Topic 6: Calculus

Differentiation

Differentiation – Introduction

  1. The gradient function dy/dx
  2. Differentiation - terms of the form axn
  3. The second derivative
  4. Exam Questions - Differentiation: introduction

Tangents and Normals

  1. Equations of tangents and normals
  2. Exam Questions - Tangents and normals

Increasing and Decreasing functions

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

Stationary Points

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

Standard Differentials

  1. Exponential function ex
  2. The natural log function, ln(x)
  3. The trig functions sin(x), cos(x) and tan(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

The Reciprocal Function of dy/dx

  1. The reciprocal function of dy/dx

Exam Questions – Differentiation

  1. Exam Questions - Differentiation methods
  2. Exam Questions - Differentiation: tangents, normals and stationary points
  3. Exam Questions - Exponential rates of change

Integration

Integration – Introduction

  1. Integration - Terms of the form axn
  2. Exam Questions - Integration: introduction

Equations of Curves

  1. Finding the equation of a curve given the gradient function
  2. Exam Questions - Equations of curves

Definite Integration

  1. Definite integration
  2. Exam Questions - Definite Integration
  3. Convergent and Divergent Integrals

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

Integration

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

Further methods of integration

Integration by Substitution

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

Kinematics

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

  1. Acceleration as a vector

Linear motion with variable acceleration

  1. Linear motion with variable acceleration
  2. Exam Questions - Linear motion with variable acceleration