Topic 6 – Core: Calculus

Differentiation – Introduction

1. The gradient function dy/dx
2. Differentiation - terms of the form axⁿ
3. The second derivative

Tangents and Normals

1. Equations of tangents and normals

Stationary Points

1. Stationary points
2. Nature of a stationary point
3. Applications of stationary points
4. Exam Questions - Applications of stationary points

Increasing and Decreasing functions

1. Increasing and decreasing functions
2. Exam Questions - Increasing and decreasing functions

Standard Differentials

1. Exponential function e^x

The Chain Rule

1. Chain rule: Polynomial to a rational power
2. Chain rule: Exponential types
3. Chain rule: Natural log types
4. Chain rule: Trigonometric types

The Product and Quotient Rules

1. The product rule
2. The quotient rule

More Standard Differentials

The Reciprocal Function of dy/dx

1. The reciprocal function of dy/dx

Implicit Functions

1. Implicit functions
2. Tangents and normals
3. Exam Questions - Implicit functions

Connected Rates of Change

1. Exam Questions - Connected rates of change

Integration – Introduction

Equations of Curves

1. Finding the equation of a curve given the gradient function

Definite Integration

Applications of Integration – Area bound by a curve

1. Area bound by a curve and x-axis
2. Exam Questions - Area bound by a curve and x-axis
3. Area bound by a curve and y-axis
4. Exam Questions - Area bound by a curve and y-axis

Applications of Integration – Volumes of revolution

1. Volume of revolution about the x-axis
2. Exam Questions - Volume of revolution about the x-axis
3. Volume of revolution about the x-axis generated between curves
4. Volume of Revolution about the y-axis
5. Exam Questions - Volume of Revolution about the y-axis
6. Volume of Revolution about the y-axis generated between curves

Integration – Common Functions

1. Exam Questions - Integration:(ax+b)ⁿ types
2. Exam Questions - Integrating exponential functions e^x, e^ax and e^(ax+b)
3. Exam Questions - Integrating reciprocal functions 1/x and 1/(ax+b)
4. Integrals of the form : f '(x)/f(x)
5. Integrals of the form : f'(x)e^f(x)

Integrals of Trigonometric Functions

1. Integrals of sin x, cos x, sec² x
2. Integrals of the form sin(ax+b), cos(ax+b), sec² (ax+b) types
3. Integrals Using Trigonometric Identities
4. sin²x and cos²x types
5. Exam Questions - Trigonometric types

Integration by Substitution

1. Integration by substitution
2. Integration of trigonometric functions by substitution
3. Integration of exponential types by substitution
4. Integration by substitution using limits
5. Integration of trigonometric functions by substitution with limits
6. Exam Questions - Integration by substitution

Integrating Products of the form f[g(x)]g'(x) by inspection

Integration by Parts

1. Integration by parts
2. Integration by parts (ln types)
3. Integration by parts using limits
4. Proof of the formula - Integration by parts
5. Exam Questions - Integration by parts

Displacement and Displacement Time Graphs

1. Displacement
2. Displacement time graphs

Motion in a Straight Line

1. Motion in a straight line with constant acceleration (suvat)
2. Horizontal motion with constant acceleration
3. Exam Questions - Horizontal motion
4. Vertical motion under gravity
5. Exam Questions - Vertical motion under gravity

Velocity Time Graphs

1. Velocity time graphs
2. Exam Questions - Velocity time graphs

Velocity Vectors

1. Calculating a speed and bearing
2. Calculating a velocity given speed and direction
3. Calculating the position vector after time t
4. Constant velocity as the rate of change of position vectors
5. Exam Questions - Velocity vectors

Constant Acceleration Vectors

Motion with variable acceleration

1. Exam Questions - Linear motion with variable acceleration