[Leejohnston]

Applied Mathematics — Modelling, Rates, Vectors & Real-World Problems

Applied mathematics connects pure maths to the real world. 
It powers physics, engineering, finance, biology, computer science, and every simulation on Earth.

This thread introduces the core ideas behind modelling, vectors, rates of change, optimisation, and solving real-world problems.

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1. What Is Applied Mathematics?

Applied maths uses:
• algebra 
• calculus 
• geometry 
• statistics 
• vectors 

…to solve practical problems such as:
• motion 
• growth and decay 
• optimisation 
• forces 
• trajectories 
• population modelling 
• computing algorithms 

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2. Mathematical Modelling

A mathematical model uses equations to represent a real situation.

Steps in modelling:
1. Identify the important features 
2. Make assumptions 
3. Build a mathematical model 
4. Solve / simulate 
5. Compare with real data 
6. Refine the model 

Examples:
• constant acceleration model: s = ut + ½at² 
• population growth: P = P₀e^{kt} 
• cooling: Newton’s Law of Cooling 
• spread of disease: SIR model 

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3. Rates of Change

Many real-world quantities change over time.

Examples:
• velocity = rate of change of distance 
• acceleration = rate of change of velocity 
• power = rate of energy transfer 
• gradient = rate of change of a graph 

If y = f(x), then:

dy/dx = rate of change of y with respect to x

Example:
If f(x) = x³ → f’(x) = 3x² 
At x = 2, the rate of change = 3(4) = 12

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4. Exponential Growth & Decay

Common in:
• populations 
• bacteria growth 
• radioactive decay 
• interest rates 
• cooling/heating 

Model form:

Code:

y = Ae^{kt}

If k > 0 → growth 
If k < 0 → decay 

Example:
A population doubles every 5 days → k = ln(2)/5

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5. Vectors in Applied Maths

Vectors represent:
• direction 
• movement 
• force 
• velocity 
• acceleration 

A vector ⟨a, b⟩ shows movement in x and y.

Magnitude (length):

Code:

|v| = √(a² + b²)

Unit vector: direction only:

Code:

v / |v|

Force addition:

Code:

F_total = F₁ + F₂

Example:
⟨3, 4⟩ + ⟨1, −2⟩ = ⟨4, 2⟩

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6. Optimisation Problems

Applied maths often asks:

• What’s the maximum value? 
• What’s the minimum cost? 
• What’s the fastest route? 
• What shape gives the most volume? 

Use calculus:
• find derivative 
• set to zero 
• test turning points 

Example:
Maximising area with fixed perimeter 
Minimising travel time in physics problems 

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7. Numerical Methods (Simple Intro)

Computers solve equations when algebra becomes too hard.

Newton’s method: 
Approximates roots of equations.

Euler’s method: 
Approximates solutions to differential equations.

Used in:
• weather models 
• AI simulations 
• fluid dynamics 
• population systems 
• astrophysics 

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8. Applied Problem Examples

1. Motion 
A ball is thrown up with velocity 12 m/s. 
Find height after 1.5s if a = −9.8:

Code:

s = ut + ½at²
= 12(1.5) + ½(−9.8)(1.5²)

2. Vectors 
A ship travels ⟨4, 3⟩ km then ⟨−1, 5⟩ km. 
Resultant = ⟨3, 8⟩

3. Growth 
Bacteria doubles every hour. 
After 6 hours: 
= 2⁶ = 64 times the original population.

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9. Common Mistakes

❌ Forgetting assumptions in modelling 
✔ Always write assumptions clearly 

❌ Mixing vector magnitude with components 
✔ magnitude = √(a² + b²) 

❌ Thinking exponential growth is linear 
✔ It accelerates over time 

❌ Incorrect units 
✔ keep time, distance, speed consistent 

❌ Misusing derivative rules 
✔ Apply power rule carefully 

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10. Practice Questions

1. A car accelerates from 0 to 18 m/s in 6 seconds. Find acceleration. 
2. Find magnitude of vector ⟨7, −4⟩. 
3. Bacteria grows exponentially: P = 300e^{0.2t}. Find P at t = 5. 
4. Solve using modelling: A tank leaks 5% per hour. Write formula for water amount after t hours. 
5. Optimise: For f(x) = x³ − 6x, find turning points. 
6. Use Euler’s method (step 0.1) to estimate next point for dy/dx = 3y at y₀ = 2.

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Summary

This thread introduced:
• modelling 
• rates of change 
• exponential growth/decay 
• vectors 
• optimisation 
• numerical methods 
• real-world examples 

Applied mathematics is where maths becomes *useful*, powering science, engineering, computing, and every simulation in the world.