§-Unit

Unit 3

13 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How are the distribution, expected value and variance of a discrete random variable defined and computed?

Discrete random variables, their probability distributions, the expected value (mean)

E(X) = \sum x P(X = x)

, the variance

\mathrm{Var}(X) = E(X^2) - [E(X)]^2

and the standard deviation

\mathrm{sd}(X) = \sqrt{\mathrm{Var}(X)}

VCE Math Methods Unit 3 key-knowledge point on discrete random variables. Probability distributions, expected value, variance and standard deviation, the linearity rule for $E(aX + b)$ , and the standard Paper 1 patterns.

8 min read →

How do transformations, composites and inverses build new functions from old, and what conditions guarantee they exist?

Transformations from

y = f(x)

y = A f(n(x - b)) + c

(dilation, reflection, translation), composite functions

(f \circ g)(x) = f(g(x))

and the conditions for their existence, and inverse functions

f^{-1}

with the link to one-to-one functions

VCE Math Methods Unit 3 key-knowledge point on building functions from old. The standard transformation form $A f(n(x-b)) + c$ , composite functions and existence conditions, inverses and one-to-one domain restriction, and standard Paper 1 patterns.

10 min read →

How do the factor and remainder theorems let us factorise and analyse polynomials by hand?

The factor theorem and the remainder theorem for polynomial functions, the method of equating coefficients, and the factorisation of cubic and quartic polynomials over the rationals

VCE Math Methods Unit 3 key-knowledge point on the factor and remainder theorems. Statement of the theorems, the trial-and-divide method, equating coefficients, and the standard Paper 1 cubic factorisation pattern.

8 min read →

What are the graphs of the sine, cosine and tangent functions and what features do they have under transformation?

Graphs of circular functions

f(x) = \sin(x)

f(x) = \cos(x)

and

f(x) = \tan(x)

, their key features (period, amplitude, asymptotes), exact values at standard angles, and graphs of the form

f(x) = a\sin(b(x - h)) + k

VCE Math Methods Unit 3 key-knowledge point on circular functions. Sine, cosine and tangent graphs, period and amplitude, exact unit-circle values, transformed trig graphs, and standard Paper 1 patterns.

9 min read →

What are the key features of exponential and logarithmic graphs, and how are they related?

Graphs of exponential functions

f(x) = a^x

(in particular

f(x) = e^x

) and logarithmic functions

f(x) = \log_a(x)

(in particular

f(x) = \ln(x)

), including their key features and the inverse relationship

VCE Math Methods Unit 3 key-knowledge point on exponential and logarithmic functions. Graphs of $e^x$ and $\ln(x)$ , transformations, log laws, the inverse relationship, and standard Paper 1 exam patterns.

9 min read →

How are polynomial, power and modulus functions defined, and what are the key features of their graphs?

Graphs of polynomial functions and key features including stationary points and points of inflection, intercepts, asymptotes, end behaviour, and the graphs of power functions

f(x) = x^n

for

n \in Q

and the modulus function

f(x) = |x|

VCE Math Methods Unit 3 key-knowledge point on polynomial, power and modulus functions. Cubic and quartic shapes, rational powers including square root and cube root, the modulus graph, and the standard exam patterns.

9 min read →

How are the first and second derivatives used to find tangent lines, classify stationary points and sketch curves?

Equations of tangents and normals to graphs of functions, stationary points and points of inflection, use of the first and second derivatives to classify stationary points, and curve sketching

Tangents, normals and stationary points for VCE Math Methods Unit 3: first and second derivative tests, points of inflection, and curve sketching.

10 min read →

What are Bernoulli trials, when does the binomial distribution apply, and how are its probabilities, mean and variance computed?

Bernoulli trials, the binomial distribution

X \sim \mathrm{Bi}(n, p)

, its probability function

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

, mean

E(X) = np

, and variance

\mathrm{Var}(X) = np(1-p)

VCE Math Methods Unit 3 key-knowledge point on the binomial distribution. Bernoulli trial conditions, the binomial probability formula, the mean and variance shortcuts, CAS usage on Paper 2, and standard Paper 1 patterns.

9 min read →

How are polynomial, exponential, logarithmic and circular equations solved exactly, especially without a calculator?

Solution of polynomial equations of low degree with real coefficients, exponential and logarithmic equations using properties such as

a^x = e^{x\ln a}

, and circular equations using exact unit-circle values

VCE Math Methods Unit 3 key-knowledge point on solving equations. Polynomial, exponential, logarithmic and circular equations using factoring, log laws, exact values, and the substitution trick. Standard Paper 1 exam patterns.

10 min read →

How is differentiation applied to optimisation problems and to interpreting rates of change?

Applications of differentiation to optimisation problems (maximising or minimising a quantity subject to constraints) and to rates of change in modelled real-world contexts

Optimisation and rates of change for VCE Math Methods Unit 3: the six-step recipe, checking endpoints, and the standard Paper 2 Section B patterns.

9 min read →

How do the product, quotient and chain rules combine with standard derivatives to differentiate any function built from polynomial, exponential, logarithmic and trigonometric pieces?

The product, quotient and chain rules of differentiation, and the derivatives of standard functions

x^n

for

n \in Q

e^x

\ln(x)

\sin(x)

\cos(x)

and

\tan(x)

VCE Math Methods Unit 3 key-knowledge point on the differentiation rules. The product, quotient and chain rules, the standard derivatives of polynomial, exponential, logarithmic and circular functions, and the standard Paper 1 patterns.

9 min read →

How are probabilities of events computed, including for combined and conditional events?

Random experiments, sample spaces, events and probabilities, including the addition rule, conditional probability

P(A|B) = \frac{P(A \cap B)}{P(B)}

, the multiplication rule, and the concept of independence

VCE Math Methods Unit 3 key-knowledge point on probability fundamentals. Sample spaces and events, the addition and multiplication rules, conditional probability and independence, and the standard Paper 1 and Paper 2 patterns.

8 min read →

What is the formal definition of the derivative, and how is it computed from the limit?

Average and instantaneous rates of change, the definition of the derivative as a limit

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

, and the use of this definition to differentiate from first principles

Differentiation from first principles for VCE Math Methods: average versus instantaneous rate of change, the limit definition of the derivative (Unit 2 content that Units 3 and 4 assume), a four-step method and worked examples.

7 min read →