Differentiation (A2 Unit 4: Applied Mathematics B) — WJEC A-Level Mathematics
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Differentiation (A2 Unit 4: Applied Mathematics B) explained
Translating contextual physical descriptions into differential equations involves expressing rates of change as derivatives.
Read the full explanation
A rate of increase of quantity y with respect to time t is written as dy/dt > 0, while a rate of decrease is −dy/dt. Proportionality statements use a positive constant k: 'rate of growth is proportional to population P' translates to dP/dt = kP; 'cooling rate is proportional to temperature difference' yields dT/dt = −k(T − T_env). In kinematics, Newton's second law F = ma translates to m dv/dt = F(v, t) or m v dv/dx = F(x). In economics, price-demand sensitivity models dQ/dP.
Your focus
- Translate verbal descriptions of rates of change into first-order differential equations.
- Introduce and interpret constants of proportionality with appropriate mathematical signs.
- Construct mathematical differential models for kinematics, biological growth, and economic demand.
Differentiation (A2 Unit 4: Applied Mathematics B) exam tips
Marking Points
- translating a described rate of change into the correct derivative notation (dy/dt or dP/dt)
- introducing a constant of proportionality k with correct sign for increase or decrease
- formulating the complete first order differential equation representing the contextual scenario
Examiner Tips
- 💡Look for keywords: 'rate of increase' means positive derivative; 'rate of decrease' requires a negative sign.
- 💡Translate 'proportional to' as '= k ×' and determine whether k should carry a positive or negative sign.
Common Mistakes
- omitting the negative sign when a quantity is stated to be decreasing or decaying at a rate proportional to its value
- confusing the rate of change dy/dt with the quantity y itself