Unit 2: Calculus
13 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.
Topic 3: Introduction to differential calculus
- Define the derivative of a function as a limit and use first principles to find the derivative of a polynomial function
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.
5 min answer → - Apply the power rule, the sum rule, and the constant-multiple rule to differentiate polynomial functions, and use the derivative to find tangent and normal line equations
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.
5 min answer → - Use the derivative to find stationary points of a polynomial function and classify them, and apply differentiation to simple optimisation problems
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).
6 min answer →
How are exponential, logarithmic and trigonometric functions extended in QCE Math Methods Unit 2?
Topic 1: Exponential functions
- Graph and analyse exponential functions of the form $y = a \cdot b^x + c$, identifying key features (intercepts, asymptote, domain, range) and applying transformations
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 $y = 3 \cdot 2^{x-1} - 4$.
5 min answer → - Model exponential growth and decay using $y = A \cdot r^t$ or $y = A e^{kt}$, including problems involving population growth, radioactive decay, depreciation and continuous compound interest
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.
6 min answer → - Recall and apply the laws of indices to simplify expressions and solve equations involving rational and negative exponents
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.
5 min answer → - Define logarithms as the inverse of exponentials, apply the laws of logarithms, and solve exponential equations using logarithms
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.
6 min answer →
How is differential calculus introduced in QCE Math Methods Unit 2?
What discrete probability distributions does QCE Math Methods Unit 2 introduce?
Topic 2: Trigonometric functions
- Define radian measure of angle and relate to arc length; evaluate exact values of sine, cosine and tangent of common angles using the unit circle
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.
5 min answer → - Sketch and analyse graphs of $y = a \sin(b(x - h)) + k$ and $y = a \cos(b(x - h)) + k$, identifying amplitude, period, phase shift and vertical translation
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.
5 min answer → - State and apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and use it together with related identities to simplify expressions and solve equations
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.
5 min answer →