IB Math AI HL · Core Practice Workbook

Topic 1 Number & Algebra — Sequences

Arithmetic: $u_n = u_1 + (n-1)d$ · $S_n = \dfrac{n}{2}(2u_1 + (n-1)d)$

Geometric: $u_n = u_1 \cdot r^{n-1}$ · $S_\infty = \dfrac{u_1}{1-r}, \;|r|<1$

Key: $d$ = common difference; $r$ = common ratio. $S_\infty$ exists only when $|r| < 1$.

Quick Example

Arithmetic: $u_1=3, d=4 \Rightarrow S_{10} = \frac{10}{2}(6+36) = 210$

Topic 1 Number & Algebra — Logarithms & Exponents

$\log_b x = y \;\Leftrightarrow\; b^y = x$

Compound Interest: $A = P\!\left(1+\dfrac{r}{n}\right)^{nt}$

Key laws: $\log(ab)=\log a + \log b$; $\log(a^n)=n\log a$; change-of-base: $\log_b a = \frac{\ln a}{\ln b}$.

Quick Example

$\log_2 x = 5 \;\Rightarrow\; x = 2^5 = 32$

Topic 2 Functions — Exponential & Transformations

$f(x) = a\,e^{kx}$ · $f'(x) = ak\,e^{kx}$

Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)$

Key: $e \approx 2.718$. Exponential functions model growth/decay. The derivative of $e^{kx}$ is always $k\,e^{kx}$.

Quick Example

$f(x)=3e^{2x} \;\Rightarrow\; f'(x)=6e^{2x}$

Topic 2 Functions — Regression & Modelling

Linear regression: $\hat{y} = ax + b$ · Pearson correlation: $r \in [-1, 1]$

Key: $|r| \geq 0.85$ → strong; $0.5 \leq |r| < 0.85$ → moderate; $|r| < 0.5$ → weak. $r = +0.92$ = strong positive.

Quick Example

If $\hat{y}=0.8x+2$ and $x=10$: predict $y = 0.8(10)+2 = 10$.

Topic 3 Geometry — Vectors & Voronoi

Dot product: $\mathbf{a}\cdot\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$

Perpendicular: $\mathbf{a}\cdot\mathbf{b} = 0$ · $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

Voronoi: Each cell contains all points closer to that site than any other site. Perpendicular bisectors form the edges.

Quick Example

$\mathbf{a}=(2,1,-1),\;\mathbf{b}=(3,-2,1)$: $\mathbf{a}\cdot\mathbf{b}=6-2-1=3$

Topic 4 Statistics — Probability & Distributions

Conditional: $P(A|B) = \dfrac{P(A \cap B)}{P(B)}$

Binomial: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$

Poisson: $P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}$

Normal: $X \sim N(\mu,\sigma^2)$. Convert to $Z = \frac{X-\mu}{\sigma}$. Use GDC/table. $P(Z < 1) \approx 0.8413$.

Quick Example

Poisson $\lambda=3$: $P(X=2)=\frac{e^{-3}\cdot9}{2}=0.2240$

Topic 4 Statistics — Chi-Squared & Matrices

$\chi^2$ degrees of freedom: $(r-1)(c-1)$

$2\times2$ determinant: $\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$

Chi-sq test: Compares observed vs expected frequencies. Reject $H_0$ if $p$-value $< \alpha = 0.05$. For a $3\times4$ table: df $=(3-1)(4-1)=6$.

Quick Example

$\det\begin{pmatrix}3&1\\2&4\end{pmatrix}=12-2=10$

Topic 5 Calculus — Differentiation & Integration

Power rule: $\frac{d}{dx}x^n = nx^{n-1}$

$\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1}+C$ · $\displaystyle\int_a^b f(x)\,dx = [F(x)]_a^b$

Separable ODE: $\frac{dy}{dx}=f(x)g(y)\;\Rightarrow\;\int\frac{dy}{g(y)}=\int f(x)\,dx$

Area between curves: $\int_a^b [f(x)-g(x)]\,dx$ where $f \geq g$ on $[a,b]$.

Quick Example

$\int_0^3 x^2\,dx = \left[\frac{x^3}{3}\right]_0^3 = 9 - 0 = 9$

Core Concepts
Practice Workbook

Key Concepts & Formulae

Worked Solutions

Core ConceptsPractice Workbook

Key Concepts & Formulae

Worked Solutions

Core Concepts
Practice Workbook