Mini-Lesson

Calculus (HL)

This HL mini-lesson develops the calculus core: differentiation (power rule, tangents, optimisation), kinematics, integration as anti-derivative and area, the area between curves, and numerical estimation with the trapezoidal rule. (Differential equations, slope fields and Euler's method are in the dedicated HL lesson.)

AI HL flavour: calculus is used for rates, totals and areas — with numerical methods when an exact integral is hard.

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.

Differentiation

The power rule & gradients

f(x) = a xⁿ ⇒ f′(x) = a n xⁿ⁻¹differentiate term by term

Worked example

f(x) = 3x² − 4x + 1 ⇒ f′(x) = 6x − 4.

f′(2) = 6×2 − 4 = 8

Calculate

Your turn — derivative value

1For f(x) = 3x² − 4x + 1, find f′(2).

Hint: f′(x) = 6x − 4.

Optimisation

Tangents, stationary points & optimisation

Increasing where f′(x) > 0; decreasing where f′(x) < 0.
Stationary points where f′(x) = 0 (max/min).
Tangent at x = a has gradient f′(a).

Optimisation: model the quantity, set f′ = 0, and verify max or min (sign of f′ either side, or the second derivative).

Quick check

Stationary points

?At a smooth maximum or minimum, the derivative f′(x) equals:

Kinematics

Displacement, velocity & acceleration

v = ds ⁄ dt · a = dv ⁄ dtreverse by integrating

Worked example

s(t) = t³ − 3t²: v(t) = 3t² − 6t, a(t) = 6t − 6.

v(4) = 48 − 24 = 24 m/s; a(4) = 24 − 6 = 18 m/s²

Calculate

Your turn — velocity

2For s(t) = t³ − 3t², find v = ds⁄dt at t = 4 (m/s).

m/s

Hint: v(t) = 3t² − 6t at t = 4.

Integration

Anti-derivative & definite integral

∫ a xⁿ dx = a xⁿ⁺¹ ⁄ (n+1) + C∫_a^b f dx = F(b) − F(a)

Worked example

∫₁³ (2x + 1) dx = [x² + x]₁³ = 12 − 2 = 10

Calculate

Your turn — definite integral

3Evaluate ∫₁³ (2x + 1) dx.

Hint: antiderivative x² + x; (9+3) − (1+1).

Area between curves

Area enclosed by two curves

The area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫_a^b [f(x) − g(x)] dx.

Worked example

Between y = x (upper) and y = x² (lower) from 0 to 1:

∫₀¹ (x − x²) dx = [x²⁄2 − x³⁄3]₀¹ = 1⁄2 − 1⁄3 = 0.1667

Calculate

Your turn — area between curves

4Find the area between y = x and y = x² from x = 0 to x = 1, to 4 decimal places.

Hint: ∫₀¹ (x − x²) dx = 1⁄2 − 1⁄3.

Trapezoidal rule

Numerical integration

Area ≈ h⁄2 [ y₀ + y_n + 2(y₁ + … + y_n−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, slight over-estimate)

Calculate

Your turn — trapezoidal rule

5Estimate ∫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)].

Sort it

Which calculus tool?

Tap a task, then whether it needs differentiation, integration, or a numerical estimate.

📉 Differentiate

📐 Integrate

≈ Numerical estimate

Quick check

Area between curves setup

?To find the area between an upper curve f and lower curve g, you integrate:

Quick check

Trapezoidal over/under

?For a concave-up curve, the trapezoidal rule gives an estimate that is:

Match it

Match the calculus idea to its meaning

Tap an item on the left, then its match on the right.

Item

Match

Applied rates

Rates of change in context

AI HL applies derivatives to real rates: marginal cost dC⁄dq, marginal revenue dR⁄dq, and optimisation of profit P = R − C where dP⁄dq = 0.

Always interpret f′ with units — e.g. dollars per unit, metres per second.

Quick check

Marginal analysis

?A firm maximises profit P(q) = R(q) − C(q). The optimal quantity satisfies:

Recap

The big ideas to take away

Derivative: power rule; f′ = gradient; f′ = 0 at stationary points

Kinematics: v = ds⁄dt, a = dv⁄dt

Integration: anti-derivative + C; definite integral = F(b) − F(a)

Area between curves: ∫(upper − lower) dx

Trapezoidal rule: area ≈ h⁄2[y₀ + y_n + 2(middle)]

Note: concave-up curves are over-estimated by trapeziums

You have worked through the whole topic. Press Finish to see your score.

🏆

Mini-lesson complete!

⭐⭐⭐

You have covered HL Calculus for AI. 🎉

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