# VCE Specialist Mathematics Units 3 and 4 – Calculus

3.1 Inverse Trigonometric Functions

3.2 Differentiation

3.3 Integration

3.4 Area by Integration

3.5 Volume by Integration

3.6 Differential Equations

3.7 Kinematics

3.8 Related Rates of Change

In this area of study, students cover advanced calculus techniques for analytic and numeric differentiation and integration of a range of functions, and combinations of functions; and their application in a variety of theoretical and practical situations, including curve sketching, evaluation of arc length, area and volume, differential equations and kinematics.

This area of study includes:

Differential and integral calculus, including:

- derivatives of inverse circular functions
- second derivatives, use of notations \( f^{\prime \prime}(x) \) and \( \displaystyle \frac{d^2y}{dx^2} \) and their application to the analysis of graphs of functions, including points of inflection and concavity
- applications of chain rule to related rates of change and implicit differentiation; for example, implicit differentiation of the relations \( x^2 + y^2 = 16 \) and \( 4xy^2 = x + y \)
- techniques of anti-differentiation and for the evaluation of definite integrals:
- anti-differentiation of \( \displaystyle \frac{1}{x} \) to obtain \( \log_e |x|\)
- anti-differentiation of \( \displaystyle \frac{1}{\sqrt{a^2-x^2}} \) and \( \displaystyle \frac{1}{a^2+x^2} \) recognition that they are derivatives of corresponding inverse circular functions
- use of the substitution \( u = g(x) \) to anti-differentiate expressions
- use of the trigonometric identities \( \sin^2 (ax) = \displaystyle \frac{1}{2}\left(1–\cos(2ax)\right), \cos^2 (ax) = \frac{1}{2} \left(1 + \cos(2ax)\right) \), in anti-differentiation techniques
- anti-differentiation using partial fractions of rational functions

- relationship between the graph of a function and the graphs of its anti-derivative functions
- numeric and symbolic integration using technology
- application of integration, arc lengths of curves, areas of regions bounded by curves and volumes of solids of revolution of a region about either coordinate axis.

Differential equations, including:

- formulation of differential equations from contexts in, for example, physics, chemistry, biology and economics, in situations where rates are involved (including some differential equations whose analytic solutions are not required, but can be solved numerically using technology)
- verification of solutions of differential equations and their representation using direction (slope) fields
- solution of simple differential equations of the form \( \displaystyle \frac{dy}{dx} = f(x), \frac{dy}{dx} = g(y) \), and in general differential equations of the form \( \displaystyle \frac{dy}{dx} = f(x) g(y) \) using separation of variables and differential equations of the form \( \displaystyle \frac{d^2y}{dx^2} = f(x) \)
- numerical solution by Euler’s method (first order approximation).

Kinematics: rectilinear motion, including:

- application of differentiation, anti-differentiation and solution of differential equations to rectilinear motion of a single particle, including the different derivative forms for acceleration \( \displaystyle a = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v \frac{dv}{dx} = \frac{d}{dx}\left( \frac{1}{2} v^2 \right) \)
- use of velocity–time graphs to describe and analyse rectilinear motion.

*source – VCE Mathematics Study Design*

VCE Specialist Mathematics Units 3 and 4 Courses

VCE Specialist Mathematics Units 3 and 4 Syllabus

## Course Content

### Inverse Trigonometric Functions

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