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*

#### Differentiation of Inverse Sine Function

- Video - Differentiation Formula of Inverse Sine Function (2:06)
- Video - Proof of Differentiation of Inverse Sine Function (2:48)
- Video - Differentiation of Inverse Sine Function (1:37)
- Video - Differentiation of Inverse Sine Function involving Chain Rule (2:36)
- Video - Differentiation of Inverse Sine Function involving Product Rule (1:38)
- Video - Differentiation of Inverse Sine Function involving Exponential Function (1:00)
- Video - Differentiation of Inverse Sine Function involving Reciprocal Function (1:36)
- Topic - Proving Differentiation of Inverse Sine
- Topic - Differentiation of Inverse Sine involving Chain Rule 1
- Topic - Differentiation of Inverse Sine involving Chain Rule 2
- Topic - Differentiation of Inverse Sine involving Chain Rule 3
- Topic - Differentiation of Inverse Sine involving Chain Rule 4
- Topic - Differentiation of Inverse Sine involving Chain Rule 5
- Topic - Differentiation of Inverse Sine involving Chain Rule 6
- Topic - Differentiation of Inverse Sine involving Chain Rule 7

#### Differentiation of Inverse Cosine Function

- Video - Differentiation Formula of Inverse Cosine Function (2:12)
- Video - Proof of Differentiation of Inverse Cosine Function (1:50)
- Video - Differentiation of Inverse Cosine Function (1:25)
- Video - Differentiation of Inverse Cosine Function involving Chain Rule (1:51)
- Video - Differentiation of Inverse Cosine Function involving Surds (1:48)
- Video - Differentiation of Inverse Cosine Function involving Sine Function (1:34)
- Topic - Proving Differentiation of Inverse Cosine
- Topic - Differentiation of Inverse Cosine involving Chain Rule 1
- Topic - Differentiation of Inverse Cosine involving Chain Rule 2
- Topic - Differentiation of Inverse Cosine involving Chain Rule 3
- Topic - Differentiation of Inverse Cosine involving Chain Rule 4
- Topic - Differentiation of Inverse Cosine involving Chain Rule 5
- Topic - Differentiation of Inverse Cosine involving Chain Rule 6
- Topic - Differentiation of Inverse Cosine involving Chain Rule 7

#### Differentiation of Inverse Tangent Function

- Video - Differentiation Formula of Inverse Tangent Function (2:27)
- Video - Proof of Differentiation of Inverse Tangent Function (1:31)
- Video - Differentiation of Inverse Tangent Function (1:40)
- Video - Differentiation of Inverse Tangent Function involving Chain Rule (3:43)
- Video - Differentiation of Inverse Tangent Function involving Logarithmic Function (2:27)
- Video - Differentiation of Inverse Tangent Function involving Sine Function (1:41)
- Topic - Proving Differentiation of Inverse Tangent
- Topic - Differentiation of Inverse Tangent involving Chain Rule 1
- Topic - Differentiation of Inverse Tangent involving Chain Rule 2
- Topic - Differentiation of Inverse Tangent involving Chain Rule 3
- Topic - Differentiation of Inverse Tangent involving Chain Rule 4
- Topic - Differentiation of Inverse Tangent involving Chain Rule 5
- Topic - Differentiation of Inverse Tangent involving Chain Rule 6
- Topic - Differentiation of Inverse Tangent involving Chain Rule 7

#### Implicit Differentiation

- Topic - Implicit Differentiation of Circles
- Topic - Implicit Differentiation of Circles in General Form 1
- Topic - Implicit Differentiation of Circles in General Form 2
- Topic - Implicit Differentiation of Parabola
- Topic - Implicit Differentiation of Hyperbola
- Topic - Implicit Differentiation of Ellipse

#### Product Rule

- Video - Fundamental of Product Rule (7:49)
- Video - Product Rule and Chain Rule 1 (3:32)
- Video - Product Rule and Chain Rule 2 (4:12)
- Video - Derivative Value using Product Rule and Chain Rule (3:26)
- Topic - Product Rule: Linear Products
- Topic - Product Rule: Non-Monic Linear Products
- Topic - Product Rule: Quadratic Products
- Topic - Product Rule: Complete Square & Linear Products
- Topic - Product Rule: Complete Cubic and Quadratic Products

#### Indefinite Integration by Recognition

- Video - Reverse Chain Rule (7:06)
- Video - Integration by Recognition: Using Derivative of Exponential Functions (3:45)
- Video - Integration by Recognition using Derivative of Sine Function (3:32)
- Video - Integration by Recognition using Derivative of Cosine Function (4:20)
- Video - Integration by Recognition using Derivative of Tangent Function (4:16)
- Video - Integration by Recognition: Using Derivative of Logarithmic Functions (4:10)
- Video - Integration by Recognition using Product Rule (6:16)
- Video - Integration by Recognition using Long Division (7:22)
- Video - Integration of cosec x by Recognition involving Log TAN (4:18)
- Video - Integration of sec x by Recognition (4:52)

#### Integration by Substitution

- Video - Integration using Substitutions 1 (3:45)
- Video - Integration using Substitutions 2 (3:05)
- Video - Integration using Substitutions 3 (2:06)
- Video - Removal of Pronumerals 1 (4:39)
- Video - Removal of Pronumerals 2 (2:28)
- Video - Removal of Pronumerals 3 (2:43)
- Topic - Integration by Substitution: Basic
- Topic - Integration by Substitution: Surds
- Topic - Integration by Substitution: Rational Expressions

#### Indefinite Integration by Parts

- Video - Understanding Integration by Parts (4:14)
- Video - Integration by Parts: x times cos x (1:33)
- Video - Integration by Parts: Square of Logarithmic Functions (2:51)
- Video - Integration by Parts: Exponential Function times cos x (3:30)
- Video - Integration by Parts: x^2 times cos x (2:12)
- Video - Integration by Parts: Inverse sin x (3:22)
- Video - Integration by Parts: Inverse tan x (1:17)
- Video - Integration by Parts: sec^3 x (4:02)
- Video - Integration by Parts involving Substitution (2:33)

#### Integration using Substitution in Right-Angled Triangles

- Video - Integration using Substitution in Right-Angled Triangles 1 (3:11)
- Video - Integration using Substitution in Right-Angled Triangles 2 (1:54)
- Video - Integration using Substitution in Right-Angled Triangles 3 (4:58)
- Video - Integration using Substitution in Right-Angled Triangles 4 (1:54)
- Video - Integration using Substitution in Right-Angled Triangles 5 (3:16)

#### Definite Integral of Rational Functions

- Video - Finding Area by Differentiating Logarithmic Functions (3:12)
- Video - Integration Rule of Rational Functions involving Natural Exponents (1:20)
- Video - Definite Integration of Rational Functions (2:10)
- Topic - Integral of Rational Functions: Basic Form
- Topic - Integral of Rational Functions: Monic Linear Denominator
- Topic - Integral of Rational Functions: Non-Monic Linear Fractions
- Topic - Integral of Rational Functions: Multi-Term Denominators
- Topic - Integral of Rational Functions: Multi-Term by Non-Monic
- Topic - Integral of Rational Functions: Non-Monic Factor by Multi-Term
- Topic - Integral of Rational Functions: Non-Monic Factor Fractions
- Topic - Integral of Rational Functions: Non-Monic Multi-Term
- Topic - Integral of Rational Functions: Non-Monic Multi-Term Fractions
- Topic - Integral of Rational Functions: Monic Quadratic Denominators
- Topic - Integral of Rational Functions: Non-Monic Quadratic Denominators
- Topic - Integral of Rational Functions: Non-Monic Cubic Denominators
- Topic - Integral of Rational Functions: Rational Expressions
- Topic - Integral of Rational Functions: Non-Monic Rational Expressions

#### Integrations Resulting Natural Logarithmic Functions

- Video - Integrations involving Natural Logarithmic functions (3:17)
- Video - Integrating Linear by Linear Resulting Natural Log Functions (3:11)
- Video - Integrating Quadratics by Linear Resulting Natural Log Functions (2:18)
- Video - Integrations of Cubic by Quadratic Resulting Natural Log Functions (2:31)
- Video - Prove Integration Formula Resulting Natural Log Functions (6:53)

#### Further Graphs of Derivatives

- Video - Understanding Derivatives of Graphs (6:14)
- Video - Graphs of Derivatives Given Graphs (8:23)
- Video - Derivative Graphs of Natural Logarithmic Graphs (2:28)
- Video - Derivative Graphs of Sine Graphs (0:51)
- Video - Derivative Graphs of Cosine Graphs (0:35)
- Video - Derivative Graphs of Rectangular Hyperbolic Graphs (1:26)
- Video - Derivative Graphs of Inverse Tangent Graphs (1:53)

#### Volumes for Two Functions

- Video - Volumes between Two Curves Rotated about x-axis (5:44)
- Video - Volumes between Two Curves (5:13)
- Topic - Volume of Revolution between Horizontal Lines & Parabolas
- Topic - Volume of Revolution between Straight Lines & Parabolas
- Topic - Volume of Revolution between Straight Lines & Surd Graphs
- Topic - Volume of Revolution about y-axis between Parabolas & Surd Graphs
- Topic - Half Volume of Revolution between Parabolas & Surd Cubic Graphs
- Topic - Volume of Revolution between Parabolas

#### Volumes using Integration

- Video - Volumes Rotated about x-axis (4:43)
- Video - Volumes Rotated about y-axis (5:58)
- Video - Volumes Rotated about x-axis and y-axis (5:48)
- Topic - Volume of Revolution bounded by Straight Lines & x-axis
- Topic - Volume of Revolution bounded by Semi-Circles & x-axis
- Topic - Volume of Revolution bounded by Surd Graphs & y-axis
- Topic - Volume of Revolution bounded by Parabolas & x-axis
- Topic - Volume of Revolution bounded by Parabolas & y-axis
- Topic - True/False of Expression of Volumes Rotating X-axis
- Topic - True/False of Expression of Volumes Rotating Y-axis

#### Volumes by Slicing Method Rotated y-axis

- Video - Volumes by Slicing Method Rotated y-axis (4:14)
- Video - Volumes by Slicing Method Rotated y-axis: y = x^2 (8:30)
- Video - Volumes by Slicing Method Rotated y-axis: y = -x^2 + 4 (3:14)
- Video - Volumes by Slicing Method Rotated y-axis: y = inverse sin x (6:54)
- Video - Volumes by Slicing Method Rotated Parallel to y-axis (1:41)
- Video - Volumes by Slicing Method Rotated Parallel to y-axis: y = 4x - x^2 (5:27)
- Video - Volumes by Slicing Method Rotated Parallel to y-axis: y = sin x (8:44)

#### Volumes by Slicing Method Rotated x-axis

- Video - Volumes by Slicing Method Rotated x-axis (2:25)
- Video - Volumes by Slicing Method Rotated x-axis: y = x^2 (2:27)
- Video - Volumes by Slicing Method Rotated x-axis: y = sin x (3:25)
- Video - Volumes by Slicing Method Rotated x-axis: Ellipse (2:32)
- Video - Volumes by Slicing Method Rotated Parallel to x-axis: y = x^2 (3:07)
- Video - Volumes by Slicing Method Rotated Parallel to x-axis: y = e^x (3:37)

#### Volumes by Cylindrical Shells Method

- Video - Volumes by Cylindrical Shells Method (4:46)
- Video - Volumes by Cylindrical Shells Method: y = ln x (6:55)
- Video - Volumes by Cylindrical Shells Rotated y-axis (1:42)
- Video - Volumes by Cylindrical Shells Rotated y-axis: y = x^2 (3:16)
- Video - Volumes by Cylindrical Shells Rotated y-axis: y = 2x - x^3 (2:18)
- Video - Volumes by Cylindrical Shells Parallel to Lines: y = sin^-1 x (8:52)
- Video - Volumes between Curves: y = x^2+2 and y = x+4 (5:23)
- Video - Volumes between Hyperbolic Curves (5:39)
- Video - Volumes of Torus by Cylindrical Shells Method (6:33)

#### Volumes of Cross Sections

- Video - Volumes of Rectangular Cross Section (3:21)
- Video - Volumes of Right Square Pyramid by Rectangular Cross Section (10:17)
- Video - Volumes of Triangular Section (6:10)
- Video - Volumes of Equilateral Triangle Cross Section (4:46)
- Video - Volumes of Triangular Cross Section (8:01)
- Video - Volumes of Circular Cross Section (12:11)
- Video - Volumes of Conical Cross Section (9:21)

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

- Topic - General Solutions of dy/dx = f(x) Polynomial Functions
- Topic - General Solutions of dy/dx = f(x) Exponential Functions
- Topic - General Solutions of dy/dx = f(x) Surd Functions
- Topic - General Solutions of dy/dx = f(x) Trigonometric Functions
- Topic - Particular Solutions of dy/dx = f(x) Substitution Method
- Topic - Particular Solutions of dy/dx = f(x) Multiple Choices

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

- Topic - General Solutions of dy/dx = g(y) Polynomials
- Topic - General Solutions of dy/dx = g(y) Trigonometric Functions
- Topic - Particular Solutions of dy/dx = g(y) Inverse Sine Functions
- Topic - General Solutions of dy/dx = g(y) Exponential Functions
- Topic - General Solutions of dy/dx = g(y) Square Root Functions
- Topic - Particular Solutions of dy/dx = g(y) Surd Functions

#### Differentiation and Displacement, Velocity and Acceleration

- Video - Velocity and Speed (3:04)
- Video - Kinematics given Velocity (9:49)
- Video - Kinematics given Displacement (10:51)
- Topic - Working with Displacement
- Topic - Moving Forward or Backward
- Topic - Obtaining Velocity from Displacement
- Topic - Obtaining Acceleration from Velocity
- Topic - Obtaining Acceleration from Displacement
- Topic - Obtaining Displacement from Velocity
- Topic - Obtaining Displacement from Acceleration
- Topic - Understanding Velocity and Speed
- Topic - Total Distance Travelled of Linear Displacement
- Topic - Total Distance Travelled of Quadratic Displacement
- Topic - Total Distance Travelled of Cubic Displacement

#### Kinematics using Integration

- Video - Expression of the Distance using Integration (2:23)
- Video - Working with Velocity in Trigonometric Functions (3:11)
- Video - Working with Acceleration in Natural Exponential Functions (5:12)
- Video - Maximum Speed using Acceleration 1 (5:38)
- Video - Maximum Speed using Acceleration 2 (2:54)
- Video - Changes in Displacement and Velocity (2:57)