In this area of study students cover graphical treatment of limits, continuity and differentiability of functions of a single real variable, and differentiation, anti-differentiation and integration of these functions. This material is to be linked to applications in practical situations.

This area of study includes:

- review of average and instantaneous rates of change, tangents to the graph of a given function and the derivative function
- deducing the graph of the derivative function from the graph of a given function and deducing the graph of an anti-derivative function from the graph of a given function derivatives of \( x^n \) , for \( n ∈ Q, e^x , \log_e(x), \sin(x), \cos(x) \) and \( \tan(x) \)
- derivatives of \( f (x) \pm g(x), f (x) \times g(x), \displaystyle \frac{f(x)}{g(x)} \) and \( f (g (x)) \) where \(f\) and \(g\) are polynomial functions, exponential, circular, logarithmic or power functions and transformations or simple combinations of these functions
- application of differentiation to graph sketching and identification of key features of graphs, identification of intervals over which a function is constant, stationary, strictly increasing or strictly decreasing, identification of the maximum rate of increase or decrease in a given application context (consideration of the second derivative is not required), identification of local maximum/minimum values over an interval and application to solving problems, and identification of interval endpoint maximum and minimum values
- anti-derivatives of polynomial functions and functions of the form \( f(ax + b) \) where \(f\) is \(x^n\) , for \( n ∈ Q, e^x , \sin(x), \cos(x) \) and linear combinations of these
- informal consideration of the definite integral as a limiting value of a sum involving quantities such as area under a curve, including examples such as distance travelled in a straight line and cumulative effects of growth such as inflation
- anti-differentiation by recognition that \( F^{\prime} (x) = f(x) \) implies \( \displaystyle \int f(x) dx = F(x) +c \)
- informal treatment of the fundamental theorem of calculus, \( \displaystyle \int_{a}^{b} f(x)dx = F(b)-F(a) \)
- properties of anti-derivatives and definite integrals
- application of integration to problems involving finding a function from a known rate of change given a boundary condition, calculation of the area of a region under a curve and simple cases of areas between curves, distance travelled in a straight line, average value of a function and other situations.

*source – VCE Mathematics Study Design*

Lessons

#### Constant Rates

- Video - Definition of Rates of Change (4:55)
- Topic - Choosing Constant Rates
- Topic - Graphs of Constant Rates
- Topic - Constant Rates of Change: Positive, Negative or Zero
- Topic - Constant Rates of Change: Behaviours
- Topic - Calculating Constant Rates of Change
- Topic - Constant Rates of Change: Rules
- Topic - Constant Rates of Change: Empty Swimming Pool

#### Limits at Constants

- Video - Understanding Limits at Constant Values (6:58)
- Topic - Limits at Constant Values
- Topic - Limits at Constant Values involving Common Factors
- Topic - Limits at Constant Values involving Differences of Squares
- Topic - Limits at Constant Values involving Factorise
- Topic - Limits at Constant Values involving Quadratic Factorise

#### Limits at Infinity

- Video - Understanding Limits at Infinity (6:20)
- Video - Calculating Limits at Infinity (7:51)
- Topic - Limits at Infinity
- Topic - Limits at Infinity involving Linear Fractions
- Topic - Limits at Infinity involving Quadratic Fractions
- Topic - Limits at Infinity involving Fractions Higher Degree Numerators

#### Basic Differentiation Rules

- Video - Differentiation of Constant Terms (2:14)
- Video - Differentiation of Linear Functions (3:16)
- Video - Methods of Differentiation (5:01)
- Video - Differentiation of Negative Exponents (5:12)
- Video - Differentiation of Non-Monic Negative Exponents (8:13)
- Video - Simplifying Expressions before Differentiation (6:37)
- Video - Differentiation of Polynomial Functions (3:48)
- Video - Differentiation of Functions in Terms of Specific Letters (2:43)
- Video - Differentiation of Functions in Terms of Specific Letters involving Additions (4:37)
- Video - Differentiation of Functions in Terms of Specific Letters involving Multiplications (6:20)
- Video - Equations involving Differentiation (1:51)

#### Chain Rule

- Video - Fundamental of Chain Rule (7:58)
- Video - Chain Rule involving Rational Functions (5:24)
- Video - Chain Rule involving Irrational Functions (6:06)
- Video - Derivative Value using Chain Rule involving Irrational Functions (1:46)
- Topic - Chain Rule: Power of Linear Expressions
- Topic - Chain Rule: Power of Quadratic Expressions
- Topic - Chain Rule: Surds 1/2 Power
- Topic - Chain Rule: Surds 3/2 Power
- Topic - Chain Rule: Surds 1/3 Power
- Topic - Chain Rule: Linear Rational Expressions
- Topic - Chain Rule: Quadratic Rational Expressions
- Topic - Chain Rule: Rational Expressions of Surds
- Topic - Chain Rule: Cubic Rational Expressions

#### 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

#### Quotient Rule

- Video - Fundamental of Quotient Rule (2:53)
- Video - Differentiations using Quotient Rule (7:53)
- Video - Finding Pronumerals involving Quotient Rule (2:42)
- Video - Derivative Value using Quotient Rule (1:37)
- Topic - Quotient Rule: Basic Rule
- Topic - Quotient Rule: Non-Monic Denominators
- Topic - Quotient Rule: Quadratic Numerators
- Topic - Quotient Rule: Quadratic Numerators and Denominators

#### Higher Derivatives

- Video - Fundamental of Higher Derivatives (5:34)
- Video - Higher Derivatives involving Chain Rule (4:59)
- Video - Higher Derivatives involving Surds (5:27)
- Video - Higher Derivatives involving Quotient Rule (2:40)
- Video - Higher Derivatives involving Product Rule (6:16)
- Topic - Higher Derivatives of Quadratic Polynomials
- Topic - Higher Derivatives of Cubic Polynomials
- Topic - Higher Derivatives using Chain Rule

#### Derivative of Sine Functions

- Video - Differentiation of Sine Function: Chain Rule 1 (3:21)
- Video - Differentiation of Sine Function: Chain Rule 2 (2:01)
- Video - Differentiation of Sine Function: Chain Rule 3 (2:33)
- Video - Differentiation of Sine Function: Chain Rule 4 (5:02)
- Video - Differentiation of Sine Function: Product Rule 1 (4:54)
- Video - Differentiation of Sine Function: Product Rule 2 (4:16)
- Video - Differentiation of Sine Function: Product Rule 3 (3:06)
- Video - Differentiation of Sine Function: Quotient Rule 1 (3:17)
- Video - Differentiation of Sine Function: Quotient Rule 2 (2:39)
- Video - Differentiation of Sine in Degrees (7:16)
- Video - Derivative Value of Sine Functions (1:33)

#### Derivatives of Cosine Functions

- Video - Differentiation of Cosine Function: Chain Rule 1 (3:17)
- Video - Differentiation of Cosine Function: Chain Rule 2 (5:10)
- Video - Differentiation of Cosine Function: Chain Rule 3 (4:10)
- Video - Differentiation of Cosine Function: Chain Rule 4 (3:02)
- Video - Differentiation of Cosine Function: Product Rule 1 (2:25)
- Video - Differentiation of Cosine Function: Product Rule 2 (2:58)
- Video - Differentiation of Cosine Function: Product Rule 3 (3:16)
- Video - Differentiation of Cosine Function: Quotient Rule (6:30)
- Video - Differentiation of Cosine in Degrees (2:55)
- Video - Stationary Points involving Cosine Functions (4:05)

#### Derivatives of Exponential Functions

- Video - Derivative of Exponential Functions (1:30)
- Video - Derivative of Exponential Functions using Chain Rule (6:29)
- Video - Derivative of Exponential Functions using Quotient Rule (7:20)
- Video - Derivative of Exponential Functions using Product Rule (6:12)
- Video - Second Derivative of Exponential Functions (6:10)
- Video - Equations of Tangent of Exponential Functions (4:46)
- Video - Stationary Point using Derivative of Exponential Functions (9:56)
- Video - Maximum Turning Point of Exponential Functions (7:18)

#### Derivatives of Logarithmic Functions

- Video - Fundamental of Derivative of Logarithmic Functions (2:41)
- Video - Derivative of Logarithmic Functions using Logarithm Law of Addition (4:06)
- Video - Derivative of Logarithmic Functions using Logarithm Law of Subtraction (1:47)
- Video - Derivative of Logarithmic Functions involving Indices (3:36)
- Video - Derivative of Mixed Logarithmic Functions (5:24)
- Video - Derivative of Non-Natural Logarithmic Functions (3:09)
- Video - Stationary Points using Derivative of Logarithmic Functions (5:32)
- Video - Derivative of Logarithmic Functions using Quotient Rule (4:40)
- Video - Tangents and Normals of Logarithmic Functions (2:15)

#### Tangents

- Video - Gradient of Tangents (4:46)
- Video - Application of Gradient of Tangents (7:49)
- Video - Equation of Tangents (6:15)
- Video - Angle and Gradient of Tangents (3:39)
- Video - Gradient of Parallel and perpendicular Tangents (8:38)
- Video - Horizontal Tangents (2:39)
- Video - Tangents and Corresponding y-intercepts (2:07)
- Video - Tangents and Corresponding x-intercepts (4:33)

#### Turning Points

- Video - Coordinates of Turning Points (7:30)
- Video - Identifying Nature of Turning Points from Derivative Curves (5:25)
- Video - Identifying Nature of Turning Points using Tables (5:08)
- Video - Identifying Stationary Points (8:17)
- Video - Understanding Nature of Stationary Points (5:31)
- Video - No Stationary Points (1:50)
- Video - Condition of Being Stationary Points (2:51)
- Video - Maximum and Minimum Turning Points (6:49)
- Video - Stationary Points by their Shapes (7:34)
- Video - Maximum and Minimum Turning Points of Quartic Functions (7:54)

#### Gradient Functions

- Video - Principle of Graphing Gradient Functions (5:08)
- Video - Gradient Function Graphs of Straight Lines (5:23)
- Video - Gradient Function Graphs of Parabola (8:13)
- Video - Gradient Function Graphs of Cubic Functions (10:46)
- Video - Gradient Function Graphs of Hyperbola (13:22)
- Video - Gradient Function Graphs of Surds (12:19)
- Video - Gradient Function Graphs of Periodic Functions (8:15)
- Video - Gradient Function Graphs of Quartic Functions (8:25)
- Video - Gradient Function Graphs of Semicircle (2:24)
- Video - Gradient Function Graphs of Exponential Functions (2:11)
- Video - Gradient Function Graphs of Logarithmic Functions (1:47)

#### Sketching Curves from Derivatives

- Video - Introduction to Sketching Curves from Derivatives (7:49)
- Video - Sketching Curves from Derivatives involving Concave Down (3:35)
- Video - Sketching Curves from Derivatives involving Point of Inflection (2:39)
- Video - Sketching Curves from Graphs of Derivatives 1 (4:33)
- Video - Sketching Curves from Graphs of Derivatives 2 (8:23)

#### Curves of Rational Functions in Quadratics by Quadratic Form

- Video - Understanding Curve Sketching involving Symmetry (1:24)
- Video - Even Functions Odd Functions (1:49)
- Video - Differentiation using Quotient Rule (2:29)
- Video - Turning Points of Rational Functions (1:17)
- Video - Nature of Turning Points of Rational Functions (3:03)
- Video - x-Intercepts of Rational Functions (0:58)
- Video - Vertical Asymptotes in Rational Functions (1:53)
- Video - Horizontal Asymptotes in Rational Functions (4:52)
- Video - Curve Sketching of Rational Functions involving Symmetry (1:53)

#### Integration of Power Functions

- Video - Principle of Integration of Power Function (2:49)
- Video - Integration of Power Functions (1:56)
- Video - Integration of Power Functions involving Fractions (2:23)
- Video - Integration of Power Functions Surds (1:06)
- Video - Integration of Power Functions (5:32)
- Video - Integration of Reciprocal Functions (1:35)
- Video - Integration of Irrational Functions (4:32)
- Video - Integration of Reciprocal Irrational Functions (4:41)

#### Particular Values

- Video - Finding Integral Constants (7:27)
- Video - Finding Integral Constants involving Point of Inflections (4:45)
- Video - Finding Integral Constants involving Second Derivatives (2:33)
- Topic - Finding Integral Constants from Linear Expressions
- Topic - Integral Constants by Non-Monic Expressions from Linearity
- Topic - Finding Integral Constants from Quadratics
- Topic - Integral Constants using Non-Monic Expressions from Quadratics
- Topic - Finding Two Integral Constants from Linear Expressions

#### Definite Integrals

- Video - Theory of Definite Integrals (5:46)
- Video - Evaluating Plain Definite Integrals (4:56)
- Video - Evaluating Definite Integrals of Higher Orders (8:05)
- Video - Differentiating Definite Integrals (4:38)
- Video - Finding Unknown Limits of Definite Integrals (5:00)
- Video - Finding Unknowns of Expressions involving Definite Integrals (4:54)
- Video - Definite Integrals involving Simplifications (3:47)
- Video - Definite Integrals Backwards (1:58)

#### Definite Integration of Power Functions

- Video - Definite Integrals of Power Functions (5:22)
- Topic – Definite integral of Power Functions in Positive Coefficients
- Topic – Definite integral of Power Functions in Negative Coefficients
- Topic – Definite integral of Power Functions in Rational Expressions
- Topic - Definite integral of Power Functions involving Surds

#### 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)