Mathematics: Applications & Interpretation SL & HL
Interactive Formula Sheet with Study Notes & FAQs — Updated for 2026 Examinations
Prior Learning SL & HL
Key Revision Points
- Area formulas — Always check units. The parallelogram, triangle, and trapezoid formulas require the perpendicular height, not the slant height.
- Circle formulas — Remember \(\pi\) is approximately 3.14159. Use your GDC's \(\pi\) key for exact answers.
- Volume formulas — Prism volume = cross-section area × length. Cylinder is a special case (circular cross-section).
- Distance & midpoint — These extend to 3D in Topic 3. Master the 2D versions here first.
💡 Exam Tip: In Paper 2, always show your substitution into
formulas before using the GDC for the final answer — this earns method marks.
SL & HL Content
Area: Parallelogram
$$A = bh$$
\(b\) = base, \(h\) = perpendicular height
Area: Triangle
$$A = \frac{1}{2}bh$$
\(b\) = base, \(h\) = height
Area: Trapezoid
$$A = \frac{1}{2}(a+b)h$$
\(a, b\) = parallel sides, \(h\) = height
Area: Circle
$$A = \pi r^2$$
\(r\) = radius
Circumference: Circle
$$C = 2\pi r$$
\(r\) = radius
Volume: Cuboid
$$V = lwh$$
\(l\) = length, \(w\) = width, \(h\) = height
Volume: Cylinder
$$V = \pi r^2 h$$
\(r\) = radius, \(h\) = height
Volume: Prism
$$V = Ah$$
\(A\) = cross-section area, \(h\) = height
Lateral Surface Area: Cylinder
$$A = 2\pi rh$$
\(r\) = radius, \(h\) = height
Distance between two points
$$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$
For points \((x_1,y_1)\) and \((x_2,y_2)\)
Coordinates of midpoint
$$\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right)$$
For endpoints \((x_1,y_1)\) and \((x_2,y_2)\)
HL Only
Quadratic formula
$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a},\quad a \neq 0$$
For \(ax^2+bx+c=0\). Listed under Prior Learning HL only in the AI
syllabus.
Topic 1: Number & Algebra
Key Revision Points
- Arithmetic sequences — The common difference \(d\) determines if the sequence increases (\(d \gt 0\)) or decreases (\(d \lt 0\)). Always identify \(u_1\) and \(d\) first.
- Geometric sequences — Check if \(|r| \lt 1\) for convergence of infinite series. The ratio \(r = u_{n+1}/u_n\) for any consecutive terms.
- Compound interest — Distinguish between nominal and effective rates. Identify \(k\) (compounding frequency) carefully.
- Logarithms (HL) — Remember: logs convert multiplication to addition and powers to multiplication. Change of base formula is essential for calculator use.
- Matrices (HL) — A matrix is invertible only if its determinant is non-zero. Practice finding eigenvalues by solving \(\det(\mathbf{A}-\lambda\mathbf{I})=0\).
💡 Exam Tip: For compound interest, always read whether the rate
is nominal or effective, and identify the compounding frequency \(k\) carefully.
SL & HL Content
Arithmetic Sequence: \(n^{th}\) term
$$u_n = u_1 + (n-1)d$$
\(u_1\) = first term, \(d\) = common difference
Arithmetic Series: Sum of \(n\) terms
$$S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$$
Geometric Sequence: \(n^{th}\) term
$$u_n = u_1 r^{n-1}$$
\(r\) = common ratio
Geometric Series: Sum of \(n\) terms
$$S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1 - r^n)}{1-r},\quad r
\neq 1$$
Compound Interest
$$FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$$
FV = future value, PV = present value, \(n\) = years, \(k\) =
compounding periods/year, \(r\%\) = nominal annual rate
Exponents & Logarithms
$$a^x = b \iff x = \log_a b$$
\(a \gt 0,\; b \gt 0,\; a \neq 1\)
Percentage Error
$$\varepsilon = \left|\frac{v_A - v_E}{v_E}\right| \times 100\%$$
\(v_A\) = approximate value, \(v_E\) = exact value
HL Only
Laws of Logarithms
$$\log_a xy = \log_a x + \log_a y$$$$\log_a \frac{x}{y} = \log_a x -
\log_a y$$$$\log_a x^m = m\log_a x$$
For \(a, x, y \gt 0\). Change of base: \(\log_a x = \frac{\log_b
x}{\log_b a}\)
Infinite Geometric Series
$$S_\infty = \frac{u_1}{1-r},\quad |r| \lt 1$$
Complex Numbers
$$z = a + bi$$
\(a, b \in \mathbb{R}\), \(i^2 = -1\)
Discriminant
$$\Delta = b^2 - 4ac$$
\(\Delta \gt 0\): two real roots, \(\Delta = 0\): one repeated root,
\(\Delta \lt 0\): complex roots
Polar / Euler Form
$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$
\(r = |z|\), \(\theta = \arg(z)\)
Determinant of a 2×2 Matrix
$$\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \implies
\det\mathbf{A} = ad - bc$$
Inverse of a 2×2 Matrix
$$\mathbf{A}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a
\end{pmatrix}$$
Provided \(\det\mathbf{A} \neq 0\)
Matrix Power (Diagonalization)
$$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}$$
\(\mathbf{P}\) = eigenvector matrix, \(\mathbf{D}\) = diagonal
eigenvalue matrix
Topic 2: Functions
Key Revision Points
- Straight lines — Know all three forms: slope-intercept \(y=mx+c\), general \(ax+by+d=0\), and point-slope \(y-y_1=m(x-x_1)\). Parallel lines have equal gradients; perpendicular lines have \(m_1 \cdot m_2 = -1\).
- Quadratic functions — The axis of symmetry \(x = -\frac{b}{2a}\) gives the x-coordinate of the vertex. Substitute back to find the y-coordinate.
- Logistic function (HL) — Models population growth with a carrying capacity \(L\). The curve is S-shaped with a horizontal asymptote at \(y = L\).
💡 Exam Tip: When modelling with functions, always state the
domain and range in context. The GDC is your best friend for finding intersections and key features.
SL & HL Content
Equations of a Straight Line
$$y = mx + c$$$$ax + by + d = 0$$$$y - y_1 = m(x - x_1)$$
Gradient Formula
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Axis of Symmetry
$$x = -\frac{b}{2a}$$
For quadratic \(f(x) = ax^2 + bx + c\)
HL Only
Logistic Function
$$f(x) = \frac{L}{1 + Ce^{-kx}}$$
\(L, k, C \gt 0\). Horizontal asymptote at \(y = L\).
Topic 3: Geometry & Trigonometry
Key Revision Points
- 3D distance and midpoint — Direct extensions of 2D formulas. Always label coordinates clearly in 3D problems.
- Sine & Cosine rules — Use sine rule when you have a side-angle pair; use cosine rule when you have SAS or SSS.
- Arc length & sector area (degrees) — SL uses degrees: \(l = \frac{\theta}{360°} \times 2\pi r\). HL also uses radians: \(l = r\theta\).
- Vectors (HL) — The dot product gives a scalar and is used for angles. The cross product gives a vector perpendicular to both inputs.
- Transformation matrices (HL) — Remember the rotation matrices use \(\theta\) for anticlockwise rotation. Clockwise swaps signs on the sine terms.
💡 Exam Tip: In triangle problems, always draw and label a
diagram. Identify which rule to use based on the given information (SSS → cosine rule, AAS → sine
rule).
SL & HL Content
3D Distance
$$d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$$
3D Midpoint
$$\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2},\;\frac{z_1+z_2}{2}\right)$$
Volume: Right Pyramid
$$V = \frac{1}{3}Ah$$
\(A\) = base area, \(h\) = height
Volume: Right Cone
$$V = \frac{1}{3}\pi r^2 h$$
Lateral Surface Area: Cone
$$A = \pi r l$$
\(l\) = slant height
Volume: Sphere
$$V = \frac{4}{3}\pi r^3$$
Surface Area: Sphere
$$A = 4\pi r^2$$
Sine Rule
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Cosine Rule
$$c^2 = a^2 + b^2 - 2ab\cos C$$$$\cos C = \frac{a^2+b^2-c^2}{2ab}$$
Area: Triangle (trig)
$$A = \frac{1}{2}ab\sin C$$
Arc Length (degrees)
$$l = \frac{\theta}{360°} \times 2\pi r$$
Sector Area (degrees)
$$A = \frac{\theta}{360°} \times \pi r^2$$
HL Only
Arc Length (radians)
$$l = r\theta$$
Sector Area (radians)
$$A = \frac{1}{2}r^2\theta$$
Pythagorean Identity
$$\cos^2\theta + \sin^2\theta = 1$$$$\tan\theta =
\frac{\sin\theta}{\cos\theta}$$
Transformation Matrices (2D)
$$\text{Rotation (anticlockwise):}\;\begin{pmatrix}\cos\theta &
-\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$$$$\text{Reflection in
}y=(\tan\theta)x:\;\begin{pmatrix}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos
2\theta\end{pmatrix}$$$$\text{Enlargement, factor }k:\;\begin{pmatrix}k & 0 \\ 0 &
k\end{pmatrix}$$
Magnitude of a Vector
$$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$
Vector Equation of a Line
$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$
\(\mathbf{a}\) = position vector, \(\mathbf{b}\) = direction vector
Scalar Product (Dot Product)
$$\mathbf{v}\cdot\mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3 =
|\mathbf{v}||\mathbf{w}|\cos\theta$$
Vector Product (Cross Product)
$$\mathbf{v}\times\mathbf{w} = \begin{pmatrix}v_2w_3-v_3w_2 \\
v_3w_1-v_1w_3 \\ v_1w_2-v_2w_1\end{pmatrix}$$$$|\mathbf{v}\times\mathbf{w}| =
|\mathbf{v}||\mathbf{w}|\sin\theta$$
Area of Parallelogram (vectors)
$$A = |\mathbf{v}\times\mathbf{w}|$$
Triangle area = \(\frac{1}{2}|\mathbf{v}\times\mathbf{w}|\)
Topic 4: Statistics & Probability
Key Revision Points
- Descriptive statistics — Know how to find mean, median, mode, IQR, and standard deviation using your GDC. Outliers are typically values beyond \(Q_1 - 1.5 \times \text{IQR}\) or \(Q_3 + 1.5 \times \text{IQR}\).
- Probability rules — Addition rule is for "or" (union), multiplication rule is for "and" (intersection). Draw Venn diagrams or tree diagrams to visualise.
- Chi-squared test — Always state hypotheses, calculate expected frequencies, find degrees of freedom, and compare the p-value with the significance level.
- Regression — Always check the correlation coefficient \(r\) before making predictions. Interpolation is reliable; extrapolation is not.
- Poisson distribution (HL) — Mean equals variance (\(E(X) = \text{Var}(X) = m\)). Used for modelling rare events in fixed intervals.
💡 Exam Tip: Statistics is GDC-heavy in Math AI. Practice entering
data into lists and using statistical tests on your specific calculator model.
SL & HL Content
Interquartile Range
$$\text{IQR} = Q_3 - Q_1$$
Mean of a Data Set
$$\bar{x} = \frac{\sum_{i=1}^{k} f_i x_i}{n}$$
where \(n = \sum f_i\) is the total frequency
Probability of Event A
$$P(A) = \frac{n(A)}{n(U)}$$
Complementary Events
$$P(A) + P(A') = 1$$
Addition Rule
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Mutually Exclusive Events
$$P(A \cup B) = P(A) + P(B)$$
When \(P(A \cap B) = 0\)
Conditional Probability
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Independent Events
$$P(A \cap B) = P(A) \cdot P(B)$$
Expected Value
$$E(X) = \sum x \cdot P(X=x)$$
Binomial Distribution \(X \sim B(n,p)\)
$$P(X=x) = \binom{n}{x}p^x(1-p)^{n-x}$$$$E(X) = np$$$$\text{Var}(X) =
np(1-p)$$
Chi-Squared Statistic
$$\chi^2_{calc} = \sum\frac{(f_o - f_e)^2}{f_e}$$
\(f_o\) = observed, \(f_e\) = expected. d.f. =
\((\text{rows}-1)(\text{cols}-1)\)
Spearman's Rank \(r_s\)
$$r_s = 1 - \frac{6\sum d^2}{n(n^2-1)}$$
For no tied ranks. Usually calculated via GDC.
Regression Line (\(y\) on \(x\))
$$y - \bar{y} = \frac{s_{xy}}{s_x^2}(x - \bar{x})$$
Coefficients found using GDC
Normal Distribution
$$X \sim N(\mu, \sigma^2)$$$$Z = \frac{X - \mu}{\sigma}$$
Probabilities found using GDC or Z-table
HL Only
Linear Transformation
$$E(aX+b) = aE(X)+b$$$$\text{Var}(aX+b) = a^2\text{Var}(X)$$
Linear Combinations (independent)
$$E(a_1X_1 \pm a_2X_2) = a_1E(X_1) \pm a_2E(X_2)$$$$\text{Var}(a_1X_1
\pm a_2X_2) = a_1^2\text{Var}(X_1) + a_2^2\text{Var}(X_2)$$
Variances always add for independent variables
Unbiased Variance Estimate
$$s_{n-1}^2 = \frac{n}{n-1}s_n^2$$
Poisson Distribution \(X \sim \text{Po}(m)\)
$$P(X=x) = \frac{e^{-m}m^x}{x!}$$$$E(X) = m,\quad \text{Var}(X) = m$$
Markov Chains
$$\mathbf{T}^n\mathbf{s}_0 = \mathbf{s}_n$$
Steady state: \(\mathbf{T}\mathbf{s} = \mathbf{s}\)
Topic 5: Calculus
Key Revision Points
- Power rule — The derivative of \(x^n\) is \(nx^{n-1}\) and the integral reverses this. This is the foundation of all SL calculus.
- Trapezoidal rule — Essential for SL. It approximates the area under a curve. More sub-intervals = better approximation.
- Chain, product, quotient rules (HL) — Practice identifying which rule to use. Chain rule for compositions, product for multiplied functions, quotient for divisions.
- Euler's method (HL) — A numerical method for solving ODEs. Smaller step size \(h\) gives better accuracy but requires more steps.
- Kinematics (HL) — Displacement = integral of velocity. Distance = integral of |velocity|. Acceleration = derivative of velocity.
💡 Exam Tip: In Math AI, calculus questions often appear in
real-world modelling contexts. Always interpret your derivative/integral answer in the context of
the problem.
SL & HL Content
Derivative of \(x^n\)
$$f(x) = x^n \implies f'(x) = nx^{n-1}$$
Integral of \(x^n\)
$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\quad n \neq -1$$
Area Under a Curve
$$A = \int_a^b |f(x)|\,dx$$
Trapezoidal Rule
$$\int_a^b y\,dx \approx \frac{h}{2}\big((y_0+y_n) +
2(y_1+y_2+\cdots+y_{n-1})\big)$$
where \(h = \frac{b-a}{n}\)
HL Only
Derivative of \(\sin x\)
$$\frac{d}{dx}(\sin x) = \cos x$$
Derivative of \(\cos x\)
$$\frac{d}{dx}(\cos x) = -\sin x$$
Derivative of \(e^x\) and \(\ln x\)
$$\frac{d}{dx}(e^x) = e^x$$$$\frac{d}{dx}(\ln x) = \frac{1}{x},\quad x
\gt 0$$
Chain Rule
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
If \(y = g(f(x))\), then \(y' = g'(f(x)) \cdot f'(x)\)
Product Rule
$$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$$
Quotient Rule
$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} -
u\frac{dv}{dx}}{v^2}$$
Standard Integrals
$$\int\frac{1}{x}\,dx = \ln|x|+C$$$$\int\sin x\,dx = -\cos
x+C$$$$\int\cos x\,dx = \sin x+C$$$$\int e^x\,dx = e^x+C$$
Volume of Revolution
$$V = \pi\int_a^b y^2\,dx$$
About the x-axis
Kinematics
$$a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$$$$\text{Distance} =
\int_{t_1}^{t_2}|v|\,dt$$$$\text{Displacement} = \int_{t_1}^{t_2}v\,dt$$
Euler's Method
$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$$$x_{n+1} = x_n + h$$
For \(\frac{dy}{dx} = f(x,y)\), \(h\) = step size
Coupled Systems (Euler)
$$x_{n+1} = x_n + h\cdot f_1(x_n,y_n,t_n)$$$$y_{n+1} = y_n + h\cdot
f_2(x_n,y_n,t_n)$$
Coupled Differential Equations
$$\mathbf{x} = Ae^{\lambda_1 t}\mathbf{p}_1 + Be^{\lambda_2
t}\mathbf{p}_2$$
\(\lambda_{1,2}\) = eigenvalues, \(\mathbf{p}_{1,2}\) = eigenvectors
Frequently Asked Questions (FAQs)
1. What is the difference between IB Math AI SL and HL?
IB Math AI SL focuses on practical mathematics — statistics, modelling, and
real-world problem solving using technology. HL extends this with complex algebra, calculus
(chain/product/quotient rules, differential equations), matrices, eigenvalues, Markov chains, and
deeper hypothesis testing including Poisson distributions.
2. How is Math AI different from Math AA?
Math AI emphasises applications, statistics, and heavy GDC use. It's ideal for
students going into social sciences, business, or design. Math AA focuses on pure mathematics,
algebraic rigour, and deeper calculus — suited for STEM fields like engineering and physics. Both
are equally valid IB courses.
3. Which formulas are provided in the exam?
The IBO provides an official formula booklet containing most key formulas.
However, you must memorise basic definitions (probability rules, derivative/integral of \(x^n\)),
understand when each formula applies, and practice navigating the booklet quickly under timed
conditions.
4. Can I use a calculator on all Math AI papers?
Yes! Unlike Math AA, both Paper 1 and Paper 2 in Math AI allow calculator use. A
graphing display calculator (GDC) is required. However, you must still show working for method marks
— never just write the final answer.
5. What GDC is best for IB Math AI?
The TI-84 Plus CE, TI-Nspire CX, and Casio fx-CG50 are all popular. The
TI-Nspire is powerful but has a steeper learning curve. The TI-84 and Casio are more
straightforward. Choose one and master its statistical and graphing functions.
6. How important is the chi-squared test?
Very important for both SL and HL. You should be able to state null/alternative
hypotheses, calculate expected values, find the test statistic (\(\chi^2 =
\sum\frac{(f_o-f_e)^2}{f_e}\)), determine degrees of freedom, and compare the p-value with the
significance level to make a decision.
7. How do I approach modelling questions?
Identify the type of model (linear, quadratic, exponential, logistic,
sinusoidal), use your GDC to find the regression equation, check the correlation coefficient \(r\)
or \(R^2\), and always discuss the limitations of your model in context. State domain restrictions
and comment on the reasonableness of predictions.
8. What are the most important SL formulas?
Key SL formulas include: arithmetic & geometric series sums, compound
interest, sine & cosine rules, area of a triangle (\(\frac{1}{2}ab\sin C\)), basic derivatives
(\(nx^{n-1}\)), the trapezoidal rule, probability rules (addition, conditional, binomial), and
descriptive statistics (mean, IQR).
9. What additional topics do HL students need?
HL adds: complex numbers and polar form, matrices (determinants, inverses,
eigenvalues), advanced calculus (chain/product/quotient rules, volumes of revolution, differential
equations), Euler's method, Poisson distribution, Markov chains, and transformation matrices for 2D
geometry.
10. How are vectors tested in Math AI HL?
HL vector questions focus on the dot product (for finding angles), cross product
(for areas), vector equations of lines, and parametric equations. You should be comfortable
converting between different forms and finding intersections of lines in 3D space.
11. What is the trapezoidal rule used for?
The trapezoidal rule approximates definite integrals by dividing the area under
a curve into trapezoids. It's especially useful when you have data points rather than a function, or
when the integral is too complex to solve analytically. More sub-intervals give a better
approximation.
12. How should I prepare for Statistics & Probability?
This is the heaviest topic in Math AI. Master your GDC for: entering data lists,
finding regression equations, running chi-squared tests, and calculating normal/binomial
probabilities. Practice interpreting results in context — examiners look for contextual conclusions,
not just numbers.
13. What are Markov chains (HL)?
Markov chains model systems that transition between states with fixed
probabilities. The transition matrix \(\mathbf{T}\) multiplied by the initial state vector gives the
next state. The steady-state vector \(\mathbf{s}\) satisfies \(\mathbf{Ts} = \mathbf{s}\) and
represents long-term probabilities.
14. Tips for the Math AI Internal Assessment (IA)?
Choose a topic you're genuinely interested in with real data. AI IAs work well
with: statistical analysis of a dataset, mathematical modelling of a real phenomenon, or
optimisation problems. Use at least 2-3 mathematical techniques at the appropriate level. Reflect
critically on limitations.
15. How do I use this formula sheet for revision?
Use the tabs to focus on one topic at a time. Read the study notes first for
context, then review each formula card. Use the search to quickly find specific formulas. Filter by
SL/HL to focus on your level. Practice using each formula with past paper questions alongside this
sheet.