Quarto Markdown (.qmd) Testing

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January 1, 2025

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Theorem 1 (Quadratic Formula) \[ x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \]

Proof. by \((a+b)^2=a^2+2ab+b^2\) \[ \begin{align*} &ax^2+bx+c=0 \ (a \neq 0) \\ &x^2+\frac{b}{a}x+\frac{c}{a}=0\\&(x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2}\\ &(x+\frac{b}{2a})^2=\frac{-4ac+b^2}{4a^2}\\ &x+\frac{b}{2a}=\pm\sqrt{\frac{b^2-4ac}{4a^2}}\\ &x=\pm\sqrt{\frac{b^2-4ac}{4a^2}}-\frac{b}{2a}\\ &x=\pm\frac{\sqrt{b^2-4ac}}{2a}-\frac{b}{2a}\\ &x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{align*} \]

Definition 1 (Metric Space) A metric space is a set \(M\)with a function \(d: M \times M \to \mathbb{R}\) such that:

  1. \(d(x, y) \geq 0\)(Non-negativity)
  2. \(d(x, y) = 0\) if and only if \(x = y\)
  3. \(d(x, y) = d(y, x)\) (Symmetry)
  4. \(d(x, z) \leq d(x, y) + d(y, z)\) (Triangle inequality)

Definition 2 (The limit of a sequence) Suppose \(a_n\) is a convergent sequence, and \(L \in \mathbb{R}\) is the limit

\[ \forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, |a_n-L| \lt \epsilon \]

Note: the order of the quantifier is \(\forall\), \(\exists\), \(\forall\)

Theorem 1 2

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What is JavaScript ?

console.log("Hello")

Fundamental Theorem of Calculus

\[ \int_a^b f'(x) \, dx = f(b) - f(a) \]

Thm: Multiple Lines Theorem

\[ \begin{align*} &ax^2+bx+c = 0 \\ &x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{align*} \tag{1}\]

Tip 1

This is a callout

Theorem 2 This is a theorem example for testing reference

Figure 1 Tip 1 Definition 1

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Google

https://quarto.org/docs/get-started/

Figure 1: alt text
A\B B1 B2
A1 A1 x B1 A1 x B2
A2 A2 x B1 A2 x B2

Footnotes

  1. You can embed pdf into your document↩︎

  2. You can use @ to refer to the sources↩︎