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Skills available for Australia year 12 maths curriculum

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11-12.1 Functions and graphs

11-12.1 Exponential functions

11-12.1 Further differentiation and applications

  • 11-12. Exponential functions:

    • 11-12.ACMMM098 estimate the limit of a to the (h –1)/h as h → 0 using technology, for various values of a > 0

    • 11-12.ACMMM099 recognise that e is the unique number a for which the above limit is 1

    • 11-12.ACMMM100 establish and use the formula d/dx (e to the x power) = e to x power

    • 11-12.ACMMM101 use exponential functions and their derivatives to solve practical problems.

  • 11-12. Trigonometric functions:

    • 11-12.ACMMM102 establish the formulas d/dx (sinx) = cosx, and d/dx (cosx) = –sinx by numerical estimations of the limits and informal proofs based on geometric constructions

    • 11-12.ACMMM103 use trigonometric functions and their derivatives to solve practical problems.

  • 11-12. Differentiation rules:

    • 11-12.ACMMM104 understand and use the product and quotient rules

    • 11-12.ACMMM105 understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions

    • 11-12.ACMMM106 apply the product, quotient and chain rule to differentiate functions such as xe to the x power, tan x, 1/xn , xsinx, e to the - x sinx and f(ax +b).

  • 11-12. The second derivative and applications of differentiation:

    • 11-12.ACMMM107 use the increments formula: δy ≅ dy/dx × δx to estimate the change in the dependent variable y resulting from changes in the independent variable x

    • 11-12.ACMMM108 understand the concept of the second derivative as the rate of change of the first derivative function

    • 11-12.ACMMM109 recognise acceleration as the second derivative of position with respect to time

    • 11-12.ACMMM110 understand the concepts of concavity and points of inflection and their relationship with the second derivative

    • 11-12.ACMMM111 understand and use the second derivative test for finding local maxima and minima

    • 11-12.ACMMM112 sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection

    • 11-12.ACMMM113 solve optimisation problems from a wide variety of fields using first and second derivatives.

11-12.1 The logarithmic function

11-12.2 Continuous random variables and the normal distribution

  • 11-12. General continuous random variables:

    • 11-12.ACMMM164 use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable

    • 11-12.ACMMM165 understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts

    • 11-12.ACMMM166 recognise the expected value, variance and standard deviation of a continuous random variable and evaluate them in simple cases

    • 11-12.ACMMM167 understand the effects of linear changes of scale and origin on the mean and the standard deviation.

  • 11-12. Normal distributions:

11-12.2 Trigonometric functions

11-12.2 Integrals

  • 11-12. Anti-differentiation:

    • 11-12.ACMMM114 recognise anti-differentiation as the reverse of differentiation

    • 11-12.ACMMM115 use the notation ∫f(x)dx for anti-derivatives or indefinite integrals

    • 11-12.ACMMM116 establish and use the formula ∫xndx = 1/(n + 1)x to the (n+1) power + c for x ≠ –1

    • 11-12.ACMMM117 establish and use the formula ∫e to the x power dx = e to the x power + c

    • 11-12.ACMMM118 establish and use the formulas ∫sin x dx = –cos x + c and ∫cos x dx = sin x + c

    • 11-12.ACMMM119 recognise and use linearity of anti-differentiation

    • 11-12.ACMMM120 determine indefinite integrals of the form ∫f(ax + b)dx

    • 11-12.ACMMM121 identify families of curves with the same derivative function

    • 11-12.ACMMM122 determine f(x), given f'(x) and an initial condition f(a)=b

    • 11-12.ACMMM123 determine displacement given velocity in linear motion problems.

  • 11-12. Definite integrals:

    • 11-12.ACMMM124 examine the area problem, and use sums of the form ∑if(xi)δxi to estimate the area under the curve y=f(x)

    • 11-12.ACMMM125 interpret the definite integral ∫(a b) f(x)dx as area under the curve y=f(x) if f(x)>0

    • 11-12.ACMMM126 recognise the definite integral ∫(a b)f(x)dx as a limit of sums of the form ∑if(xi)δxi

    • 11-12.ACMMM127 interpret ∫(a b)f(x)dx as a sum of signed areas

    • 11-12.ACMMM128 recognise and use the additivity and linearity of definite integrals.

  • 11-12. Fundamental theorem:

    • 11-12.ACMMM129 understand the concept of the signed area function F(x)=∫(a x)f(t)dt

    • 11-12.ACMMM130 understand and use the theorem: F'(x)= d/dx (∫(a x)f(t)dt) = f(x) and illustrate its proof geometrically

    • 11-12.ACMMM131 understand the formula ∫(a b)f(x)dx = F(b) - F(a) and use it to calculate definite integrals.

  • 11-12. Applications of integration:

    • 11-12.ACMMM132 calculate the area under a curve

    • 11-12.ACMMM133 calculate total change by integrating instantaneous or marginal rate of change

    • 11-12.ACMMM134 calculate the area between curves in simple cases

    • 11-12.ACMMM135 determine positions given acceleration and initial values of position and velocity.

11-12.2 Arithmetic and geometric sequences and series

11-12.3 Interval estimates for proportions

  • 11-12. Random sampling:

    • 11-12.ACMMM171 understand the concept of a random sample

    • 11-12.ACMMM172 discuss sources of bias in samples, and procedures to ensure randomness

    • 11-12.ACMMM173 use graphical displays of simulated data to investigate the variability of random samples from various types of distributions, including uniform, normal and Bernoulli.

  • 11-12. Sample proportions:

    • 11-12.ACMMM174 understand the concept of the sample proportion p as a random variable whose value varies between samples, and the formulas for the mean p and standard deviation √(p(1–p)/n of the sample proportion p

    • 11-12.ACMMM175 examine the approximate normality of the distribution of p for large samples

    • 11-12.ACMMM176 simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to illustrate the distribution of p and the approximate standard normality of p - p /√(p(1–p)/n where the closeness of the approximation depends on both n and p.

  • 11-12. Confidence intervals for proportions:

    • 11-12.ACMMM177 the concept of an interval estimate for a parameter associated with a random variable

    • 11-12.ACMMM178 use the approximate confidence interval p–z√(p(1–p)/n, p+z√(p(1–p)/n, as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution

    • 11-12.ACMMM179 define the approximate margin of error E = z√(p(1–p)/n and understand the trade-off between margin of error and level of confidence

    • 11-12.ACMMM180 use simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain p.

11-12.3 Introduction to differential calculus

  • 11-12. Rates of change:

    • 11-12.ACMMM077 interpret the difference quotient f(x+h)–f(x)h as the average rate of change of a function f

    • 11-12.ACMMM078 use the Leibniz notation δx and δy for changes or increments in the variables x and y

    • 11-12.ACMMM079 use the notation δy/δx for the difference quotient f(x+h)–f(x)h where y = f(x)

    • 11-12.ACMMM080 interpret the ratios f(x+h)–f(x)/h and δy/δx as the slope or gradient of a chord or secant of the graph of y = f(x).

  • 11-12. The concept of the derivative:

    • 11-12.ACMMM081 examine the behaviour of the difference quotient f(x+h)–f(x)/h as h → 0 as an informal introduction to the concept of a limit

    • 11-12.ACMMM082 define the derivative f'(x) as limh→0 f(x+h)–f(x)/h

    • 11-12.ACMMM083 use the Leibniz notation for the derivative: dy/dx = limδx→0 δy/δx and the correspondence dy/dx = f'(x) where y = f(x)

    • 11-12.ACMMM084 interpret the derivative as the instantaneous rate of change

    • 11-12.ACMMM085 interpret the derivative as the slope or gradient of a tangent line of the graph of y = f(x).

  • 11-12. Computation of derivatives:

    • 11-12.ACMMM086 estimate numerically the value of a derivative, for simple power functions

    • 11-12.ACMMM087 examine examples of variable rates of change of non-linear functions

    • 11-12.ACMMM088 establish the formula d/dx(xn) = nx to the (n–1) power for positive integers n by expanding (x + h)n or by factorising (x + h)n -xn.

  • 11-12. Properties of derivatives:

    • 11-12.ACMMM089 understand the concept of the derivative as a function

    • 11-12.ACMMM090 recognise and use linearity properties of the derivative

    • 11-12.ACMMM091 calculate derivatives of polynomials and other linear combinations of power functions.

  • 11-12. Applications of derivatives:

    • 11-12.ACMMM092 find instantaneous rates of change

    • 11-12.ACMMM093 find the slope of a tangent and the equation of the tangent

    • 11-12.ACMMM094 construct and interpret position-time graphs, with velocity as the slope of the tangent

    • 11-12.ACMMM095 sketch curves associated with simple polynomials; find stationary points, and local and global maxima and minima; and examine behaviour as x → ∞ and x → –∞

    • 11-12.ACMMM096 solve optimisation problems arising in a variety of contexts involving simple polynomials on finite interval domains.

  • 11-12. Anti-derivatives:

    • 11-12.ACMMM097 calculate anti-derivatives of polynomial functions and apply to solving simple problems involving motion in a straight line.

11-12.3 Discrete random variables

11-12.3 Counting and probability