Global K-12 Mathematics Curriculum Framework

This set of flashcards provides a dense technical summary of the global K-12 mathematics curriculum, covering Arithmetic Axioms, Algebraic Functions, Analytic Geometry, and Trigonometric Identities.

What is the Epsilon-Delta $(\epsilon, \delta)$ definition of a limit?

$$\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$$

State the Completeness Axiom for the set of real numbers $\mathbb{R}$.

$\forall S \subseteq \mathbb{R}$ nonempty and bounded above, $\exists \sup(S) \in \mathbb{R}$ (Analogous for infimum)

What is the Triangle Inequality for absolute values?

$$||x| - |y|| \leq |x \pm y| \leq |x| + |y|$$

Under what condition does the modular inverse $[a]^{-1}$ exist in $\mathbb{Z}_n$?

$$\text{gcd}(a, n) = 1$$

State the Fundamental Theorem of Algebra regarding polynomials.

A polynomial $P(x)$ of degree $n$ has exactly $n$ roots over $\mathbb{C}$, counting multiplicities: $P(x) = a_n \prod_{i=1}^n (x - r_i)$

According to Vieta's formulas, what is the sum of the roots $\sum r_i$ of a polynomial?

$$\sum r_i = -\frac{a_{n-1}}{a_n}$$

What is the limit definition of the constant $e$?

$$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$$

How is the eigenstructure of a matrix $A$ defined?

$A\mathbf{v} = \lambda \mathbf{v}$, where $\lambda$ is an eigenvalue found via $\det(A - \lambda I) = 0$ and $\mathbf{v}$ is the corresponding eigenvector.

What is the iterative formula for Newton's Method for root-finding?

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

In Analytic Geometry, what are the parametric equations for a circle centered at $(h, k)$?

$$x = h + r \cos(t), y = k + r \sin(t)$$

What does the discriminant $\Delta = B^2 - 4AC$ indicate about a general conic section?

$\Delta < 0 \implies \text{ellipse}$; $\Delta = 0 \implies \text{parabola}$; $\Delta > 0 \implies \text{hyperbola}$

What is the formula for the projection of vector $\mathbf{u}$ onto vector $\mathbf{v}$ ($\text{proj}_\mathbf{v} \mathbf{u}$)?

$$\text{proj}_\mathbf{v} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{ |\mathbf{v}|^2 } \mathbf{v}$$

State the three forms of the cosine double-angle identity ($\cos(2\theta)$).

$$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$$

What is the Law of Cosines for a triangle with sides $a, b, c$ and angle $C$?

$$c^2 = a^2 + b^2 - 2ab \cos(C)$$

What are the two fundamental trigonometric limits used in calculus?

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

Define the Squeeze Theorem for limits.

If $g(x) \leq f(x) \leq h(x)$ near $a$, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$