Unit 2: Calculus and further functions

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.

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.

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 $y = 3 \cdot 2^{x-1} - 4$.

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.

How the graph of a logarithmic function relates to its exponential inverse, how to find the vertical asymptote, domain, intercepts and key points for $y=\log_b(x-h)+k$, and how to solve logarithmic equations and check for extraneous solutions. Includes worked examples and practice questions.

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.

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.

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.

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.

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.

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).

How to apply the chain rule, product rule and quotient rule to polynomial and power functions in Unit 2, how to choose the right rule, how to factorise the result, and how to find gradients at a point. Checked worked examples and practice questions with model answers.

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.

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.

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.