Формулы-графики-чертежи

Математика LaTeX

1. Корни:

$$
	x=\frac{-b \pm \sqrt{ b^{2}-4ac }}{2a}
$$

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

2. Производная:

$$
	\frac{d}{dx}\left(x^{2}\right)=2x
$$

$$\frac{d}{dx}\left(x^2\right)=2x$$

3. Интеграл:

$$
	\int_{0}^{\infty} e^{-x^{2}}dx=\frac{\sqrt{ \pi }}{2}
$$

$${\int }_0^{\infty }{e}^{-x^2}dx=\frac{\sqrt{\pi }}{2}$$

4. Сумма:

$$
	\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}
$$

$$\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

5. Система уравнений:

$$
	\left\{ \begin{aligned} x + y &= 5 \\ 2x - y &= 1
	\end{aligned} \right.
$$

$$\{\begin{array}{rl}x+y& =5\\ 2x-y& =1\end{array}$$

6. Матрица:

$$
	\begin{pmatrix}a & b \\c & d\end{pmatrix}
$$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

7. Кусочная функция:

$$
f(x) = \begin{cases}
x^2, & x \ge 0 \\
-x, & x < 0
\end{cases}
$$

$$f(x)=\{\begin{array}{ll}{x}^2,& x\ge 0\\ -x,& x<0\end{array}$$

8. Сложный интеграл:

$$
	\mathcal{L} = \int_{0}^{1} \left( \frac{1}{2} \dot{q}^2 - V(q) \right) dt
$$

$$\mathcal{L}={\int }_0^1\left(\frac{1}{2}{\dot{q}}^2-V(q)\right)dt$$

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Отображение графиков функций

$y=x$:

<i1frame src="https://www.desmos.com/calculator/el8xt0724c?embed" width="100%" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

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$y=\sin x$

<iframe src="https://www.desmos.com/calculator/lznqpwoqzo?embed" width="100%" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

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$y=\frac{\sin (x)}{x}+e^{-x^2}\cos (2\pi x)$

<iframe src="https://www.desmos.com/calculator/edfmxb4ssd?embed" width="100%" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

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$\{\begin{array}{l}y=\frac{1}{x}\\ y=x^2\\ x^{\frac{2}{4}}+y^{\frac{2}{9}}=1\end{array}$

<iframe src="https://www.desmos.com/calculator/evqctp2dta?embed" width="100%" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

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Построение чертежей

Движение тела по окружности:

https://www.desmos.com/geometry/s1ni6prioc

Баллистика:

https://www.desmos.com/geometry/iq4wkvnskl