VCE Specialist Mathematics Units 3 and 4 Calculus
Course

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

55 Lessons

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

INVERSE TRIGONOMETRIC FUNCTIONS

Differentiation of Inverse Sine Function

Differentiation of Inverse Cosine Function

Differentiation of Inverse Tangent Function

DIFFERENTIATION

Points of Inflection and Concavity

Implicit Differentiation

Product Rule

INTEGRATION

Indefinite Integration by Recognition

Integration by Substitution

Integration by Partial Fractions

Indefinite Integration by Parts

Further Integration of Trigonometric Functions by Substitution

Further Integration of Trigonometric Functions using Properties

Integration using Substitution in Right-Angled Triangles

Definite Integration by Substitution

Integration of Rational Functions

Integrations Resulted in Natural Log Functions by Substitutions

Definite Integral of Rational Functions

Integrations Resulting Natural Logarithmic Functions

Further Graphs of Derivatives

Definite Integrals of Even and Odd Functions

AREA BY INTEGRATION

Area Under a Curve

Signed Areas

Area between Two Functions

Area between Two Functions involving Signed Areas

Area under Trigonometric Curves

Problem Solving by Integration

Area Between Y-axis

VOLUME BY INTEGRATION

Volumes for Two Functions

Volumes using Integration of Trigonometric Functions

Volumes using Integration

Volumes by Slicing Method Rotated y-axis

Volumes by Slicing Method Rotated x-axis

Volumes by Cylindrical Shells Method

Volumes of Cross Sections

Volumes by Slicing Method by Removing Hollow

DIFFERENTIAL EQUATIONS

Differential Equations of the Form dy/dx = f(x)

Differential Equations of the Form dy/dx = g(y)

Separable Differential Equations

Setting Up Differential Equations

KINEMATICS

Kinematics using Differentiation

Velocity and Acceleration

Differentiation and Displacement, Velocity and Acceleration

Kinematics using Integration

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