This HL-only mini-lesson covers the calculus extensions: differential equations (rate models such as Newton's cooling and population growth), slope fields, Euler's method for numerical solutions, and using the trapezoidal rule when integrals cannot be found exactly.
AI HL flavour: when a rate equation has no neat formula solution, you approximate it numerically with Euler's method or read behaviour from a slope field.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.
Differential equations
Modelling with rates of change
A differential equation links a quantity to its rate of change. Two classics:
Exponential growth/decay: dy⁄dt = k y (solution y = y₀ ekt).
Newton's cooling: dT⁄dt = −k(T − Troom) — cools fast then slows.
A differential equation describes how something changes; solving it (analytically or numerically) gives the quantity over time.
Quick check
Recognise the model
?A cup of tea obeys dT⁄dt = −k(T − 20). This is:
Slope fields
Reading a slope field
A slope field draws a short segment of gradient dy⁄dx at many points. A solution curve follows the segments like a boat following a current — it lets you see behaviour without solving.
Along an isocline (a curve where dy⁄dx is constant) all the segments have the same slope — a quick way to sketch the field.
Euler's method
Euler's method for numerical solutions
Euler's method steps forward in small increments h using the local gradient:
yn+1 = yn + h · f(xn, yn)xn+1 = xn + h
Worked example
dy⁄dx = x + y, y(0) = 1, step h = 0.1. Find y(0.3).
1For dy⁄dx = x + y with y(0) = 1 and h = 0.1, find y after one step (at x = 0.1), to 3 decimal places.
Hint: y + h(x + y) = 1 + 0.1(0 + 1).
Calculate
Your turn — Euler to x = 0.3
2Continue Euler's method for dy⁄dx = x + y, y(0) = 1, h = 0.1, to find y(0.3) to 3 decimal places.
Hint: after step 1 (1.100) and step 2 (1.220), do step 3: 1.220 + 0.1(0.2 + 1.220).
Quick check
Euler accuracy
?To make Euler's method more accurate, you should:
Numerical integration
The trapezoidal rule
When an integral has no elementary form, estimate the area with the trapezoidal rule.
Area ≈ h⁄2 [ y₀ + yn + 2(y₁ + … + yn−1) ]h = (b − a) ⁄ n
Worked example
∫₀⁴ x² dx, n = 4, y = 0,1,4,9,16:
≈ 1⁄2 [0 + 16 + 2(1 + 4 + 9)] = 22 (exact 21.33)
Calculate
Your turn — trapezoidal rule
3Estimate ∫₀⁴ x² dx with the trapezoidal rule, n = 4 (h = 1), y = 0, 1, 4, 9, 16.
Hint: 1⁄2 [0 + 16 + 2(1 + 4 + 9)].
Exponential solution
Solving dy/dt = ky
The growth/decay equation dy⁄dt = k y has the exact solution y = y₀ ekt.
Worked example
y₀ = 100, k = 0.08, t = 10: y = 100 e^(0.8) = 222.55
Calculate
Your turn — exact solution
4The equation dy⁄dt = 0.08y with y(0) = 100 has solution y = 100 e^(0.08t). Find y at t = 10, to 2 decimal places.
Hint: 100 × e^(0.08 × 10) = 100 e^(0.8).
Calculate
Your turn — trapezoidal exactness gap
5The trapezoidal estimate of ∫₀⁴ x² dx is 22 and the exact value is 21.33. Find the error (estimate − exact), to 2 decimal places.
Hint: 22 − 21.33.
Sort it
Exact or numerical?
Tap a task, then the method that fits.
✏️ Solve exactly
🔢 Euler's method
≈ Trapezoidal rule
Quick check
Slope field solutions
?A solution curve drawn on a slope field must:
Match it
Match the method to its situation
Tap an item on the left, then its match on the right.
Item
Match
Coupled systems
Coupled differential equations
Some models link two changing quantities — e.g. predator–prey populations. These coupled systems dx⁄dt and dy⁄dt are stepped together with Euler's method.
At each step, update both variables using the current values before moving on.
Quick check
Equilibrium
?A population model dP⁄dt = 0.5P(1 − P⁄1000) has an equilibrium (dP⁄dt = 0) at P =
Recap
The big ideas to take away
Differential equations: dy⁄dt = ky (growth); Newton cooling dT⁄dt = −k(T − T_room)
Slope fields: segments show dy⁄dx; solutions follow the field
Euler's method: y_{n+1} = y_n + h·f(x_n, y_n); smaller h ⇒ more accurate
Trapezoidal rule: area ≈ h⁄2[y₀ + y_n + 2(middle)]
Exact solution: dy⁄dt = ky ⇒ y = y₀ e^(kt)
Choice: use exact methods when possible, numerical when not
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