Applications and Interpretation HL covers everything in AI SL and extends it with more advanced statistics, calculus and modelling. The IB formula booklet gives you the formulas with no explanation. Here are all 83 that apply to AI HL, with what each one means and when to use it.
This is an annotated companion to the booklet you are given in the exam, not a copy of it. Formulas are grouped by topic: Prior learning, Number & Algebra, Functions, Geometry & Trigonometry, Statistics & Probability, Calculus.
Base times perpendicular height.
When: Any parallelogram.
Half of base times perpendicular height.
When: When you know a base and its height.
Average of the two parallel sides times the height between them.
When: Four-sided shapes with one pair of parallel sides.
π times the radius squared.
When: Circles and disc-shaped regions.
The distance around a circle.
When: Perimeter of a circle.
Length × width × height.
When: Rectangular boxes.
Circle area times height.
When: Cylinders / tubes.
Cross-section area times length.
When: Any solid with a constant cross-section.
Unroll the side into a rectangle: circumference × height.
When: Surface area of the curved part.
Pythagoras on the horizontal and vertical gaps.
When: Length of a segment from coordinates.
Average the x-coordinates and the y-coordinates.
When: The point halfway between two points.
Start at the first term and add the common difference once per step.
When: Sequences that go up/down by a fixed amount.
Use when you know the first term and the common difference.
When: Totalling an arithmetic sequence from \(u_1\) and \(d\).
Use when you know the first and last term.
When: Totalling an arithmetic sequence from \(u_1\) and \(u_n\).
Multiply the first term by the ratio once per step.
When: Growth/decay by a fixed factor.
Shortcut for adding terms that keep multiplying by r.
When: Repeated percentage growth (e.g. savings).
If terms shrink, infinitely many add to a finite total.
When: Only when |r| < 1.
PV = start; r% = yearly rate; k = compounds per year; n = years.
When: Investments/loans compounding more than once a year.
How far an approximate value is from the exact one, as a %.
When: Rounded/measured vs true value.
Logs undo exponentials; use a log when the unknown is a power.
When: Solving for an exponent.
Products → sums, quotients → differences, powers come out front.
When: Combining/splitting logs to solve equations.
Modulus = distance from origin; argument = angle from positive real axis.
When: Polar form, Argand diagrams.
Two compact ways to write a complex number using its modulus and angle.
When: Multiplying, dividing and powering complex numbers.
Cross-multiply diagonals and subtract; inverse swaps the diagonal and negates the off-diagonal.
When: Solving systems, transformations (AI HL).
Diagonalise a matrix to raise it to a power quickly.
When: Markov chains, repeated transformations (AI HL).
The vertical line through the vertex - halfway between the roots.
When: Max/min of a quadratic.
Solves any quadratic; the ± gives the two roots.
When: When it won't factorise nicely.
Δ>0 → two roots; Δ=0 → one; Δ<0 → none.
When: How many real roots a quadratic has.
Three equivalent forms: gradient–intercept, general, and point–gradient.
When: Lines, gradients and intercepts.
Rise over run between two points.
When: Finding the slope from two points.
An S-shaped growth curve that levels off at a carrying capacity.
When: Limited-growth models (AI HL).
Grows with the cube of the radius.
When: Balls, domes.
Exactly four circle-areas.
When: Outer area of a sphere.
A third of the cylinder with the same base and height.
When: Cones.
Uses the slant height l, not the vertical height.
When: Surface of the sloping part.
A third of base area times height.
When: Pyramids.
The three ratios linking an angle to the sides of a right triangle.
When: Right-angled triangles.
Pairs each side with its opposite angle.
When: A side + its opposite angle are known.
Pythagoras plus an angle-correction term.
When: Two sides + included angle, or all three sides.
Half the product of two sides times the sine of the angle between.
When: No perpendicular height available.
The fraction θ/360 of the full circumference.
When: AI - angles in degrees.
The fraction θ/360 of the full circle area.
When: AI - angles in degrees.
Radius times the angle (θ in radians).
When: AA - angles in radians.
The 'pizza slice' area (θ in radians).
When: AA - angles in radians.
Lets you swap between sin and cos.
When: Given one of sinθ/cosθ, find the other.
Definition of tan in terms of sin and cos.
When: Simplifying / solving trig equations.
Magnitude = length; the dot product gives the angle between vectors.
When: Angles, projections, perpendicularity.
Gives a vector perpendicular to both; its magnitude is the area of the parallelogram they span.
When: Normals, areas, planes (AA HL).
Pythagoras in three dimensions.
When: Length of a segment in 3-D space.
Average each coordinate of the two endpoints.
When: Midpoint in 3-D space.
Matrices that stretch, enlarge, rotate or reflect points about the origin.
When: Geometric transformations (AI HL).
A known point plus multiples of a direction vector trace out the line.
When: Lines in 2-D / 3-D (HL).
Each value weighted by how often it occurs.
When: Grouped or repeated data.
Favourable outcomes over total outcomes.
When: Equally-likely outcomes.
Add the chances, subtract the overlap.
When: P(A or B).
Chance of A once B is known - sample space shrinks to B.
When: 'Given that…' problems.
Multiply when one event doesn't affect the other.
When: Both happen; also the test for independence.
Long-run average: outcomes weighted by probability.
When: Fair games, average payoff.
n independent trials, success chance p; on average np successes.
When: Fixed number of yes/no trials.
Line of best fit; r measures strength/direction of linear association.
When: Bivariate data, prediction.
Counts of random events at average rate λ.
When: Events per interval (calls/hour etc.).
Compares observed with expected frequencies.
When: Tests of independence / goodness-of-fit (AI).
The spread of the middle 50% of the data.
When: Spread; outlier boundaries.
An event and its non-occurrence have probabilities that sum to 1.
When: Finding \(P(\text{not } A)\).
When two events cannot both happen, just add their probabilities.
When: Events with no overlap.
Scaling and shifting: the mean scales and shifts; the variance scales by \(a^2\).
When: Transforming a random variable (HL).
For independent variables: means add; variances add with squared weights.
When: Sums of independent variables (AI HL).
Corrects the sample variance so it doesn't underestimate the population.
When: Estimating population spread from a sample (AI HL).
Apply a transition matrix \(n\) times to find the state after \(n\) steps.
When: Markov chains (AI HL).
Bring the power down, reduce it by one.
When: Differentiating powers of x.
The common functions and their derivatives.
When: Differentiating trig/exponential/log (AA).
Differentiate the outside, times the derivative of the inside.
When: Function inside a function.
For products and fractions of two functions.
When: Differentiating e.g. x²eˣ or x/(x+1).
Reverse of differentiating: raise the power, divide by it.
When: Integrating powers of x (don't forget +C).
The common antiderivatives.
When: Integrating exponential/log/trig (AA).
A definite integral sums thin strips into a signed area.
When: Area, and (with velocity) distance.
Rotate a region about the x-axis and sum disc volumes.
When: Solids of revolution (AA HL).
Differentiate to go displacement→velocity→acceleration; integrate to reverse.
When: Motion problems.
Estimate area with trapezia of width h.
When: Approximating an integral (AI).
Integrate \(x\) with respect to \(y\) for a region against the \(y\)-axis.
When: Regions measured along \(y\) (HL).
Step forward along a differential equation using the gradient at each point.
When: Numerical solution of \(\tfrac{dy}{dx} = f(x,y)\) (HL).
Euler's method applied to two equations at once.
When: Coupled differential equations (AI HL).
Built from the eigenvalues and eigenvectors of the system.
When: Solving coupled linear systems (AI HL).
Knowing a formula is not the same as using it under exam pressure. Practise every AI HL topic with auto-marked questions and full worked solutions, and see the GDC guides for the calculator steps that go with them.