IB Math Analysis and Approaches HL Study Guide

Comprehensive vocabulary and formula flashcards covering Algebra, Functions, Vectors, Trigonometry, and Calculus for IB Mathematics Analysis and Approaches HL.

Arithmetic Sequence

A sequence where each term is the previous number plus a common difference $d$, calculated using the formula $u_n = u_1 + (n - 1)d$.

Geometric Sequence

A sequence where each term is the previous number multiplied by a common ratio $r$, with the $n ext{th}$ term given by $u_n = u_1 imes r^{n-1}$.

Compound Interest Formula

The calculation for future value $FV$ given present value $PV$, nominal annual rate $r$, compounding periods $k$, and years $n$: FV = PV imes (1 + rac{r}{100k})^{kn}.

Laws of Logarithms

Four primary rules: I: $ext{log}_c(a) + ext{log}_c(b) = ext{log}_c(a imes b)$; II: ext{log}_c(a) - ext{log}_c(b) = ext{log}_c(rac{a}{b}); III: $n ext{log}_c(a) = ext{log}_c(a^n)$; IV: ext{log}_b(a) = rac{ ext{log}_c(a)}{ ext{log}_c(b)}.

Binomial Expansion Formula

The expansion of $(a + b)^n$ where each term is described as ${n}C_r a^{n-r} b^r$ and ${n}C_r$ is the binomial coefficient.

Mathematical Induction

A three-step indirect proof method: 1. Show true for $n = 1$; 2. Assume true for $n = k$; 3. Prove true for $n = k + 1$ using the assumption.

Complex Number Forms

Cartesian form is $z = a + bi$ (with real part $a$ and imaginary part $b$); Polar form is $z = r( ext{cos}( heta) + i ext{sin}( heta))$ or $r ext{cis}( heta)$.

Complex Conjugate

If $z = a + bi$, the conjugate $z^*$ is defined as $a - bi$, which is useful for dividing complex numbers in Cartesian form.

Euler’s Theorem

A theorem stating the relationship $e^{ix} = ext{cos}(x) + i ext{sin}(x)$, linking trigonometric functions and complex exponentials.

De Moivre’s Theorem

The identity used for raising complex numbers to powers: $z^n = (r( ext{cos}(x) + i ext{sin}(x)))^n = r^n( ext{cos}(nx) + i ext{sin}(nx))$.

Partial Fractions

A method to split a single fraction with distinct linear terms in the denominator into a sum of distinct smaller fractions.

Discriminant ($\Delta$)

Calculated as $b^2 - 4ac$ for a quadratic; it determines the number of roots ($2$ if posite, $1$ if zero, and no real roots if negative).

Oblique Asymptote

A non-vertical, non-horizontal line a function approaches when the degree of the numerator is larger than the degree of the denominator.

Even Function

A function where $f(x) = f(-x)$, resulting in a graph that is symmetrical over the $y$-axis.

Odd Function

A function where $-f(x) = f(-x)$, resulting in a graph with rotational symmetry with respect to the origin.

Fundamental Theorem of Algebra

States that any polynomial of degree $n$ has exactly $n$ roots, which can include double, triple, or complex roots.

Unit Vector ($\hat{a}$)

A vector with a magnitude of $1$, calculated by dividing a vector by its magnitude: \hat{a} = rac{\mathbf{a}}{|\mathbf{a}|}.

Dot (Scalar) Product

The calculation $\mathbf{c} \cdot \mathbf{d} = |\mathbf{c}||\mathbf{d}|\text{cos}( heta) = c_1d_1 + c_2d_2 + c_3d_3$, used to find the angle between vectors.

Cross (Vector) Product

A product of two vectors that results in a third vector perpendicular to both original vectors, such that $|\mathbf{u} \times \mathbf{v}|$ is the area of the parallelogram formed by them.

Normal Vector

A vector that is perpendicular to a plane, often found by taking the cross product of the two direction vectors defining the plane.

L’HBpital’s Rule

If the limit of a quotient of functions results in $0/0$ or $\infty/\infty$, the limit equals the limit of the quotient of their derivatives: $\lim_{x \to c} \frac{f'(x)}{g'(x)}$.

Points of Inflection

Points where the second derivative $f''(x) = 0$ and the concavity of the function changes.

Maclaurin Series

A way to approximate a function around $x = 0$ using an infinite polynomial sum derived from the function's $n ext{th}$ derivatives at zero.

Volume of Revolution

The volume generated by rotating a curve $360^\circ$ around an axis, calculated about the $x$-axis using $V = \pi \int_a^b y^2 \,dx$.

Euler’s Method

A numerical method to approximate solutions to differential equations using the recursive formula $y_{n+1} = y_n + h \times f(x_n, y_n)$, where $h$ is the step size.

Binomial Distribution ($X \sim B(n, p)$)

A discrete probability distribution for $n$ trials with success probability $p$, used where there are only two possible outcomes.

Normal Distribution ($X \sim N(\mu, \sigma^2)$)

A continuous probability distribution with a bell-shaped curve determined by mean $\mu$ and standard deviation $\sigma$.

Pearson’s Correlation Coefficient ($r$)

A standardised value between $-1$ and $1$ describing the relationship between two variables; $r = 0$ implies no correlation while $|r| = 1$ is perfect correlation.

Stratified Sampling

A sampling technique where the population is split into smaller groups called strata, and a random sample is selected from each stratum.

Quota Sampling

Non-random sampling where the population is divided into groups and samples are taken in proportion to the size of each group (strata).