Unit 2: Calculus

A focused answer to the QCE Math Methods Unit 2 dot point on the derivative as a limit. Sets up the difference quotient, evaluates the limit as $h \to 0$, and works the QCAA-style first-principles problem for $f(x) = 3x^2 - 5x$ from EA Paper 1.

A focused answer to the QCE Math Methods Unit 2 dot point on the power rule and combined-rule differentiation of polynomials. States the rules, applies them to a fourth-degree polynomial, and works the QCAA-style tangent-line problem at a specified point.

A focused answer to the QCE Math Methods Unit 2 dot point on stationary points. Locates stationary points by solving $f'(x) = 0$, classifies them as maxima, minima or stationary points of inflection using the first-derivative sign test, and works the QCAA-style optimisation problem (maximising the area of a fenced rectangle).

A focused answer to the QCE Math Methods Unit 2 subject-matter point on extended functions. Exponential growth and decay models, logarithmic functions, trigonometric functions (unit circle, exact values, graphs and transformations), and applications.

A focused answer to the QCE Math Methods Unit 2 dot point on exponential functions. Sketches $y = b^x$ for $b > 1$ and $0 < b < 1$, identifies the y-intercept, horizontal asymptote, domain and range, and works the QCAA-style transformation problem $y = 3 \cdot 2^{x-1} - 4$.

A focused answer to the QCE Math Methods Unit 2 dot point on exponential growth and decay. Sets up models from worded scenarios, switches between $y = A r^t$ and $y = A e^{kt}$, and works the QCAA-style continuous compound interest and radioactive-decay problems from IA1 and EA Paper 2.

A focused answer to the QCE Math Methods Unit 2 dot point on the laws of indices. Lists the seven exponent laws, applies them to rational and negative powers, and works the QCAA-style equation $a^{2x+1} = a^{x-3}$ style problem used in IA1 and EA.

A focused answer to the QCE Math Methods Unit 2 dot point on logarithms. States the definition $\log_b x = y \iff b^y = x$, derives the laws (product, quotient, power, change of base), and works the QCAA-style exponential equation $5^x = 28$ using logs.

A focused answer to the QCE Math Methods Unit 2 subject-matter point on differential calculus. The gradient as the slope of a tangent, the derivative as a function, the power rule $d/dx(x^n) = nx^{n-1}$, and applications to tangent lines and stationary points; foundation for Unit 3 calculus.

A focused answer to the QCE Math Methods Unit 2 subject-matter point on discrete probability distributions. Probability mass functions, expected value $E(X) = \sum x P(X=x)$, variance $\text{Var}(X) = E(X^2) - [E(X)]^2$, and the Bernoulli distribution; foundation for Unit 3 binomial.

A focused answer to the QCE Math Methods Unit 2 dot point on radian measure. Defines $1$ radian as the angle subtending an arc equal to the radius, converts between degrees and radians, derives arc length $s = r\theta$, and tabulates the exact values of sine, cosine and tangent at common unit-circle angles.

A focused answer to the QCE Math Methods Unit 2 dot point on trig graphs. Sketches $y = \sin x$ and $y = \cos x$, identifies amplitude $|a|$, period $2\pi/b$, horizontal phase shift $h$ and vertical translation $k$ in the transformed forms, and works the QCAA-style modelling problem with periodic temperature.

A focused answer to the QCE Math Methods Unit 2 dot point on trig identities. States the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, derives the tangent identity, and works the QCAA-style "given $\sin\theta$, find $\cos\theta$ and $\tan\theta$" problem with quadrant reasoning.