Calculus (ME-C1, C2, C3)

11 dot points across 11 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we model projectile motion in two dimensions, and what quantities (range, maximum height, time of flight) can we extract?

Model projectile motion in two dimensions using parametric equations and find range, maximum height, time of flight and trajectory equation

Projectile motion. The parametric equations for position, velocity and acceleration under gravity, the Cartesian trajectory equation, and standard quantities (range, maximum height, time of flight), with worked examples.

13 min read →

How do we describe the motion of a particle in the plane with a position vector, and what do its derivatives tell us about velocity and acceleration?

Describe motion in the plane using a position vector r(t), differentiate to get velocity v(t)=r'(t) and acceleration a(t)=r''(t), and use the dot product to analyse the motion (speed, perpendicular velocities, angle between v and a)

Motion as a vector function. The position vector r(t)=x(t)i+y(t)j, differentiating to velocity v(t)=r'(t) and acceleration a(t)=r''(t), speed |v|, the path, and the dot product as a tool: the angle between v and a, when speed increases or decreases, perpendicular velocities at a collision, and closest approach, with worked examples.

16 min read →

What are the derivatives and antiderivatives of the inverse trigonometric functions?

Differentiate inverse trigonometric functions and integrate functions that involve them

Calculus of inverse trig functions. The derivatives of $\arcsin$ , $\arccos$ and $\arctan$ , their graphs and domains, their chain-rule extensions, and integrals leading back to inverse trig, built up stage by stage with worked examples.

14 min read →

How do we differentiate the inverse of a function from the function itself, and what is the gradient of the inverse at a point?

Differentiate an inverse function using (f^-1)'(x) = 1/f'(f^-1(x)) (equivalently dy/dx = 1/(dx/dy)), including the second derivative

Differentiating an inverse function: the rule $(f^{-1})'(x) = 1/f'(f^{-1}(x))$ , equivalently $dy/dx = 1/(dx/dy)$ , where it comes from (chain rule and reflection in $y=x$ ), evaluating the gradient of an inverse at a matching point, the second derivative of an inverse, and the inverse-trig and log derivatives as special cases.

15 min read →

How do we model and solve problems involving exponential growth and decay using $\frac{dN}{dt} = k N$ ?

Model unrestricted growth and decay with

\frac{dN}{dt} = k N

and solve the resulting separable differential equation

Exponential growth and decay. The differential equation $\frac{dN}{dt} = k N$ , its solution $N = N_0 e^{kt}$ , doubling time, half-life, the modified law $\frac{dN}{dt} = k(N - P)$ built up stage by stage, and applications, with worked examples.

14 min read →

How do we use the substitution method to evaluate integrals that arise from the reverse chain rule?

Apply integration by substitution to evaluate definite and indefinite integrals, including reverse chain rule cases

Integration by substitution. Choosing the right substitution, transforming the integrand and differential, changing limits for definite integrals, and standard reverse chain rule patterns, with worked examples.

13 min read →

How do we link the rate of change of one quantity to the rate of change of another using the chain rule?

Related rates. Linking two changing quantities through an equation, differentiating implicitly with respect to time, and substituting instantaneous values, with worked examples.

12 min read →

How do we solve a first-order differential equation when the variables separate?

Solve separable first-order differential equations of the form

\frac{dy}{dx} = f(x) g(y)

by separating variables and integrating both sides

Separable differential equations. The separation-of-variables method, integration on both sides, applying initial conditions, and standard models including Newton's law of cooling, with worked examples.

12 min read →

How do we compute the volume of a solid generated by rotating a region around an axis?

Calculate volumes of revolution about the x-axis and y-axis using the disc method

Volumes of revolution. The disc method for rotation about the x-axis and y-axis built up stage by stage, the washer method, the integral setup, and the handling of regions bounded by curves and lines, with worked examples.

15 min read →

How do we find the gradient of a curve when x and y are both given as functions of a parameter, and use it for tangents, normals and turning points?

Find the derivative of a function defined parametrically using the chain rule

The Year 12 Extension 1 answer on differentiating parametric functions, new in the 2024 syllabus: why the chain rule gives dy/dx as dy/dt divided by dx/dt, when tangents are horizontal or vertical, and tangents, normals and flight-path angles for parametric curves such as circles, parabolas and projectiles, with checked worked examples.

12 min read →

Which integrals lead to inverse trigonometric antiderivatives, and how do we recognise them?

Integrate functions whose antiderivative involves

\arcsin

\arccos

\arctan

Inverse-trig integrals. The standard inverse-trig antiderivatives, recognising the $a^2 - x^2$ and $a^2 + x^2$ patterns, completing the square to fit them shown stage by stage, and linear substitutions, with worked examples.

14 min read →