? y=x^2

y = x^{2}

$$y = x^2$$ ? y=1/x

y = \frac{1}{x}

$$y = \frac{1}{x}$$ ? y=sqrt(x)

y = \sqrt{x}

$$y = \sqrt{x}$$ ? E=mc^(3 + e^(ipi)

E = m c^{3 + e^{i π}}

$$E = m c^{3 + e^{i \pi}}$$ ? a^2+b^2=c^2

a^{2} + b^{2} = c^{2}

$$a^2 + b^2 = c^2$$ ? AA x in CC (sin^2x+cos^2x=1)

\forall x \in ℂ ({sin}^{2} x + {cos}^{2} x = 1)

$$\forall x \in \mathds{C} \left( \sin^2 x + \cos^2 x = 1 \right)$$ ? (AA x: x in CC: sin^2x+cos^2x=1)

(\forall x : x \in ℂ : {sin}^{2} x + {cos}^{2} x = 1)

$$\left( \forall x : x \in \mathds{C} : \sin^2 x + \cos^2 x = 1 \right)$$ ? sum_(i=1)^ni^3=(sum_(i=1)^ni^2)^2

\sum_{i = 1}^{n} i^{3} = {(\sum_{i = 1}^{n} i^{2})}^{2}

$$\sum_{i = 1}^n i^3 = \left( \sum_{i = 1}^n i^2 \right)^2$$ ? (a,b)

(a, b)

$$( a , b )$$ ? f

# f # #

$$f$$ ? Delta x=(b-a)/n

Δ x = \frac{b - a}{n}

$$\Delta x = \frac{b - a}{n}$$ ? int_a^b f(x)dx=lim_(n->oo)sum_[i=1]^n f(x_i^(**))Delta x

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{⋆}) Δ x

$$\int_a^b f ( x ) d x = \lim_{n \to \infty} \sum_{i = 1}^n f \left( x_i^{\star} \right) \Delta x$$ ? x_i=a+iDeltax

x_{i} = a + i Δ x

$$x_i = a + i \Delta x$$ ? x_i^(**)in[x_[i-1],x_i]

x_{i}^{⋆} \in [x_{i - 1}, x_{i}]

$$x_i^{\star} \in \left[ x_{i - 1} , x_i \right]$$ ? \int_0^oo e^{-x^2}dx = 1/2\sqrt{pi}.

\int_{0}^{\infty} e^{- x^{2}} d x = \frac{1}{2} \sqrt{π} .

$$\int_0^{\infty} e^{- x^2} d x = \frac{1}{2} \sqrt{\pi} .$$ ? x/x=(1 if x!=0)

\frac{x}{x} = (1 if x \neq 0)

$$\frac{x}{x} = ( 1 \mbox{if } x \ne 0 )$$ ? int_0^pi sinxdx=-cosx]_0^pi=-cospi-(-cos0)=-(-1)-(-1)=2

\int_{0}^{π} sin x d x = - cos x]_{0}^{π} = - cos π - (- cos 0) = - (- 1) - (- 1) = 2

$$\int_0^{\pi} \sin x d x = - \cos x ]_0^{\pi} = - \cos \pi - ( - \cos 0 ) = - ( - 1 ) - ( - 1 ) = 2$$ ? -0.123.456

- 0.123 .456

$$- 0.123 .456$$ ? epsilon=.001 quad h=-.01 quad pi~~3.14159 quad

ε = .001 h = - .01 π \approx 3.14159

$$\epsilon = .001 \,\, h = - .01 \,\, \pi \approx 3.14159 \,\,$$ ? u.v

u . v

$$u . v$$ ? RR = uuu_{n=0}^oo[-n,n]

ℝ = ⋃_{n = 0}^{\infty} [- n, n]

$$\mathds{R} = \bigcup_{n = 0}^{\infty} [ - n , n ]$$ ? {0} = nnn_{n=1}^oo(- 1/n,1/n)

{0} = ⋂_{n = 1}^{\infty} (- \frac{1}{n}, \frac{1}{n})

$$\{ 0 \} = \bigcap_{n = 1}^{\infty} \left( - \frac{1}{n} , \frac{1}{n} \right)$$ ? ^^^_{i=1}^nphi_i = phi_1 ^^ phi_2 ^^ cdots ^^ phi_n

⋀_{i = 1}^{n} φ_{i} = φ_{1} \land φ_{2} \land \dots \land φ_{n}

$$\bigwedge_{i = 1}^n \phi_i = \phi_1 \wedge \phi_2 \wedge \cdots \wedge \phi_n$$ ? vvv_{i=1}^nphi_i = phi_1 vv phi_2 vv cdots vv phi_n

⋁_{i = 1}^{n} φ_{i} = φ_{1} \lor φ_{2} \lor \dots \lor φ_{n}

$$\bigvee_{i = 1}^n \phi_i = \phi_1 \vee \phi_2 \vee \cdots \vee \phi_n$$ ? pi~~3.141592653589793

π \approx 3.141592653589793

$$\pi \approx 3.141592653589793$$ ? int_-1^1 sqrt(1-x^2)dx = pi/2

\int_{- 1}^{1} \sqrt{1 - x^{2}} d x = \frac{π}{2}

$$\int_{- 1}^1 \sqrt{1 - x^2} d x = \frac{\pi}{2}$$ ? lim_(x->a) f(x)=l <=> AA epsi > 0 EE delta > 0 : 0 < {:|x-a|:} < delta => {:|f(x) - l|:} < epsi

lim_{x \to a} f (x) = l \Leftrightarrow \forall ε > 0 \exists δ > 0 : 0 < | x - a | < δ \Rightarrow | f (x) - l | < ε

$$\lim_{x \to a} f ( x ) = l \Leftrightarrow \forall \epsilon > 0 \exists \delta > 0 : 0 < | x - a | < \delta \Rightarrow | f ( x ) - l | < \epsilon$$ ? 1/(1+1/(1+...))

\frac{1}{1 + \frac{1}{1 + ...}}

$$\frac{1}{1 + \frac{1}{1 + \ldots}}$$ ? x := y

x := y

$$x := y$$ ? int vec{A} cdot vec{dl} = int int vec{B} cdot vec{dS}

\int \vec{A} \cdot \vec{d l} = \int \int \vec{B} \cdot \vec{d S}

$$\int \vec{A} \cdot \vec{d l} = \int \int \vec{B} \cdot \vec{d S}$$ ? 1/(1+1/(1+1/(1+1/(1+...))))=(sqrt5-1)/2

\frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}} = \frac{\sqrt{5} - 1}{2}

$$\frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ldots}}}} = \frac{\sqrt{5} - 1}{2}$$ ? [a_0, a_1...a_(n-1)]

[a_{0}, a_{1} ... a_{n - 1}]

$$\left[ a_0 , a_1 \ldots a_{n - 1} \right]$$ ? [d_0, d_1...d_(n-1)]

[d_{0}, d_{1} ... d_{n - 1}]

$$\left[ d_0 , d_1 \ldots d_{n - 1} \right]$$ ? n

n

$$n$$ ? o

o

$$o$$ ? o = sum_(i=0)^n(a_i * prod_(j=0)^(i-1)d_j)

o = \sum_{i = 0}^{n} (a_{i} \cdot \prod_{j = 0}^{i - 1} d_{j})

$$o = \sum_{i = 0}^n \left( a_i \cdot \prod_{j = 0}^{i - 1} d_j \right)$$ ? a = [1,1,1]; n = 3; d = [2,3,2]; o = sum_(i=0)^3(a_i * prod_(j=0)^(i-1)d_j) = (1) + (1 * (2)) + (1 * (2*3)) = 1 + 2 + 6 = 9

a = [1, 1, 1]; n = 3; d = [2, 3, 2]; o = \sum_{i = 0}^{3} (a_{i} \cdot \prod_{j = 0}^{i - 1} d_{j}) = (1) + (1 \cdot (2)) + (1 \cdot (2 \cdot 3)) = 1 + 2 + 6 = 9

$$a = [ 1 , 1 , 1 ] ; n = 3 ; d = [ 2 , 3 , 2 ] ; o = \sum_{i = 0}^3 \left( a_i \cdot \prod_{j = 0}^{i - 1} d_j \right) = ( 1 ) + ( 1 \cdot ( 2 ) ) + ( 1 \cdot ( 2 \cdot 3 ) ) = 1 + 2 + 6 = 9$$ ? o = i_0 + d_0[i_1 + d_1[...[i_(n-1)]]]

o = i_{0} + d_{0} [i_{1} + d_{1} [... [i_{n - 1}]]]

$$o = i_0 + d_0 \left[ i_1 + d_1 \left[ \ldots \left[ i_{n - 1} \right] \right] \right]$$ ? g o f = Id_(e)

g o f = I d_{e}

$$g o f = I d_e$$ ? max

max

$$\max$$