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IB Maths: Applications & Interpretation: Calculus

Rates of change, optimisation, integration, differential equations (HL). 18 subtopics, 61 key terms.

In this topic: 5.1 · 5.2 · 5.3 · 5.4 · 5.5 · 5.6 · 5.7 · 5.8 · 5.9 · 5.10 · 5.11 · 5.12 · 5.13 · 5.14 · 5.15 · 5.16 · 5.17 · 5.18

5.1Limits and the derivative as a rate of change

Key terms

Chord

A straight line joining two points on a curve.

Tangent

A line that touches the curve at one point with the same gradient.

Limit

The value an expression approaches as h gets closer and closer to 0.

Derivative dy/dx or f′(x)

The gradient of the curve at a point = instantaneous rate of change.

As the second point of a chord moves towards the first, the chord's gradient approaches the gradient of the tangent.

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: When a question asks you to interpret a derivative, include the units and the words “rate of change of … with respect to …”.

5.2Increasing and decreasing functions

Key terms

Increasing

f′(x) > 0: the graph goes up from left to right.

Decreasing

f′(x) < 0: the graph goes down from left to right.

Stationary

f′(x) = 0.

Read intervals from the graph of \(f'\): where \(f'\) is above the \(x\)-axis, \(f\) increases; below, \(f\) decreases.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Sketch f′ quickly on your GDC: the sign of f′ is visible at a glance.

5.3Differentiating powers of x

Key terms

Power rule

d/dx(axⁿ) = anxⁿ⁻¹, n ∈ ℤ

Constant

The derivative of a constant is 0.

Sum rule

Differentiate each term separately.

Rewrite fractions as negative powers first: \(\dfrac{5}{x^{2}}=5x^{-2}\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Negative powers: multiply by the power, then subtract 1 (so −2 becomes −3, not −1).

5.4Tangents and normals

Key terms

Tangent at x = a

Line through (a, f(a)) with gradient f′(a).

Normal at x = a

Perpendicular to the tangent: gradient −1 ÷ f′(a).

Three steps: find the point \((a,f(a))\), find the gradient \(f'(a)\), use \(y-y_1=m(x-x_1)\).

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: Use the y-value of the curve, f(a), not the gradient, as y₁. Mixing them up is the most common error.

5.5Introduction to integration

Key terms

Anti-differentiation

Reversing differentiation.

Indefinite integral

∫ f(x) dx = F(x) + C

Definite integral

∫ₐᵇ f(x) dx: a number; area under the curve when f ≥ 0.

Boundary condition

A known point used to find C.

Always add \(+C\) to an indefinite integral.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: For an area question, write the integral with its limits, e.g. ∫₀³ (6 − 2x) dx, before giving the GDC value.

5.6Stationary points: maximum and minimum

Key terms

Stationary point

A point where f′(x) = 0.

Local maximum

f′ changes from + to −.

Local minimum

f′ changes from − to +.

Solve \(f'(x)=0\) to find the \(x\)-coordinates, then substitute into \(f\) for the \(y\)-coordinates.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: “Justify” means show the sign change of f′ (or use the second derivative in HL). The GDC alone is not a justification.

5.7Optimisation

Key terms

Optimisation

Finding the maximum or minimum of a quantity in context.

Constraint

A condition linking the variables (fixed perimeter, fixed volume…).

Objective function

The quantity to maximise or minimise.

Method: (1) write the quantity in terms of one variable using the constraint; (2) differentiate; (3) solve \(=0\); (4) check it is a max/min; (5) answer in context with units.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Write the domain of x before differentiating. It helps you reject impossible solutions.

5.8The trapezoidal rule

Key terms

Trapezoidal rule

Estimates an area by splitting it into trapezoids.

Strip width h

h = (b − a) ÷ n for n strips.

Overestimate / underestimate

Depends on whether the curve is concave up or down.

More strips give a better estimate.

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: Write out every y-value in a table. Missing one strip is the classic mistake.

5.9Further differentiation: standard derivatives and rules HL

Key terms

Chain rule

dy/dx = dy/du × du/dx: for a function of a function.

Product rule

(uv)′ = u′v + uv′

Quotient rule

(u/v)′ = (u′v − uv′) ÷ v²

Related rates

Link rates with the chain rule: dV/dt = dV/dr × dr/dt.

Trig functions are differentiated in radians.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Write u, v, u′ and v′ separately before applying the product or quotient rule.

5.10Second derivative and concavity HL

Key terms

Second derivative f″(x)

The rate of change of the gradient.

Concave up

f″ > 0: the gradient is increasing (bends like ∪).

Concave down

f″ < 0: the gradient is decreasing (bends like ∩).

Point of inflection

Concavity changes: f″ = 0 and changes sign.

Second derivative test: if \(f'(a)=0\) and \(f''(a)\lt0\), local maximum; if \(f''(a)\gt0\), local minimum; if \(f''(a)=0\), the test is inconclusive.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: To justify an inflection point, show that f″ changes sign, not just that f″ = 0.

5.11Further integration and substitution HL

Key terms

∫ 1/x dx

ln|x| + C

∫ eˣ dx

eˣ + C

Integration by inspection

Spot that the integrand is f(g(x))g′(x).

Substitution

Let u = g(x) to simplify ∫ f(g(x))g′(x) dx.

Inspection: the derivative of the “inside” function appears as a factor.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Check any integral by differentiating your answer: you should get back the integrand.

5.12Areas and volumes of revolution HL

Key terms

Area with the y-axis

∫ x dy between y-limits.

Volume about the x-axis

V = π∫ y² dx

Volume about the y-axis

V = π∫ x² dy

For rotation about the \(y\)-axis, rearrange to get \(x\) in terms of \(y\) and use \(y\)-limits.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Square y before integrating: π∫y² dx, not π(∫y dx)².

5.13Kinematics HL

Key terms

Displacement s

Position relative to the start (can be negative).

Velocity v

ds/dt: speed with a direction.

Acceleration a

dv/dt

Distance travelled

∫|v| dt: always positive.

The particle is at rest when \(v=0\); it changes direction when \(v\) changes sign.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: “Distance” and “displacement” are different questions. If v changes sign, they have different answers.

5.14Differential equations: setting up and separating variables HL

Key terms

Differential equation

An equation involving a derivative, e.g. dy/dx = ky.

General solution

Contains an arbitrary constant.

Particular solution

The constant is found from an initial condition.

Separation of variables

Rearrange to g(y) dy = f(x) dx, then integrate both sides.

“The rate of change of \(N\) is proportional to \(N\)” means \(\frac{dN}{dt}=kN\), with solution \(N=Ae^{kt}\).

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Write the general solution before using the initial condition. That line is usually worth a mark.

5.15Slope fields HL

Key terms

Slope field

Short line segments showing the gradient dy/dx at many points.

Solution curve

A curve that follows the slope field everywhere.

Isocline

A curve along which dy/dx has the same value.

To draw a slope field for \(\frac{dy}{dx}=f(x,y)\), calculate \(f\) at each grid point and draw a short segment with that gradient.

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: In a sketch, make the solution curve tangent to the segments it passes, and continue it to the edges of the field.

5.16Euler's method HL

Key terms

Euler's method

A step-by-step numerical solution of a differential equation.

Step length h

The increase in x at each step.

Coupled system

Two equations, each with dx/dt and dy/dt depending on x and y.

Each step follows the tangent line for a short distance.

The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.

Exam tip: Set out a table with columns n, xₙ, yₙ, f(xₙ, yₙ), yₙ₊₁. It's clear and earns method marks for every row.

5.17Phase portraits of coupled linear systems HL

Key terms

Coupled linear system

dx/dt = ax + by, dy/dt = cx + dy, written ẋ = Mx.

Phase portrait

Trajectories in the (x, y) plane showing how the system evolves.

Equilibrium point

Where dx/dt = dy/dt = 0: the origin for these systems.

Stable / unstable

Trajectories approach / move away from equilibrium.

Real eigenvalues, both negative: stable (sink). Both positive: unstable (source). Opposite signs: saddle point.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Draw the eigenvector lines first, with arrows showing in or out, then sketch the curved trajectories between them.

5.18Second-order differential equations HL

Key terms

Second-order DE

Contains d²x/dt², e.g. ẍ + aẋ + bx = 0.

Reduce to a system

Let y = dx/dt; then dy/dt = d²x/dt².

Damped oscillation

Oscillations that die away over time.

Once written as a coupled system, use Euler's method (5.16) to approximate solutions, or eigenvalues (5.17) to analyse behaviour.

The full notes, worked example and check questions for this subtopic are in the app.

Exam tip: Update x and y using the values from the same step (xₙ, yₙ), not the new ones.

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