Core Topics
Essential Concepts & Key Formulae
Number & Algebra
$S_n = \dfrac{n}{2}(2a + (n-1)d)$
$u_n = a + (n-1)d$
$u_n = ar^{n-1},\; S_\infty = \dfrac{a}{1-r}$
$u_n = a + (n-1)d$
$u_n = ar^{n-1},\; S_\infty = \dfrac{a}{1-r}$
Arithmetic & geometric sequences. Compound interest: $A = P(1+r/n)^{nt}$
Functions
$f(x) = ax^2 + bx + c$
$y = A e^{kx}$, $\;y = A(b^x)$
Logistic: $N = \dfrac{L}{1+Ce^{-rt}}$
$y = A e^{kx}$, $\;y = A(b^x)$
Logistic: $N = \dfrac{L}{1+Ce^{-rt}}$
Exponential growth/decay; logistic carrying capacity $L$
Calculus
$f'(x^n) = nx^{n-1}$
$\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$
$\dfrac{dy}{dx} = ky \Rightarrow y = Ae^{kx}$
$\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$
$\dfrac{dy}{dx} = ky \Rightarrow y = Ae^{kx}$
Differentiation for rate of change; definite integrals for area
Statistics & Probability
$P(A|B) = \dfrac{P(A\cap B)}{P(B)}$
$X \sim B(n,p)$: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$
$X \sim N(\mu, \sigma^2)$
$X \sim B(n,p)$: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$
$X \sim N(\mu, \sigma^2)$
Binomial, Normal, Poisson distributions; conditional probability
Statistical Analysis
$r \in [-1, 1]$: Pearson's PMCC
$\chi^2$ df $= (r-1)(c-1)$
$H_0$: reject if $p < \alpha$
$\chi^2$ df $= (r-1)(c-1)$
$H_0$: reject if $p < \alpha$
Regression lines; hypothesis testing ($\chi^2$, $t$-test, $z$-test)
Geometry & Trigonometry
$\text{Area} = \tfrac{1}{2}ab\sin C$
Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$
$f(t) = A\sin(Bt+C)+D$
Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$
$f(t) = A\sin(Bt+C)+D$
Sinusoidal models; period $= \frac{2\pi}{B}$; amplitude $= |A|$
Matrices
$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc$
$A^{-1} = \dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
$A^{-1} = \dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
Inverse exists iff $\det A \neq 0$; solve $A\mathbf{x}=\mathbf{b}$ via $\mathbf{x}=A^{-1}\mathbf{b}$
Vectors
$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$
$|\mathbf{a}| = \sqrt{a_1^2+a_2^2+a_3^2}$
$|\mathbf{a}| = \sqrt{a_1^2+a_2^2+a_3^2}$
Perpendicular: $\mathbf{a}\cdot\mathbf{b}=0$; angle between vectors
Probability Distributions
Poisson: $P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}$
$E(X) = \lambda$ (Poisson)
$E(X) = np$ (Binomial)
$E(X) = \lambda$ (Poisson)
$E(X) = np$ (Binomial)
Poisson: rare events over time/space; mean = variance = $\lambda$
Optimisation & LP
Maximise/minimise $Z = ax+by$
subject to linear constraints
Evaluate $Z$ at corner points
subject to linear constraints
Evaluate $Z$ at corner points
Feasible region corner points give optimal value; always check all vertices
Worked Example
Normal Distribution — Exam Style
The heights of students at a school are normally distributed with mean $\mu = 170\,\text{cm}$ and standard deviation $\sigma = 8\,\text{cm}$. Find the probability that a randomly selected student has height less than $178\,\text{cm}$.
Standardise: $z = \dfrac{178-170}{8} = 1$
Using standard normal tables: $P(Z < 1) = 0.8413$
Answer: $P(X < 178) \approx 0.8413$
Using standard normal tables: $P(Z < 1) = 0.8413$
Answer: $P(X < 178) \approx 0.8413$
Practice Examination — 20 Questions
IB AI HL Style
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