification of probability distributions for discrete random variables using graphs, tables and
probability functions;
– interpretation of mean (μ) median, mode, variance (σ2
) and standard deviation of a discrete
random variable and their use;
– the binomial distribution, Bi(n, p), as an example of a probability distribution for a discrete
random variable (students are expected to be familiar with the binomial theorem and related
binomial expansions);
– the effect of variation in the value(s) of defining parameters on the graph of a given probability
function for a discrete random variable;
– 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 continuous functions of a real
variable;
– specification of probability distributions for continuous random variables using probability
density functions;
– calculation using technology and interpretation of mean (μ) median, mode, variance (σ2) and
standard deviation 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 (use as an approximation to the
binomial distribution is not required);
– the effect of variation in the value(s) of defining parameters on the graph of a given probability density function for a continuous random variable;
– probabilities for intervals defined in terms of a random variable, including conditional probability (students should be familiar with the use of definite integrals, evaluated by hand or using
technology, for a probability density function to calculate probabilities; while not required,
teachers may also choose to relate this to the notion of a cumulative distribution function).
