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The formula booklet

Mathematics: Applications and Interpretation · SL and HL · booklet version 1.1

Every formula the IB gives you in the examinations for Applications and Interpretation, in the order the booklet itself uses, with a note from me wherever the printed line is not the whole story. Version 1.1 is the current one. It is a shorter booklet than the AA one, and deliberately so — in AI a great deal of what AA prints as a formula is expected to come from your GDC.

✅What this page is

  • The formulas listed in the IB’s own booklet for this course — version 1.1, first examinations 2021 — restated here so you can search them on a phone.
  • Not the booklet itself. There is no cover page and no IB branding here: the booklet is an IB publication and this is our reference sheet. Work from the clean copy you are issued in the exam, and ask your teacher or me for the official PDF.
  • A formula is a fact, and nobody owns \( u_n = u_1 + (n-1)d \). What is ours on this page is the commentary: notes in this grey are mine, not the IB’s.
  • Still current: this version is the one in use for examinations up to and including November 2028. The revised DP mathematics courses are first taught in 2027 and first assessed in May 2029, and they will bring a new booklet.

🧠How to use it — and what it will not do for you

Students who lose marks to the booklet almost never lose them because a formula was missing. They lose them because they did not know a line existed, or because they knew it existed and stopped thinking once they had copied it.

Practise with it open, from day one. The booklet in the exam is clean, so anything you have written in the margin of yours you will have to carry in your head. Knowing which page a formula is on is worth more under time pressure than knowing the formula.

The codes are the index. Each line is tagged with the sub-topic it belongs to — \(\text{SL }1.6\) is percentage error, \(\text{AHL }4.19\) is transition matrices. When you cannot find something, work out which sub-topic it belongs to and look there; that is how the booklet is organised, not alphabetically.

What is given tells you what is examined. A formula that is printed is one you are never asked to derive; a formula that is absent is one you are expected to know, build, or get from your GDC. The list at the bottom of this page is the one worth reading twice.

🔢Prior learning · SL and HL

—
Area of a parallelogram
\[ A = bh \]where \(b\) is the base, \(h\) is the height
Area of a triangle
\[ A = \dfrac{1}{2}(bh) \]where \(b\) is the base, \(h\) is the height
Area of a trapezoid
\[ A = \dfrac{1}{2}(a+b)h \]where \(a\) and \(b\) are the parallel sides, \(h\) is the height
Area of a circle
\[ A = \pi r^{2} \]where \(r\) is the radius
Circumference of a circle
\[ C = 2\pi r \]where \(r\) is the radius
Volume of a cuboid
\[ V = lwh \]where \(l\) is the length, \(w\) is the width, \(h\) is the height
Volume of a cylinder
\[ V = \pi r^{2}h \]where \(r\) is the radius, \(h\) is the height
Volume of prism
\[ V = Ah \]where \(A\) is the area of cross-section, \(h\) is the height
Area of the curved surface of a cylinder
\[ A = 2\pi rh \]where \(r\) is the radius, \(h\) is the height
Distance between two points \((x_1,\,y_1)\) and \((x_2,\,y_2)\)
\[ d = \sqrt{(x_1-x_2)^{2} + (y_1-y_2)^{2}} \]
Coordinates of the midpoint of a line segment with endpoints \((x_1,\,y_1)\) and \((x_2,\,y_2)\)
\[ \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right) \]

🔢Prior learning · HL only

—
Solutions of a quadratic equation
\[ \text{the solutions of } ax^{2}+bx+c = 0 \text{ are } x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a},\quad a \ne 0 \]
In AI this sits in prior learning, and for HL only: at SL you are expected to solve quadratics on the GDC rather than by formula.

🔢Topic 1: Number and algebra · SL and HL

SL
1.2
The \(n\)th term of an arithmetic sequence
\[ u_n = u_1 + (n-1)d \]
The sum of \(n\) terms of an arithmetic sequence
\[ S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)\,;\quad S_n = \dfrac{n}{2}(u_1 + u_n) \]
Two forms of the same sum. The second is the faster one whenever you already know the last term \(u_n\).
SL
1.3
The \(n\)th term of a geometric sequence
\[ u_n = u_1 r^{\,n-1} \]
The sum of \(n\) terms of a finite geometric sequence
\[ S_n = \dfrac{u_1(r^{n}-1)}{r-1} = \dfrac{u_1(1-r^{n})}{1-r},\quad r \ne 1 \]
SL
1.4
Compound interest
\[ FV = PV \times \left(1 + \dfrac{r}{100k}\right)^{kn} \]where \(FV\) is the future value, \(PV\) is the present value, \(n\) is the number of years, \(k\) is the number of compounding periods per year, \(r\%\) is the nominal annual rate of interest
The \(100\) is already in the formula, so \(r\) goes in as a percentage (\(3.5\), not \(0.035\)). \(k\) is compounding periods per year and \(n\) is years, not periods. Depreciation is the same line with a negative \(r\).
SL
1.5
Exponents and logarithms
\[ a^{x} = b \iff x = \log_a b \]where \(a > 0\), \(b > 0\), \(a \ne 1\)
SL
1.6
Percentage error
\[ \varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\% \]where \(v_E\) is the exact value and \(v_A\) is the approximate value of \(v\)
The exact value is the denominator. Swapping them is the single most common slip in this sub-topic, and the modulus means your answer is never negative.

🔢Topic 1: Number and algebra · HL only

AHL
1.9
Laws of logarithms
\[ \log_a xy = \log_a x + \log_a y \]
\[ \log_a \dfrac{x}{y} = \log_a x - \log_a y \]
\[ \log_a x^{m} = m\log_a x \]for \(a,\,x,\,y > 0\)
Note what is not here: the AI booklet gives no change-of-base formula, unlike AA.
AHL
1.11
The sum of an infinite geometric sequence
\[ S_\infty = \dfrac{u_1}{1-r},\quad |r| < 1 \]
The condition is part of the formula — it is what makes a sum to infinity exist at all.
AHL
1.12
Complex numbers
\[ z = a + b\mathrm{i} \]
Discriminant
\[ \Delta = b^{2} - 4ac \]
In AI the discriminant appears here, next to complex numbers, because that is where it matters: \(\Delta < 0\) is the case with no real roots.
AHL
1.13
Modulus-argument (polar) and exponential (Euler) form
\[ z = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta \]
AHL
1.14
Determinant of a \(2 \times 2\) matrix
\[ \boldsymbol{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \det \boldsymbol{A} = |\boldsymbol{A}| = ad - bc \]
Inverse of a \(2 \times 2\) matrix
\[ \boldsymbol{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \boldsymbol{A}^{-1} = \dfrac{1}{\det \boldsymbol{A}} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\quad ad \ne bc \]
Only the \(2 \times 2\) case is given. Larger determinants and inverses are GDC work, and no formula for eigenvalues is printed either — you set up \(\det(\boldsymbol{A} - \lambda \boldsymbol{I}) = 0\) yourself.
AHL
1.15
Power formula for a matrix
\[ \boldsymbol{M}^{n} = \boldsymbol{P}\boldsymbol{D}^{n}\boldsymbol{P}^{-1} \]where \(\boldsymbol{P}\) is the matrix of eigenvectors and \(\boldsymbol{D}\) is the diagonal matrix of eigenvalues
\(\boldsymbol{D}\) is diagonal, so \(\boldsymbol{D}^{n}\) is just each eigenvalue raised to \(n\). That is the whole point of diagonalising.

🔢Topic 2: Functions · SL and HL

SL
2.1
Equations of a straight line
\[ y = mx + c\,;\quad ax + by + d = 0\,;\quad y - y_1 = m(x-x_1) \]
Gradient formula
\[ m = \dfrac{y_2-y_1}{x_2-x_1} \]
SL
2.5
Axis of symmetry of the graph of a quadratic function
\[ f(x) = ax^{2} + bx + c \Rightarrow \text{axis of symmetry is } x = -\dfrac{b}{2a} \]
This is the only function formula SL gets. Every other model in Topic 2 — exponential, sinusoidal, cubic — is fitted on the GDC, which is the AI way of working and why the list is so short.

🔢Topic 2: Functions · HL only

AHL
2.9
Logistic function
\[ f(x) = \dfrac{L}{1 + C\mathrm{e}^{-kx}},\quad L,\,k,\,C > 0 \]
\(L\) is the carrying capacity — the value the model approaches. Naming it in context is usually worth a mark on its own.

🔢Topic 3: Geometry and trigonometry · SL and HL

SL
3.1
Distance between two points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\)
\[ d = \sqrt{(x_1-x_2)^{2} + (y_1-y_2)^{2} + (z_1-z_2)^{2}} \]
Coordinates of the midpoint of a line segment with endpoints \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\)
\[ \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2},\ \dfrac{z_1+z_2}{2}\right) \]
Volume of a right-pyramid
\[ V = \dfrac{1}{3}Ah \]where \(A\) is the area of the base, \(h\) is the height
Volume of a right cone
\[ V = \dfrac{1}{3}\pi r^{2}h \]where \(r\) is the radius, \(h\) is the height
Area of the curved surface of a cone
\[ A = \pi r l \]where \(r\) is the radius, \(l\) is the slant height
Volume of a sphere
\[ V = \dfrac{4}{3}\pi r^{3} \]where \(r\) is the radius
Surface area of a sphere
\[ A = 4\pi r^{2} \]where \(r\) is the radius
SL
3.2
Sine rule
\[ \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \]
Nothing here warns you about the ambiguous case: when you use the sine rule to find an angle, check whether the obtuse solution also fits.
Cosine rule
\[ c^{2} = a^{2} + b^{2} - 2ab\cos C\,;\quad \cos C = \dfrac{a^{2}+b^{2}-c^{2}}{2ab} \]
Area of a triangle
\[ A = \dfrac{1}{2}ab\sin C \]
The angle must be the one between the two sides — and in AI these three lines carry a lot of the three-dimensional and Voronoi work too.
SL
3.4
Length of an arc
\[ l = \dfrac{\theta}{360} \times 2\pi r \]where \(\theta\) is the angle measured in degrees, \(r\) is the radius
Area of a sector
\[ A = \dfrac{\theta}{360} \times \pi r^{2} \]where \(\theta\) is the angle measured in degrees, \(r\) is the radius
Degrees at SL. These are the AI SL forms; the radian versions \(l = r\theta\) and \(A = \tfrac{1}{2}r^{2}\theta\) are AHL 3.7. Using the wrong pair is the classic AI slip — check which mode your GDC is in as well.

🔢Topic 3: Geometry and trigonometry · HL only

AHL
3.7
Length of an arc
\[ l = r\theta \]where \(r\) is the radius, \(\theta\) is the angle measured in radians
Area of a sector
\[ A = \dfrac{1}{2}r^{2}\theta \]where \(r\) is the radius, \(\theta\) is the angle measured in radians
AHL
3.8
Identities
\[ \cos^{2}\theta + \sin^{2}\theta = 1 \]
\[ \tan\theta = \dfrac{\sin\theta}{\cos\theta} \]
Two identities, and that is all of trigonometric identity work in AI — there are no compound or double angle formulas in this booklet.
AHL
3.9
Transformation matrices
\[ \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} \]reflection in the line \(y = (\tan\theta)x\)
\[ \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \]horizontal stretch / stretch parallel to the \(x\)-axis with a scale factor of \(k\)
\[ \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \]vertical stretch / stretch parallel to the \(y\)-axis with a scale factor of \(k\)
\[ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \]enlargement, with a scale factor of \(k\), centre \((0,\,0)\)
\[ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \]anticlockwise / counter-clockwise rotation of angle \(\theta\) about the origin \((\theta > 0)\)
\[ \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \]clockwise rotation of angle \(\theta\) about the origin \((\theta > 0)\)
Six matrices, all given — so the marks are in composing them in the right order (the last transformation applied is the leftmost matrix) and in reading \(\theta\) correctly.
AHL
3.10
Magnitude of a vector
\[ |\boldsymbol{v}| = \sqrt{v_1^{2}+v_2^{2}+v_3^{2}}, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \]
AHL
3.11
Vector equation of a line
\[ \boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b} \]
Parametric form of the equation of a line
\[ x = x_0 + \lambda l,\ \ y = y_0 + \lambda m,\ \ z = z_0 + \lambda n \]
AI gives no Cartesian form of a line and nothing at all about planes — those are AA.
AHL
3.13
Scalar product
\[ \boldsymbol{v}\cdot\boldsymbol{w} = v_1w_1 + v_2w_2 + v_3w_3, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \]
\[ \boldsymbol{v}\cdot\boldsymbol{w} = |\boldsymbol{v}||\boldsymbol{w}|\cos\theta \]where \(\theta\) is the angle between \(\boldsymbol{v}\) and \(\boldsymbol{w}\)
Angle between two vectors
\[ \cos\theta = \dfrac{v_1w_1+v_2w_2+v_3w_3}{|\boldsymbol{v}||\boldsymbol{w}|} \]
That perpendicular vectors are the case \(\boldsymbol{v}\cdot\boldsymbol{w} = 0\) is not printed, and it is what most of these questions turn on.
Vector product
\[ \boldsymbol{v}\times\boldsymbol{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \]
Copy the middle row carefully: it is \(v_3w_1 - v_1w_3\), the one that looks “backwards”.
\[ |\boldsymbol{v}\times\boldsymbol{w}| = |\boldsymbol{v}||\boldsymbol{w}|\sin\theta \]where \(\theta\) is the angle between \(\boldsymbol{v}\) and \(\boldsymbol{w}\)
Area of a parallelogram
\[ A = |\boldsymbol{v}\times\boldsymbol{w}| \]where \(\boldsymbol{v}\) and \(\boldsymbol{w}\) form two adjacent sides of a parallelogram

🔢Topic 4: Statistics and probability · SL and HL

SL
4.2
Interquartile range
\[ \text{IQR} = Q_3 - Q_1 \]
SL
4.3
Mean, \(\overline{x}\), of a set of data
\[ \overline{x} = \dfrac{\sum\limits_{i=1}^{k} f_i x_i}{n}, \quad \text{where } n = \sum_{i=1}^{k} f_i \]
Standard deviation is not given at SL in AI: it is a one-variable-statistics button on the GDC, and that is how the papers expect you to get it.
SL
4.5
Probability of an event \(A\)
\[ \text{P}(A) = \dfrac{n(A)}{n(U)} \]
Complementary events
\[ \text{P}(A) + \text{P}(A') = 1 \]
SL
4.6
Combined events
\[ \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) \]
Mutually exclusive events
\[ \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) \]
Conditional probability
\[ \text{P}(A \mid B) = \dfrac{\text{P}(A \cap B)}{\text{P}(B)} \]
Independent events
\[ \text{P}(A \cap B) = \text{P}(A)\text{P}(B) \]
The two you have to choose between are the last two: multiplying probabilities is only legal once independence is given or shown.
SL
4.7
Expected value of a discrete random variable \(X\)
\[ \text{E}(X) = \sum x\,\text{P}(X = x) \]
In AI this line is usually a fair-game or expected-profit question: work out \(\text{E}(X)\), then say what it means in context.
SL
4.8
Binomial distribution
\[ X \sim \text{B}(n,\,p) \]
Mean
\[ \text{E}(X) = np \]
Variance
\[ \operatorname{Var}(X) = np(1-p) \]
No normal-distribution formulas appear anywhere in the AI booklet, and no \(z = \dfrac{x-\mu}{\sigma}\) either — normal probabilities and inverse-normal values are GDC work.

🔢Topic 4: Statistics and probability · HL only

AHL
4.14
Linear transformation of a single random variable
\[ \text{E}(aX+b) = a\text{E}(X) + b\,;\quad \operatorname{Var}(aX+b) = a^{2}\operatorname{Var}(X) \]
The \(b\) disappears from the variance: shifting data does not spread it.
Linear combinations of \(n\) independent random variables, \(X_1,\,X_2,\,\dots,\,X_n\)
\[ \text{E}(a_1X_1 \pm a_2X_2 \pm \dots \pm a_nX_n) = a_1\text{E}(X_1) \pm a_2\text{E}(X_2) \pm \dots \pm a_n\text{E}(X_n) \]
\[ \operatorname{Var}(a_1X_1 \pm a_2X_2 \pm \dots \pm a_nX_n) = a_1^{2}\operatorname{Var}(X_1) + a_2^{2}\operatorname{Var}(X_2) + \dots + a_n^{2}\operatorname{Var}(X_n) \]
Note the variances always add, even for a difference. Writing \(\operatorname{Var}(X-Y) = \operatorname{Var}(X) - \operatorname{Var}(Y)\) is the mistake this line exists to prevent.
Sample statistics
Unbiased estimate of population variance \(s_{n-1}^{2}\)
\[ s_{n-1}^{2} = \dfrac{n}{n-1}s_n^{2} \]
Your GDC reports both \(s_n\) and \(s_{n-1}\). This line is how you convert the one it gave you into the one the question wants.
AHL
4.17
Poisson distribution
\[ X \sim \operatorname{Po}(m) \]
Mean
\[ \text{E}(X) = m \]
Variance
\[ \operatorname{Var}(X) = m \]
Mean and variance equal is the giveaway: a data set whose mean and variance are close is the hint that a Poisson model is being asked for.
AHL
4.19
Transition matrices
\[ \boldsymbol{T}^{n}\boldsymbol{s}_0 = \boldsymbol{s}_n \]where \(\boldsymbol{s}_0\) is the initial state

🔢Topic 5: Calculus · SL and HL

SL
5.3
Derivative of \(x^{n}\)
\[ f(x) = x^{n} \Rightarrow f'(x) = nx^{\,n-1} \]
SL
5.5
Integral of \(x^{n}\)
\[ \int x^{n}\,\mathrm{d}x = \dfrac{x^{\,n+1}}{n+1} + C,\quad n \ne -1 \]
Area of region enclosed by a curve \(y = f(x)\) and the \(x\)-axis, where \(f(x) > 0\)
\[ A = \int_a^b y\,\mathrm{d}x \]
Note the condition \(f(x) > 0\). At SL the AI booklet gives no modulus version — HL gets one at AHL 5.12.
SL
5.8
The trapezoidal rule
\[ \int_a^b y\,\mathrm{d}x \approx \dfrac{1}{2}h\bigl((y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1})\bigr),\quad \text{where } h = \dfrac{b-a}{n} \]
\(n\) is the number of strips, so there are \(n+1\) ordinates \(y_0 \dots y_n\). Counting them wrong is the usual lost mark; the two end values are not doubled.

🔢Topic 5: Calculus · HL only

AHL
5.9
Derivative of \(\sin x\)
\[ f(x) = \sin x \Rightarrow f'(x) = \cos x \]
Derivative of \(\cos x\)
\[ f(x) = \cos x \Rightarrow f'(x) = -\sin x \]
Derivative of \(\tan x\)
\[ f(x) = \tan x \Rightarrow f'(x) = \dfrac{1}{\cos^{2}x} \]
AI writes this as \(\dfrac{1}{\cos^{2}x}\), not \(\sec^{2}x\) — there are no reciprocal functions in this booklet.
Derivative of \(\mathrm{e}^{x}\)
\[ f(x) = \mathrm{e}^{x} \Rightarrow f'(x) = \mathrm{e}^{x} \]
Derivative of \(\ln x\)
\[ f(x) = \ln x \Rightarrow f'(x) = \dfrac{1}{x} \]
Chain rule
\[ y = g(u), \text{ where } u = f(x) \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}u} \times \dfrac{\mathrm{d}u}{\mathrm{d}x} \]
Product rule
\[ y = uv \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = u\dfrac{\mathrm{d}v}{\mathrm{d}x} + v\dfrac{\mathrm{d}u}{\mathrm{d}x} \]
Quotient rule
\[ y = \dfrac{u}{v} \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{v\dfrac{\mathrm{d}u}{\mathrm{d}x} - u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^{2}} \]
All of this is HL in AI: at SL the only derivative you are given is \(x^{n}\).
AHL
5.11
Standard integrals
\[ \int \dfrac{1}{x}\,\mathrm{d}x = \ln|x| + C \]
\[ \int \sin x\,\mathrm{d}x = -\cos x + C \]
\[ \int \cos x\,\mathrm{d}x = \sin x + C \]
\[ \int \dfrac{1}{\cos^{2}x}\,\mathrm{d}x = \tan x + C \]
\[ \int \mathrm{e}^{x}\,\mathrm{d}x = \mathrm{e}^{x} + C \]
Five integrals, and no integration by parts and no substitution anywhere in AI. If an integral will not yield to these, it is a GDC question.
AHL
5.12
Area of region enclosed by a curve and \(x\) or \(y\)-axes
\[ A = \int_a^b |y|\,\mathrm{d}x \quad \text{or} \quad A = \int_a^b |x|\,\mathrm{d}y \]
Volume of revolution about the \(x\) or \(y\)-axes
\[ V = \int_a^b \pi y^{2}\,\mathrm{d}x \quad \text{or} \quad V = \int_a^b \pi x^{2}\,\mathrm{d}y \]
The limits belong to the variable you are integrating with respect to — for the \(y\)-axis version they are \(y\)-values.
AHL
5.13
Acceleration
\[ a = \dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}} = v\dfrac{\mathrm{d}v}{\mathrm{d}s} \]
The third form, \(v\dfrac{\mathrm{d}v}{\mathrm{d}s}\), is the one to reach for when acceleration is given in terms of displacement rather than time. AA does not print it.
Distance travelled from \(t_1\) to \(t_2\)
\[ \text{distance} = \int_{t_1}^{t_2} |v(t)|\,\mathrm{d}t \]
Displacement from \(t_1\) to \(t_2\)
\[ \text{displacement} = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t \]
The modulus signs are the entire difference between these two lines.
AHL
5.16
Euler's method
\[ y_{n+1} = y_n + h \times f(x_n,\,y_n)\,;\quad x_{n+1} = x_n + h \]where \(h\) is a constant (step length)
Euler's method for coupled systems
\[ \begin{aligned} x_{n+1} &= x_n + h \times f_1(x_n,\,y_n,\,t_n) \\ y_{n+1} &= y_n + h \times f_2(x_n,\,y_n,\,t_n) \\ t_{n+1} &= t_n + h \end{aligned} \]where \(h\) is a constant (step length)
Both use the values from the same step on the right-hand side: update \(x\) and \(y\) from the old pair, never from the \(x\) you just computed.
AHL
5.17
Exact solution for coupled linear differential equations
\[ \boldsymbol{x} = A\mathrm{e}^{\lambda_1 t}\boldsymbol{p}_1 + B\mathrm{e}^{\lambda_2 t}\boldsymbol{p}_2 \]
\(\lambda_1,\,\lambda_2\) are the eigenvalues and \(\boldsymbol{p}_1,\,\boldsymbol{p}_2\) the eigenvectors of the system matrix. Their signs are what you describe the long-term behaviour from.

⚠What is not in the booklet

These come up every session and none of them are printed. This is the part of the page to reread the night before.

  • No standard deviation formula at SL — and none needed: one-variable statistics on the GDC gives you \( s_n \). AHL 4.14 only tells you how to turn it into \( s_{n-1} \).
  • Nothing about the normal distribution. No density, no \( z = \dfrac{x-\mu}{\sigma} \). Every normal and inverse-normal value comes from the GDC, and writing the working as \( \text{P}(X < 12) = 0.894 \) is what earns the method mark.
  • None of the statistical tests. \( \chi^{2} \) goodness-of-fit and independence, the \( t \)-test, the correlation coefficient \( r \), the regression lines and Spearman’s \( r_s \) are all examinable and all absent — you need to know which test a context asks for, the degrees of freedom, and how to read the \( p \)-value against the significance level.
  • No formula for eigenvalues or eigenvectors, although AHL 1.15 and AHL 5.17 both depend on them. You set up \( \det(\boldsymbol{A} - \lambda\boldsymbol{I}) = 0 \) yourself.
  • Nothing about Voronoi diagrams — perpendicular bisectors, the toxic-waste-dump problem and nearest-neighbour interpolation are SL content with no printed formula. The tools are the gradient and midpoint lines in SL 2.1 and prior learning.
  • No change of base for logarithms (AA has one; AI does not), no compound or double angle identities, and no integration by parts or substitution.
  • Nothing about the Exploration, and nothing about percentage uncertainty beyond the percentage-error line at SL 1.6.

✏Find it in the booklet

Six drills. The skill being practised is not algebra — it is deciding which line you need, and noticing when there isn’t one. Try each before opening the answer.

A sector of a circle of radius \( 8\text{ cm} \) subtends an angle of \( 1.2 \) radians. You are at SL. Can you use the booklet as printed?

Not the SL line, no. SL 3.4 is in degrees: \( A = \dfrac{\theta}{360}\times\pi r^{2} \). Either convert (\( 1.2 \text{ rad} = 68.8^\circ \)) or use the HL forms at AHL 3.7, \( A = \tfrac{1}{2}r^{2}\theta = \tfrac{1}{2}(64)(1.2) = 38.4\text{ cm}^{2} \). At SL an angle in radians is a sign the question has been mis-set or that you have misread it — and check your GDC's angle mode before anything else.

A question asks whether the number of tourists per day fits a Poisson model, given a sample mean of \( 4.1 \) and a sample variance of \( 4.3 \). What does the booklet contribute?

AHL 4.17: \( \text{E}(X) = m \) and \( \operatorname{Var}(X) = m \). The argument is that the mean and variance are close, which is consistent with Poisson — and that sentence, in context, is the mark. The \( \chi^{2} \) goodness-of-fit test that would actually test it is examinable but nowhere in the booklet.

Heights are normally distributed with \( \mu = 168 \), \( \sigma = 7 \). Find \( \text{P}(X > 180) \). Which line do you use?

None — there are no normal-distribution formulas in the AI booklet at all, not even \( z = \dfrac{x-\mu}{\sigma} \). This is normal cdf on the GDC: \( 0.0432 \). Write \( X \sim \text{N}(168,\,7^{2}) \) and \( \text{P}(X > 180) \) as the method, then the value. Being sure a formula is missing is as useful as knowing where one is.

Estimate \( \displaystyle\int_1^5 \ln x\,\mathrm{d}x \) with four strips. Which line, and how many ordinates do you need?

SL 5.8, the trapezoidal rule, with \( h = \dfrac{5-1}{4} = 1 \). Four strips means five ordinates, at \( x = 1,\,2,\,3,\,4,\,5 \). Then \( \tfrac{1}{2}(1)\bigl((0 + 1.6094) + 2(0.6931 + 1.0986 + 1.3863)\bigr) = 3.99 \) (3 s.f.). Only the interior values are doubled.

A \( \$ 20\,000 \) car depreciates at \( 12\% \) per year. Its value after 5 years? Is there a depreciation formula?

There is not — you use the compound interest line at SL 1.4 with a negative rate: \( FV = 20\,000\left(1 - \dfrac{12}{100}\right)^{5} = \$ 10\,560 \) (3 s.f.), taking \( k = 1 \). Recognising depreciation as negative-rate compounding is the step being tested.

HL. A system has \( \boldsymbol{M} \) with eigenvalues \( 3 \) and \( -1 \). You are asked for \( \boldsymbol{M}^{10} \) and for the long-term behaviour of \( \dot{\boldsymbol{x}} = \boldsymbol{M}\boldsymbol{x} \). What is printed, and what is not?

Printed: AHL 1.15, \( \boldsymbol{M}^{n} = \boldsymbol{P}\boldsymbol{D}^{n}\boldsymbol{P}^{-1} \), and AHL 5.17, \( \boldsymbol{x} = A\mathrm{e}^{\lambda_1 t}\boldsymbol{p}_1 + B\mathrm{e}^{\lambda_2 t}\boldsymbol{p}_2 \). Not printed: how to get the eigenvalues and eigenvectors in the first place (\( \det(\boldsymbol{M} - \lambda\boldsymbol{I}) = 0 \)), and the interpretation — one positive and one negative eigenvalue means a saddle: the solution grows along \( \boldsymbol{p}_1 \) and decays along \( \boldsymbol{p}_2 \).

🔗Where to get the official booklet

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • My IB → Programme Resource Centre — the authoritative copy of Mathematics: applications and interpretation formula booklet, version 1.1. Your school’s IB login reaches it; ask me if you cannot.
  • Your teacher — schools are licensed to give you a copy, and most will print one. That is the simplest route and the one I would take.
  • ibo.org → Diploma Programme → Mathematics — the public subject pages, for the guides, subject briefs and specimen material.