Mathematics: Applications & Interpretation SL & HL
Interactive Formula Sheet – First Examinations 2021 – Updated Version 1.1
Prior Learning
SL & HL
Area: Parallelogram
$$A = bh$$
b = base, h = 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
Area: Cylinder curve (lateral surface)
$$A = 2\pi rh$$
r = radius, h = height
Distance between two points \((x_1, y_1), (x_2, y_2)\)
$$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$
Coordinates of midpoint
$$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$
for endpoints \((x_1, y_1), (x_2, y_2)\)
HL only
Solutions of a quadratic equation \(ax^2+bx+c=0\)
$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}, \quad a \neq 0$$
This is usually considered prior learning for SL as well, but specifically listed under "Prior Learning HL only" in the AI sheet.
Topic 1: Number and algebra
SL & HL
The \(n^{th}\) term of an arithmetic sequence
$$u_n = u_1 + (n-1)d$$
Sum of \(n\) terms of an arithmetic sequence
$$S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$$
The \(n^{th}\) term of a geometric sequence
$$u_n = u_1 r^{n-1}$$
Sum of \(n\) terms of a finite geometric sequence
$$S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1 - r^n)}{1-r}, \quad r \neq 1$$
Compound interest
$$FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$$
FV: future value, PV: present value, n: number of years, k: compounding periods per year, r%: nominal annual rate of interest
Exponents & logarithms (definition)
$$a^x = b \iff x = \log_a b$$
\(a > 0, b > 0, a \neq 1\)
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 \(a, x, y > 0\). (Change of base \(\log_a x = \frac{\log_b x}{\log_b a}\) is also common but not explicitly here)
The sum of an infinite geometric sequence
$$S_\infty = \frac{u_1}{1-r}, \quad |r| < 1$$
Complex numbers
$$z = a + bi$$
\(a, b \in \mathbb{R}\), \(i^2 = -1\)
Discriminant (for quadratic equations)
$$\Delta = b^2-4ac$$
Modulus-argument (polar) & Exponential (Euler) form
$$z = r(\cos\theta + i\sin\theta) = re^{i\theta} = r\text{cis}\theta$$
\(r = |z|\), \(\theta = \arg(z)\)
Determinant of a \(2 \times 2\) matrix
$$\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \implies \det\mathbf{A} = |\mathbf{A}| = ad-bc$$
Inverse of a \(2 \times 2\) matrix
$$\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \implies \mathbf{A}^{-1} = \frac{1}{\det\mathbf{A}}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$
Provided \(\det\mathbf{A} \neq 0\)
Power formula for a matrix (diagonalization)
$$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}$$
Where \(\mathbf{P}\) is the matrix of eigenvectors and \(\mathbf{D}\) is the diagonal matrix of eigenvalues.
Topic 2: Functions
SL & HL
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 of a quadratic function \(f(x) = ax^2+bx+c\)
$$x = -\frac{b}{2a}$$
HL only
Logistic function
$$f(x) = \frac{L}{1+Ce^{-kx}}$$
L, k, C > 0
Topic 3: Geometry and trigonometry
SL & HL
Distance between 2 points \((x_1,y_1,z_1), (x_2,y_2,z_2)\)
$$d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}$$
Coordinates of midpoint of a line with endpoints \((x_1,y_1,z_1), (x_2,y_2,z_2)\)
$$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$$
Volume: Right-pyramid
$$V = \frac{1}{3}Ah$$
A = base area, h = height
Volume: Right cone
$$V = \frac{1}{3}\pi r^2 h$$
r = radius, h = height
Area: Cone curve (lateral surface)
$$A = \pi r l$$
r = radius, l = slant height
Volume: Sphere
$$V = \frac{4}{3}\pi r^3$$
r = radius
Surface area: Sphere
$$A = 4\pi r^2$$
r = radius
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
$$A = \frac{1}{2}ab\sin C$$
Length of an arc (degrees)
$$l = \frac{\theta}{360^\circ} \times 2\pi r$$
\(\theta\) = angle in degrees, r = radius
Area of a sector (degrees)
$$A = \frac{\theta}{360^\circ} \times \pi r^2$$
\(\theta\) = angle in degrees, r = radius
HL only
Length of an arc (radians)
$$l = r\theta$$
r = radius, \(\theta\) = angle in radians
Area of a sector (radians)
$$A = \frac{1}{2}r^2\theta$$
r = radius, \(\theta\) = angle in radians
Trigonometric Identities
$$\cos^2\theta + \sin^2\theta = 1$$ $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$
Transformation Matrices (2D)
$$\text{Reflection in y = (tan}\theta)x: \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$$
$$\text{Horizontal stretch, factor k}: \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}$$
$$\text{Vertical stretch, factor k}: \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}$$
$$\text{Enlargement, factor k, centre (0,0)}: \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$$
$$\text{Anticlockwise rotation about origin by }\theta (>0): \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
$$\text{Clockwise rotation about origin by }\theta (>0): \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$$
Magnitude of a vector \(\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)
$$|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$$
Vector equation of a line
$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$
\(\mathbf{a}\) is position vector of a point on line, \(\mathbf{b}\) is direction vector
Parametric form of the equation of a line
$$x = x_0 + \lambda l, \quad y = y_0 + \lambda m, \quad z = z_0 + \lambda n$$
\((x_0, y_0, z_0)\) is a point, direction vector \(\mathbf{b} = \begin{pmatrix} l \\ m \\ n \end{pmatrix}\)
Scalar product (dot product)
$$\mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3$$ $$\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}||\mathbf{w}|\cos\theta$$
\(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\)
Angle between two vectors
$$\cos\theta = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|} = \frac{v_1w_1 + v_2w_2 + v_3w_3}{|\mathbf{v}||\mathbf{w}|}$$
Vector product (cross product)
$$\mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}$$ $$|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}||\mathbf{w}|\sin\theta$$
\(\theta\) is the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Resulting vector is perpendicular to both \(\mathbf{v}\) and \(\mathbf{w}\).
Area of a parallelogram
$$A = |\mathbf{v} \times \mathbf{w}|$$
\(\mathbf{v}\) and \(\mathbf{w}\) form two adjacent sides. Area of triangle with these sides is \(\frac{1}{2}|\mathbf{v} \times \mathbf{w}|\).
Topic 4: Statistics and probability
SL & HL
Interquartile range
$$\text{IQR} = Q_3 - Q_1$$
Mean, \(\bar{x}\), of a set of data
$$\bar{x} = \frac{\sum_{i=1}^k f_i x_i}{n}$$
where \(n = \sum_{i=1}^k f_i\) is the total frequency.
Probability of an event A
$$P(A) = \frac{n(A)}{n(U)}$$
Complementary events
$$P(A) + P(A') = 1$$
Combined events (Addition Rule)
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Mutually exclusive events
$$P(A \cup B) = P(A) + P(B)$$
If A and B are mutually exclusive, \(P(A \cap B) = 0\).
Conditional probability
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Independent events
$$P(A \cap B) = P(A)P(B)$$
Expected value of a discrete random variable X
$$E(X) = \sum x P(X=x)$$
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 test statistic \(\chi^2_{calc}\)
$$\chi^2_{calc} = \sum \frac{(f_o - f_e)^2}{f_e}$$
\(f_o\) = observed frequencies, \(f_e\) = expected frequencies. Degrees of freedom \(\nu = (\text{rows}-1)(\text{cols}-1)\) for contingency tables, or \(\nu = k-1-m\) for goodness of fit.
Spearman's rank correlation coefficient \(r_s\)
Usually calculated using technology (GDC). Formula \( r_s = 1 - \frac{6 \sum d^2}{n(n^2-1)} \) for no tied ranks exists but is less emphasized with GDC focus.
Pearson's product-moment correlation coefficient \(r\)
Usually calculated using technology (GDC).
Equation of regression line \(y\) on \(x\)
$$y - \bar{y} = \frac{s_{xy}}{s_x^2}(x-\bar{x})$$
Or \(y = ax+b\), where \(a, b\) are found using technology (GDC).
Normal Distribution \(X \sim N(\mu, \sigma^2)\)
Probabilities found using GDC or Z-table for \(Z = \frac{X-\mu}{\sigma}\).
Student's t-test
Used for hypothesis testing of a single mean or difference of two means (unpaired/paired). Test statistic and p-value typically found using technology (GDC).
HL only
Linear transformation of a single random variable X
$$E(aX+b) = aE(X)+b$$ $$\text{Var}(aX+b) = a^2\text{Var}(X)$$
Linear combinations of \(n\) independent random variables \(X_1, X_2, \dots, X_n\)
$$E(a_1X_1 \pm a_2X_2 \pm \dots \pm a_nX_n) = a_1E(X_1) \pm a_2E(X_2) \pm \dots \pm a_nE(X_n)$$ $$\text{Var}(a_1X_1 \pm a_2X_2 \pm \dots \pm a_nX_n) = a_1^2\text{Var}(X_1) + a_2^2\text{Var}(X_2) + \dots + a_n^2\text{Var}(X_n)$$
Note: Variances always add for independent variables.
Unbiased estimate of population variance \(s_{n-1}^2\)
$$s_{n-1}^2 = \frac{n}{n-1}s_n^2$$
Where \(s_n^2\) is the sample variance (often with divisor \(n\)). Most GDCs provide \(s_{n-1}\) (often denoted as \(s_x\)) directly.
Poisson distribution \(X \sim Po(m)\)
$$P(X=x) = \frac{e^{-m}m^x}{x!}, \quad x=0,1,2,\dots$$ $$E(X) = m$$ $$\text{Var}(X) = m$$
\(m\) is the average number of events in a given interval.
Transition matrices for Markov chains
$$\mathbf{T}^n \mathbf{s}_0 = \mathbf{s}_n$$
\(\mathbf{T}\) is transition matrix, \(\mathbf{s}_0\) is initial state vector, \(\mathbf{s}_n\) is state vector after \(n\) steps. Steady state \(\mathbf{s}\) satisfies \(\mathbf{T}\mathbf{s} = \mathbf{s}\).
Topic 5: Calculus
SL & HL
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 enclosed by a curve \(y=f(x)\) and the x-axis from \(x=a\) to \(x=b\)
$$A = \int_a^b |f(x)| dx$$
If \(f(x) \ge 0\) on \([a,b]\), \(A = \int_a^b f(x) dx\).
The trapezoidal rule
$$\int_a^b y dx \approx \frac{1}{2}h((y_0+y_n) + 2(y_1+y_2+\dots+y_{n-1}))$$
Where \(h = \frac{b-a}{n}\) and \(y_i = f(x_i)\). Divides \([a,b]\) into \(n\) sub-intervals.
HL only
Derivative of \(\sin x\)
$$f(x) = \sin x \implies f'(x) = \cos x$$
Derivative of \(\cos x\)
$$f(x) = \cos x \implies f'(x) = -\sin x$$
Derivative of \(\tan x\)
$$f(x) = \tan x \implies f'(x) = \frac{1}{\cos^2 x} = \sec^2 x$$
Derivative of \(e^x\)
$$f(x) = e^x \implies f'(x) = e^x$$
Derivative of \(\ln x\)
$$f(x) = \ln x \implies f'(x) = \frac{1}{x}, \quad x > 0$$
Chain rule
$$\text{If } y = g(u) \text{ and } u=f(x), \text{ then } \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$
Or if \(y = g(f(x))\), then \(y' = g'(f(x))f'(x)\).
Product rule
$$\text{If } y = uv, \text{ then } \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$
Quotient rule
$$\text{If } y = \frac{u}{v}, \text{ then } \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$
Standard integrals
$$\int \frac{1}{x} dx = \ln|x| + C$$ $$\int \sin x dx = -\cos x + C$$ $$\int \cos x dx = \sin x + C$$ $$\int \frac{1}{\cos^2 x} dx = \tan x + C$$ $$\int e^x dx = e^x + C$$
Area enclosed by a curve and x or y-axes
$$A = \int_a^b |y| dx \quad \text{or} \quad A = \int_c^d |x| dy$$
Volume of revolution about x or y-axes
$$V_x = \pi \int_a^b y^2 dx \quad (\text{about x-axis})$$ $$V_y = \pi \int_c^d x^2 dy \quad (\text{about y-axis})$$
Acceleration
$$a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} = v\frac{dv}{ds}$$
\(s(t)\) is displacement, \(v(t)\) is velocity.
Distance & Displacement travelled from \(t_1\) to \(t_2\)
$$\text{Total Distance} = \int_{t_1}^{t_2} |v(t)| dt$$ $$\text{Displacement} = \int_{t_1}^{t_2} v(t) dt$$
Euler's method (single first-order ODE)
$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$ $$x_{n+1} = x_n + h$$
For \( \frac{dy}{dx} = f(x,y) \), \(h\) is step length.
Euler's method for coupled systems
$$x_{n+1} = x_n + h \cdot f_1(x_n, y_n, t_n)$$ $$y_{n+1} = y_n + h \cdot f_2(x_n, y_n, t_n)$$ $$t_{n+1} = t_n + h$$
For \(\frac{dx}{dt}=f_1(x,y,t)\) and \(\frac{dy}{dt}=f_2(x,y,t)\).
Exact solution for coupled linear differential equations
$$\mathbf{x} = A e^{\lambda_1 t}\mathbf{p}_1 + B e^{\lambda_2 t}\mathbf{p}_2$$
For systems like \(\frac{d\mathbf{x}}{dt} = \mathbf{M}\mathbf{x}\), where \(\lambda_1, \lambda_2\) are eigenvalues and \(\mathbf{p}_1, \mathbf{p}_2\) are corresponding eigenvectors of matrix \(\mathbf{M}\). A and B are constants.
Why You Can't Just "Have" the Booklet
Senior year, I stuffed the old AI booklet into my back pocket like a lucky charm—then froze when a modeling question morphed a friendly quadratic into a snarling logistic curve. Moral of the story? Owning ≠ Knowing. This post bridges that gap:
- Official 2025 PDF, fresh off the IBO press.
- An interactive graph panel so you can see each formula breathe.
- Examples ripped from the real world (yes, TikTok algorithm growth counts as stats).
- Bite-sized hacks my tutees swear cut revision time in half.
Five Equations Students Trip Over—and How to Sidestep the Face-Plant
| Topic | Trip-Wire | Life-Saver |
|---|---|---|
| Statistical Models | Mixing r and R² in conclusions | Quote both: "r = 0.87, so 76% variance explained." |
| Sequences | Forgetting the n–1 in geometric mean | Tape "mind the gap: n–1" on your laptop. |
| Binomial to Normal | Skipping np ≥ 5 check | Run np=μ, σ=√npq in your GDC first. |
| Trig Models | Leaving calculator in radians for degree tasks | Write "deg?" on exam page top; glance before punching sin. |
| Chi-Squared | Using population frequencies, not expected | Build a quick table in the test booklet—saves silly slips. |
FAQ—Fast Answers Before You Dive In
Is the 2025 booklet different from 2023?
Yes. Minor layout tweaks, clearer cumulative distribution notation, and a new variance shortcut line (page 7).
Do I need to memorise formulas?
No—know when to use them and practise the GDC workflow. Examiners test insight, not recall.
HL student here. Any extra pages?
Same booklet, HL questions simply string formulas together in nastier combos. Focus on inter-topic links.
Final Pep Talk
Formulas are tools, not talismans. Print, annotate, experiment, repeat—and remember: every IB question is just real life dressed in algebra.
