Functions

Working with Functions

- f(x) notation
- Domain and range 1
- Domain and range 2
- Combination of functions | Example 1 | Example 2
- The inverse of a function | Example 1 | Example 2 | Example 3
- Graphical relationship between f(x) and its inverse | Example 1 | Example 2

Transformations of Graphs (C1 Revision)

- Basic graphs used in transformations
- Translations
- y = f(x + a) and y = f(x) + a
- Why does y = f(x + a) work in this way?
- Why does y = f(x) + a work in this way?
- Reflections
- Stretches
- What are asymptotes?

Trigonometry

Secθ, Cosecθ and Cotθ

Inverse Trigonometric Functions

- arcsin(x) or sin
^{-1}(x) - arccos(x) or cos
^{-1}(x) - arctan(x) or tan
^{-1}(x) - Examples using inverse trigonometric functions

Indentities and Equations: Pythagorean Type

- sin
^{2}x + cos^{2}x = 1, 1 + tan^{2}x = sec^{2}x, 1 + cot^{2}x = cosec^{2}x - Proving Pythagorean identities
- Solving equations using Pythagorean identities

Logarthmic and Exponential Functions

The Exponential Function e^{x} and Natural Log Functions

- Exponential Functions
- Transformations of exponential graphs
- The Natural Log Function: ln(x)

Differentiation

Standard Differentials

Differentiating with the Chain Rule

- Functions to a power
- Exponential functions
- Natural log functions
- Trigonometric functions
- Trigonometric functions to a power

The Product Rule and the Quotient Rule

- Product Rule
- Quotient Rule

More Standard Differentials

Reciprocal Function of dy/dx

Integration

Common Functions

- (ax + b)
^{n}type functions - Summary Exercise: (ax + b)
^{n}type functions - Exponential functions: e
^{x}, e^{ax}and e^{(ax + b)} - Reciprocal functions 1/x and 1/(ax + b)
- Integrals of the form: f'(x)/f(x)
- Integrals of the form: f'(x)e
^{f(x)}

Trigonometric Functions

- Integrals of sin(x), cos(x) and sec
^{2} - Integrals of the form sin(ax + b), cos(ax + b) and sec
^{2}(ax + b) - Identity types: 1 | 2 | 3
- sin
^{2}types - cos
^{2}types

Partial Fractions

Integration by Substitution

- Examples: 1 | 2
- Square root types (Method 1)
- Square root types (Method 2)
- Integrating trig. functions
- Integrating exponential types
- Integration by substitution with limits
- Integrating trig. functions with limits

Integration by Parts

- Introduction
- Applying integration by parts twice over
- Worked Example
- Natural Log types: ln(x)
- Integration by parts with limits
- Proof of the formula

Mixed Examples & Exam Questions on Integration

Application: Area Bound by a Curve

Application: Volumes of Revolution

- Introduction
- Example question
- Area enclosed by several curves
- Volume of revolution for a parametric curve

Numerical Integration

Numerical Solution of Equations

Solving Equations with Numerical Methods