Learning Goal Breakdown

Unit 3: Further calculus

Topic 1: The logarithmic function

• Establish and use logarithmic laws and definitions
• Interpret and use logarithmic scales such as:
  • Decibels in acoustics
  • The Richter scale for earthquake magnitude
  • Octaves in music
  • pH in chemistry
• Solve equations involving indices with and without technology
• Recognise the qualitative features of the graph of y = loga(x) (a > 1), including asymptotes, and of its translations:
  • y = loga(x) + b
  • y = loga(x + c)
• Solve equations involving logarithmic functions with and without technology
• Identify contexts suitable for modelling by logarithmic functions and use them to:
  • Solve practical problems
  • Verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

Topic 2: Further differentiation and applications

Calculus of exponential functions

• Estimate the limit of (ah - 1)/h as h → 0 using technology, for various values of a > 0
• Recognise that e is the unique number a for which the above limit is 1
• Define the exponential function ex
• Establish and use the formulas:
  • d/dx(ex) = ex
  • d/dx(ef(x)) = f'(x)ef(x)
• Identify contexts suitable for mathematical modelling by exponential functions and their derivatives and use the model to:
  • Solve practical problems
  • Verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

Calculus of logarithmic functions

• Define the natural logarithm ln(x) = loge(x)
• Recognise and use the inverse relationship of the functions:
  • y = ex
  • y = ln(x)
• Establish and use the formulas:
  • d/dx(ln(x)) = 1/x
  • d/dx(ln(f(x))) = f'(x)/f(x)
• Use logarithmic functions and their derivatives to solve practical problems

Calculus of trigonometric functions

• Establish the formulas:
  • d/dx(sin(x)) = cos(x)
  • d/dx(cos(x)) = -sin(x)
  by numerical estimations of the limits and informal proofs based on geometric constructions
• Identify contexts suitable for modelling by trigonometric functions and their derivatives and use the model to:
  • Solve practical problems
  • Verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis
• Use trigonometric functions and their derivatives to solve practical problems, including functions of the form:
  • y = sin(f(x))
  • y = cos(f(x))

Differentiation rules

• Select and apply:
  • Product rule
  • Quotient rule
  • Chain rule
  to differentiate functions; express derivatives in simplest and factorised form

Topic 3: Integrals

Anti-differentiation

• Recognise anti-differentiation as the reverse of differentiation
• Use the notation ∫𝑓(𝑥)𝑑𝑥 for anti-derivatives or indefinite integrals
• Establish and use the formula ∫𝑥ⁿ𝑑𝑥 = (1/(𝑛+1))𝑥ⁿ⁺¹ + 𝑐 for 𝑛 ≠ −1
• Establish and use the formula ∫𝑒ˣ𝑑𝑥 = 𝑒ˣ + 𝑐
• Establish and use the formulas:
  • ∫1/𝑥 𝑑𝑥 = ln|𝑥| + 𝑐 for 𝑥 > 0
  • ∫1/(𝑎𝑥 + 𝑏) 𝑑𝑥 = (1/𝑎) ln|𝑎𝑥 + 𝑏| + 𝑐
• Establish and use the formulas:
  • ∫sin(𝑥)𝑑𝑥 = −cos(𝑥) + 𝑐
  • ∫cos(𝑥)𝑑𝑥 = sin(𝑥) + 𝑐
• Understand and use the formula for indefinite integrals of the form ∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫𝑓(𝑥)𝑑𝑥 + ∫𝑔(𝑥)𝑑𝑥
• Determine indefinite integrals of the form ∫𝑓(𝑎𝑥 + 𝑏)𝑑𝑥
• Determine 𝑓(𝑥), given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏
• Determine the integral of a function using information about the derivative of the given function (integration by recognition)
• Determine displacement given velocity in linear motion problems

Fundamental theorem of calculus and definite integrals

• Examine the area problem, and use sums of the form ∑𝑓(𝑥ᵢ) Δ𝑥ᵢ to estimate the area under the curve 𝑦 = 𝑓(𝑥)
• Use the trapezoidal rule for the approximation of the value of a definite integral numerically
• Interpret the definite integral ∫[𝑎,𝑏] 𝑓(𝑥)𝑑𝑥 as the area under the curve 𝑦 = 𝑓(𝑥) if 𝑓(𝑥) > 0
• Recognise the definite integral ∫[𝑎,𝑏] 𝑓(𝑥)𝑑𝑥 as a limit of sums of the form ∑𝑓(𝑥ᵢ) Δ𝑥ᵢ
• Understand the formula ∫[𝑎,𝑏] 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎), where 𝐹 is an anti-derivative of 𝑓, and use it to calculate definite integrals

Applications of integration

• Calculate the area under a curve
• Calculate total change by integrating instantaneous or marginal rates of change
• Calculate the area between curves with and without technology
• Determine displacements given acceleration and initial values of displacement and velocity

Unit 4: Further functions and statistics

Topic 1: Further differentiation and applications 3

The second derivative and applications of differentiation

• Understand the concept of the second derivative as the rate of change of the first derivative function.
• Recognise acceleration as the second derivative of displacement position with respect to time.
• Understand the concepts of concavity and points of inflection and their relationship with the second derivative.
• Understand and use the second derivative test for finding local maxima and minima.
• Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection.
• Solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is both given and developed.

Topic 2: Trigonometric functions 2

Cosine and sine rules

• Recall sine, cosine, and tangent as ratios of side lengths in right-angled triangles.
• Understand the unit circle definition of cos(𝜃), sin(𝜃), and tan(𝜃), including periodicity using degrees and radians.
• Establish and use the sine rule (ambiguous case is required), cosine rule, and the formula for the area of a triangle: area = 1/2bcsin(A).
• Construct mathematical models using the sine and cosine rules in two- and three-dimensional contexts (including bearings in two-dimensional contexts).
• Use the model to solve problems and verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis.

Topic 3: Discrete random variables 2

Bernoulli distributions

• Use a Bernoulli random variable as a model for two-outcome situations.
• Identify contexts suitable for modelling by Bernoulli random variables.
• Recognise and determine the mean p and variance p(1 - p) of the Bernoulli distribution with parameter p.
• Use Bernoulli random variables and associated probabilities to model data and solve practical problems.

Binomial distributions

• Understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in n independent Bernoulli trials, with the same probability of success p in each trial.
• Identify contexts suitable for modelling by binomial random variables.
• Determine and use the probabilities P(X = r) = C(n, r) * p^r * (1 - p)^(n - r) associated with the binomial distribution with parameters n and p.
• Calculate the mean np and variance np(1 - p) of a binomial distribution using technology and algebraic methods.
• Identify contexts suitable to model binomial distributions and associated probabilities to solve practical problems, including the language of ‘at most’ and ‘at least’.

Topic 4: Continuous random variables and the normal distribution

General continuous random variables

• Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable.
• 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.
• Calculate the expected value, variance, and standard deviation of a continuous random variable in simple cases.
• Understand standardised normal variables (z-values, z-scores) and use these to compare samples.

Normal distributions

• Identify contexts, such as naturally occurring variations, that are suitable for modelling by normal random variables.
• Recognise features of the graph of the probability density function of the normal distribution with mean μ and standard deviation σ, and the use of the standard normal distribution.
• Calculate probabilities and quantiles associated with a given normal distribution using technology and use these to solve practical problems.

Topic 5: Interval estimates for proportions

Random sampling

• Understand the concept of a random sample.
• Discuss sources of bias in samples, and procedures to ensure randomness.
• Investigate the variability of random samples from various types of distributions, including uniform, normal, and Bernoulli, using graphical displays of real and simulated data.

Sample proportions

• 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).
• Consider the approximate normality of the distribution of p for large samples.
• 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 (p̂ - p)/√(p̂(1 − p̂)/n), where the closeness of the approximation depends on both n and p.

Confidence intervals for proportions

• Understand the concept of an interval estimate for a parameter associated with a random variable.
• Use the approximate confidence interval [p̂ - z√(p̂(1−p̂)/n), p̂ + z√(p̂(1−p̂)/n)] as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution.
• Define the approximate margin of error E = z√(p̂(1−p̂)/n) and understand the trade-off between margin of error and level of confidence.
• Use simulation to illustrate variations in confidence intervals between samples and to show that most, but not all, confidence intervals contain p.