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Glossary
Chapter 1: What mathematical modelling is
- Overview
- 1.1 The problem-solving cycle
- 1.2 Formulating a problem and decomposing it
- 1.3 Assumptions and what they cost you
- 1.4 Translating a situation into mathematics
- 1.5 Computing and interpreting a solution
- 1.6 Evaluating, refining and reporting
- 1.7 How to write a modelling report
- Review
Chapter 2: Networks and graphs
- Overview
- 2.1 Networks and their terminology
- 2.2 Connected, disconnected, directed and undirected graphs
- 2.3 Representing a network: adjacency matrices
- 2.4 Matrix algebra and the product of adjacency matrices
- 2.5 Trees and spanning trees
- 2.6 Minimum spanning trees: Kruskal's algorithm
- 2.7 Minimum spanning trees: Prim's algorithm
- 2.8 Shortest paths and Dijkstra's algorithm
- 2.9 Multi-stage problems and dynamic programming
- 2.10 Bellman's principle of optimality
- 2.11 Comparing the algorithms and choosing one
- Review
Chapter 3: Describing motion
- Overview
- 3.1 Displacement, velocity and acceleration
- 3.2 Motion with constant acceleration
- 3.3 Motion graphs
- 3.4 Differentiation and the chain rule
- 3.5 The fundamental theorem of calculus
- 3.6 Deriving the equations of motion by calculus
- 3.7 Projectile motion
- 3.8 Time of flight and range
- 3.9 Projectiles launched at an angle
- Review
Chapter 4: Forces and Newton's laws
- Overview
- 4.1 Forces and Newton's laws
- 4.2 Resultant force and equilibrium
- 4.3 Motion on a smooth horizontal plane
- 4.4 Motion on an inclined plane
- 4.5 Tension and connected particles
- 4.6 Pulleys
- 4.7 Newton's laws in vector form
- Review
Chapter 5: Friction and resistance
- Overview
- 5.1 Friction and the normal reaction
- 5.2 The coefficient of friction
- 5.3 Motion on rough planes
- 5.4 Statics on a rough plane
- 5.5 Drag proportional to a power of speed
- 5.6 Terminal velocity
- Review
Chapter 6: Momentum, impulse and collisions
- Overview
- 6.1 Momentum and impulse
- 6.2 Conservation of momentum
- 6.3 Collisions in one dimension
- 6.4 Newton's experimental law of restitution
- 6.5 Elastic and inelastic collisions
- 6.6 Collisions in two dimensions
- 6.7 Successive collisions and rebounds
- Review
Chapter 7: Circular motion and oscillation
- Overview
- 7.1 Angular velocity and period
- 7.2 Centripetal acceleration and force
- 7.3 Circular motion in context
- 7.4 Conical pendulum and banked tracks
- 7.5 Hooke's law and elastic strings
- 7.6 Simple harmonic motion
- Review
Chapter 8: Modelling a changing world
- Overview
- 8.1 Rates of change and differential equations
- 8.2 Exponential growth and decay
- 8.3 Recurrence relations as discrete models
- 8.4 First-order difference equations
- 8.5 Feedback and limited growth
- 8.6 The logistic model
- 8.7 Solving separable differential equations
- 8.8 Comparing a discrete and a continuous model
- Review
Chapter 9: The modelling project
- Overview
- 9.1 Reading the brief and choosing a problem
- 9.2 Building the model and stating assumptions
- 9.3 Testing the model against data
- 9.4 Iterating and improving
- 9.5 Communicating the result
- 9.6 Critical evaluation of models
- Review
About this book

8.6 The logistic model — Practise with help

Stub. Scaffolded practice with progressive exposure: the student attempts each step, then reveals a hint or a fuller worked step only if needed. Tier: H · refs: S4