Knowledge organisers

Calculus: IB Maths AI HL knowledge organiser

Everything to know about calculus on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Differential equation: An equation linking a function with its derivatives, such as dy/dx = ky.
Euler's method: Steps along the curve with yₙ₊₁ = yₙ + h × f(xₙ, yₙ) to estimate a solution.
Slope field: A grid of short line segments showing the gradient dy/dx at each point.
Volume of revolution: V = π∫ y² dx when a region is turned through 2π about the x-axis.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

DerivativesIn the formula booklet	\((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((\tan x)'=\tfrac1{\cos^2x},\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Chain, product, quotientIn the formula booklet	\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx},\) \((uv)'=uv'+vu',\) \(\Big(\frac uv\Big)'=\frac{vu'-uv'}{v^2}\)
Tangent; normal	\(y-f(a)=f'(a)(x-a);\) \(m_\text{n}=-\tfrac1{f'(a)}\)
Max / min; inflexion	\(f'=0,\ f''<0\ /\ f''>0;\) \(f''=0\text{ with sign change}\)
Related rates	\(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\)
IntegralsIn the formula booklet	\(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C,\) \(\int\tfrac1x\,dx=\ln\|x\|+C,\) \(\int\sin x\,dx=-\cos x+C\)
More integralsIn the formula booklet	\(\int\cos x\,dx=\sin x+C,\) \(\int\tfrac1{\cos^2x}\,dx=\tan x+C,\) \(\int e^x\,dx=e^x+C\)
Linear inside; by inspection	\(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C;\) \(\int g'(x)f'(g(x))\,dx=f(g(x))+C\)
AreasIn the formula booklet	\(\int_a^b\|y\|\,dx,\) \(\int_c^d\|x\|\,dy\)
Volumes of revolutionIn the formula booklet	\(V=\int_a^b\pi y^2\,dx,\) \(V=\int_c^d\pi x^2\,dy\)
Trapezoidal rule, \(h=\tfrac{b-a}n\)In the formula booklet	\(\int_a^by\,dx\) \({}\approx\tfrac12h\big[(y_0+y_n)\) \({}+2(y_1+\cdots+y_{n-1})\big]\)
KinematicsIn the formula booklet	\(v=\tfrac{ds}{dt},\) \(a=\tfrac{dv}{dt}=\tfrac{d^2s}{dt^2}=v\tfrac{dv}{ds};\) \(\text{distance }\int_{t_1}^{t_2}\|v\|\,dt\)
Separable	\(\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx\)
Euler's methodIn the formula booklet	\(y_{n+1}=y_n+h\,f(x_n,y_n),\) \(x_{n+1}=x_n+h\)
Euler, coupled systemsIn the formula booklet	\(x_{n+1}=x_n+h\,f_1(x_n,y_n,t_n),\) \(y_{n+1}=y_n+h\,f_2(x_n,y_n,t_n),\) \(t_{n+1}=t_n+h\)
Coupled linear system \(\dot{\boldsymbol x}=M\boldsymbol x\)In the formula booklet	\(\boldsymbol x=Ae^{\lambda_1t}\boldsymbol p_1+Be^{\lambda_2t}\boldsymbol p_2\)

More formulas are on the full IB Maths AI HL formula sheet.

Worked example

dy/dx = x + y with y = 1 when x = 0. Use Euler's method with step 0.1 to estimate y when x = 0.2.

y₁ = 1 + 0.1 × (0 + 1) = 1.1
y₂ = 1.1 + 0.1 × (0.1 + 1.1) = 1.22

Answer: y(0.2) ≈ 1.22

Common mistakes

HL algebra: product rule, brackets, logs and exact solutions
Volumes of revolution: about the y-axis, π, squaring, limits
Confusing average rate of change (a gradient of a chord) with instantaneous rate of change (the derivative)
Using the tangent gradient m instead of −1/m for the normal

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Compare average and instantaneous rates of change and interpret the derivative as the gradient function of a curve and as a rate of change.
Find stationary points by solving f′(x) = 0 and classify them as local maxima or minima, checking with technology.
Write differential equations from statements about rates of change, such as a population growing in proportion to its size.
Find volumes of revolution when a region is rotated 360° about the x-axis or the y-axis, using technology where needed.
Solve coupled linear systems using the eigenvalues and eigenvectors of the coefficient matrix and write the general solution.
Use coupled differential equation models for populations, predator–prey interactions and combat, interpreting equilibrium points and long-term behaviour.

The printable sheet

Calculus knowledge organiser for IB Maths AI HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Calculus knowledge organiser (IB Maths AI HL), A4. Download the PDF.

Revise it next

Other IB Maths AI HL topics: Number and algebra · Functions · Geometry and trigonometry · Statistics and probability · All IB Maths AI HL organisers