User:MathsIsFun

My goal is to make mathematics more accessible and fun for everyone, and a big part of that is to explain mathematics using "easy language", but this requires a balancing act between precision and comprehension.

Let me explain: there is an educational concept called the spiral, which roughly means that a subject comes around again and again, always at a higher level. For example, a young person is taught that multiplication is just repeated addition. But then a year later the subject is revisited and multiplying by negatives is taught, then decimals come along ...

[[image:multiply-p2n3.gif|frame|

This is an illustration of 2 times -3. Observe that our toddler is (according to him) moving forward two paces at a time, but he does this three times in a negative direction. If he were stepping backwards two paces at a time while facing forwards, that would be -2 times 3. Have a look at http://www.mathsisfun.com/multiplying-negatives.html Multiplying by Negatives for a longer description.]]

And that is why I have developed ([http://www.mathsisfun.com Math is Fun], or "Maths is Fun" in British English), to be a place where mathematics can be explained in a more "user-friendly" manner.

And like all people who embark on explaining Science to the general public I must at times leave out details which would only confuse, but it can be very hard to know where to draw the line.

So please forgive me, fellow Wikipedians, when I over-simplify! And correct me gently, but do correct me!

Use this [http://www.mathsisfun.com/contact.php Contact Form]

or leave a message on the [http://www.mathisfunforum.com Math is Fun Forum]

$\backslash chi^2\; =\; \backslash sum\; \backslash frac\{(O\; -\; E)^2\}\{E\}$

$f(x)\; =\; f(a)+\backslash frac\; \{f\text{'}(a)\}\{1!\}\; (x-a)+\; \backslash frac\{f$(a)}{2!} (x-a)^2+\frac{f'(a)}{3!}(x-a)^3+ \cdots.

$e^\{x\}\; =\; 1\; +\; x\; +\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash cdots$

$e^\{x\}\; =\; \backslash sum^\{\backslash infin\}\_\{n=0\}\; \backslash frac\{x^n\}\{n!\}$

$\backslash sin\; x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash cdots$

$\backslash sin\; x\; =\; \backslash sum^\{\backslash infty\}\_\{n=0\}\; \backslash frac\{(-1)^n\}\{(2n+1)!\}\; x^\{2n+1\}$

$\backslash cos\; x\; =\; 1\; -\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^4\}\{4!\}\; -\; \backslash cdots$

$\backslash cos\; x\; =\; \backslash sum^\{\backslash infin\}\_\{n=0\}\; \backslash frac\{(-1)^n\}\{(2n)!\}\; x^\{2n\}$

$e^\{x\}\; =\; \backslash sum^\{\backslash infin\}\_\{n=0\}\; \backslash frac\{x^n\}\{n!\}\; =\; 1\; +\; x\; +\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^4\}\{4!\}\; +\; \backslash frac\{x^5\}\{5!\}\; +\; \backslash cdots$

$e^\{ix\}\; =\; 1\; +\; ix\; +\; \backslash frac\{(ix)^2\}\{2!\}\; +\; \backslash frac\{(ix)^3\}\{3!\}\; +\; \backslash frac\{(ix)^4\}\{4!\}\; +\; \backslash frac\{(ix)^5\}\{5!\}\; +\; \backslash cdots$

$e^\{ix\}\; =\; 1\; +\; ix\; -\; \backslash frac\{x^2\}\{2!\}\; -\; \backslash frac\{ix^3\}\{3!\}\; +\; \backslash frac\{x^4\}\{4!\}\; +\; \backslash frac\{ix^5\}\{5!\}\; -\; \backslash cdots$

$e^\{ix\}\; =\; \backslash left(\; 1\; -\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^4\}\{4!\}\; -\; \backslash cdots\; \backslash right)\; +\; i\backslash left(\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash cdots\; \backslash right)$

$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x$

Help:Displaying_a_formula

r_{xy}=\frac{n\sum x_iy_i-\sum x_i\sum y_i}

{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.

:$f(ta+(1-t)b)\backslash geq\; t\; f(a)+(1-t)f(b)$

|A|

= a \cdot \begin{vmatrix} e & f \\ h & i \end{vmatrix}

- b \cdot \begin{vmatrix} d & f \\ g & i \end{vmatrix}

+ c \cdot \begin{vmatrix} d & e \\ g & h \end{vmatrix}

|A|

= a \cdot \begin{vmatrix} f & g & h \\ j & k & l \\ n & o & p \end{vmatrix}

- b \cdot \begin{vmatrix} e & g & h \\ i & k & l \\ m & o & p \end{vmatrix}

+ c \cdot \begin{vmatrix} e & f & h \\ i & j & l \\ m & n & p \end{vmatrix}

- d \cdot \begin{vmatrix} e & f & g \\ i & j & k \\ m & n & o \end{vmatrix}

$\backslash Rightarrow\; \backslash iff\; \backslash approx$

r_{xy}=\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}

{\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}},

\begin{align}

\text{Variance: } \sigma^2 & = \frac{206^2 + 76^2 + (-224)^2 + 36^2 + (-94)^2}{5} \\

& = \frac{42,436 + 5,776+ 50,176+ 1,296 + 8,836}{5} \\

& = \frac{108,520}{5} = 21,704\\

\end{align}

$\backslash sum\; f\; =\; 15+27+8+5\; =\; 55$

$\backslash sum\; fx\; =\; 15\; \backslash times\; 1\; +\; 27\; \backslash times\; 2\; +\; 8\; \backslash times\; 3\; +\; 5\; \backslash times\; 4\; =\; 113$

$\backslash bar\{x\}\; =\; \backslash frac\{\backslash sum\; fx\}\{\backslash sum\; f\}\; =\; \backslash frac\{15\; \backslash times\; 1\; +\; 27\; \backslash times\; 2\; +\; 8\; \backslash times\; 3\; +\; 5\; \backslash times\; 4\}\{15+27+8+5\}\; =\; 2.05...$

$\backslash sum\_\{k=m\}^n\; ca\_k\; =\; c\backslash sum\_\{k=m\}^n\; a\_k$

$\backslash sum\_\{k=m\}^n\; 6k^2\; =\; 6\backslash sum\_\{k=m\}^n\; k^2$

$\backslash sum\_\{k=m\}^n\; (a\_k+b\_k)\; =\; \backslash sum\_\{k=m\}^n\; a\_k\; +\; \backslash sum\_\{k=m\}^n\; b\_k$

$\backslash sum\_\{k=m\}^n\; (a\_k-b\_k)\; =\; \backslash sum\_\{k=m\}^n\; a\_k\; -\; \backslash sum\_\{k=m\}^n\; b\_k$

$\backslash sum\_\{k=m\}^n\; (k+k^2)\; =\; \backslash sum\_\{k=m\}^n\; k\; +\; \backslash sum\_\{k=m\}^n\; k^2$

$\backslash sum\_\{n=1\}^\{4\}\; (\; 2n+1)\; =\; 3+5+7+9=24$

$\backslash sum\_\{i=2\}^\{14\}\; \backslash left\; (\; \backslash \$7\; \backslash times\; 4\; (i-1)\; +\; \backslash \$11\; \backslash times\; (i-2)^2\; \backslash right\; )$

$\backslash sum\_\{i=2\}^\{14\}\; \backslash \$7\; \backslash times\; 4\; (i-1)\; +\; \backslash sum\_\{i=2\}^\{14\}\; \backslash \$11\; \backslash times\; (i-2)^2$

$\backslash \$7\; \backslash times\; 4\; \backslash sum\_\{i=2\}^\{14\}\; (i-1)\; +\; \backslash \$11\; \backslash times\; \backslash sum\_\{i=2\}^\{14\}\; (i-2)^2$

$\backslash \$7\; \backslash times\; 4\; \backslash sum\_\{j=1\}^\{13\}\; j\; +\; \backslash \$11\; \backslash sum\_\{k=1\}^\{12\}\; k^2$

$\backslash \$7\; \backslash times\; 4\; \backslash times\; \backslash frac\{13\; \backslash times\; 14\}\{2\}\; +\; \backslash \$11\; \backslash times\; \backslash frac\{12\; \backslash times\; 13\; \backslash times\; 25\}\{6\}$

$\backslash \$7\; \backslash times\; 4\; \backslash times\; 91\; +\; \backslash \$11\; \backslash times\; 650$

$\backslash \$7\; \backslash times\; 364\; +\; \backslash \$11\; \backslash times\; 650$

$\backslash \$2548\; +\; \backslash \$7150\; =\; \backslash \$9698\; \backslash ,$

$\backslash sum\_\{k=1\}^n\; 1\; =\; n$

$\backslash sum\_\{k=1\}^n\; c\; =\; nc$

$\backslash sum\_\{k=1\}^n\; k\; =\; \backslash frac\{n(n+1)\}\{2\}$

$\backslash sum\_\{k=1\}^n\; k^2\; =\; \backslash frac\{n(n+1)(2n+1)\}\{6\}$

$\backslash sum\_\{k=1\}^n\; k^3\; =\; \backslash left\; (\backslash frac\{n(n+1)\}\{2\}\; \backslash right\; )^2$

$\backslash sum\_\{k=1\}^n\; k^3\; =\; \backslash left\; (\backslash sum\_\{k=1\}^n\; k\; \backslash right\; )^2$

$\backslash sum\_\{k=1\}^n\; (2k-1)\; =\; n^2$

$\backslash sum\_\{k=1\}^\{14\}\; k^2\; =\; \backslash frac\{14(14+1)(2\backslash cdot\; 14+1)\}\{6\}\; =\; 1015$

$\backslash sum\_\{k=0\}^\{n-1\}\; (a+kd)\; =\; \backslash frac\{n\}\{2\}\; (2a+(n-1)d)$

$\backslash sum\_\{k=0\}^\{10-1\}\; (1+k\; \backslash cdot\; 3)\; =\; \backslash frac\{10\}\{2\}\; (2\; \backslash cdot\; 1\; +(10-1)\backslash cdot\; 3)$

$\backslash sum\_\{k=0\}^\{n-1\}\; (ar^k)\; =\; a\; \backslash left\; (\backslash frac\{1-r^n\}\{1-r\}\backslash right\; )$

$\backslash sum\_\{k=0\}^\{4-1\}\; (10\; \backslash cdot\; 3^k)\; =\; 10\; \backslash left\; (\backslash frac\{1-3^4\}\{1-3\}\backslash right\; )\; =\; 400$

$\backslash sum\_\{k=0\}^\{10-1\}\; \backslash tfrac\{1\}\{2\}(\backslash tfrac\{1\}\{2\})^k\; =\; \backslash frac\{1\}\{2\}\; \backslash left\; (\backslash frac\{1\; -\; (\backslash frac\{1\}\{2\})^\{10\}\}\{1-\backslash frac\{1\}\{2\}\}\backslash right\; )\; =\; \backslash frac\{1\}\{2\}\; \backslash left\; (\backslash frac\{1\; -\; \backslash frac\{1\}\{1024\}\}\{\backslash frac\{1\}\{2\}\}\backslash right\; )\; =\; 1\; -\; \backslash frac\{1\}\{1024\}$

$\backslash sum\_\{k=0\}^\{64-1\}\; (1\; \backslash cdot\; 2^k)\; =\; 1\; \backslash left\; (\backslash frac\{1-2^\{64\}\}\{1-2\}\backslash right\; )$

$\backslash sum\; n$

$\backslash sum\_\{n=1\}^4\; n$

$\backslash sum\_\{n=1\}^4\; n\; =\; 1\; +\; 2\; +\; 3\; +\; 4\; =\; 10$

$\backslash sum\_\{n=1\}^4\; n^2\; =\; 1^2\; +\; 2^2\; +\; 3^2\; +\; 4^2\; =\; 30$

$\backslash sum\_\{i=1\}^3\; i(i+1)\; =\; 1\backslash cdot\; 2\; +\; 2\backslash cdot\; 3\; +\; 3\backslash cdot\; 4\; =\; 20$

$\backslash sum\_\{i=3\}^5\; \backslash frac\{i\}\{i+1\}\; =\; \backslash frac\{3\}\{4\}\; +\; \backslash frac\{4\}\{5\}\; +\; \backslash frac\{5\}\{6\}$

$(a+b)^n=\backslash sum\_\{k=0\}^n\{n\; \backslash choose\; k\}a^\{n-k\}b^\{k\}$

$\{n\; \backslash choose\; k\}(\backslash tfrac\{1\}\{n\})^\{k\}\; =\; \backslash frac\{n!\}\{k!(n-k)!\}\; \backslash cdot\; \backslash frac\{1\}\{n^k\}$

\begin{align}

\sum_{k=0}^\infty{\frac{1}{k!}} &= \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... \\

&= 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + ... \\

\end{align}

\begin{align}

(a+b)^3 & = \sum_{k=0}^3{3 \choose k}a^{3-k}b^{k} \\

& = {3 \choose 0}a^{3-0}b^{0} + {3 \choose 1}a^{3-1}b^{1} + {3 \choose 2}a^{3-2}b^{2} + {3 \choose 3}a^{3-3}b^{3} \\

& = 1 \cdot a^3b^0 + 3 \cdot a^2b^1 + 3 \cdot a^1b^2 + 1 \cdot a^0b^3 \\

& = a^3 + 3a^2b + 3ab^2 + b^3 \\

\end{align}

\begin{align}

(1+\tfrac{1}{n})^n & = \sum_{k=0}^n{n \choose k}1^{n-k}(\tfrac{1}{n})^{k} \\

& = \sum_{k=0}^n{n \choose k}(\tfrac{1}{n})^{k} \\

& = \sum_{k=0}^n~\frac{n!}{k!~(n-k)!} \cdot \frac{1}{n^k} \\

\end{align}

\begin{align}

(x+5)^4 & = \sum_{k=0}^4{4 \choose k}x^{4-k}5^{k} \\

& = {4 \choose 0}x^{4-0}5^{0} + {4 \choose 1}x^{4-1}5^{1} + {4 \choose 2}x^{4-2}5^{2} + {4 \choose 3}x^{4-3}5^{3} + {4 \choose 4}x^{4-4}5^{4} \\

& = 1 \cdot x^45^0 + 4 \cdot x^35^1 + 6 \cdot x^25^2 + 4 \cdot x^15^3 + 1 \cdot x^05^4 \\

& = x^4 + 4x^35 + 6x^25^2 + 4\cdot 5^3 + 5^4 \\

\end{align}

\begin{align}

\sum_{k=0}^{10-1} \tfrac{1}{2}(\tfrac{1}{2})^k & = \frac{1}{2} \left (\frac{1 - (\frac{1}{2})^{10}}{1-\frac{1}{2}}\right ) \\

& = \frac{1}{2} \left (\frac{1 - \frac{1}{1024}}{\frac{1}{2}}\right ) \\

& = 1 - \tfrac{1}{1024} \\

& = 0.9990234375 \\

\end{align}

$\backslash sigma\; =\; \backslash sqrt\{\backslash frac\{1\}\{N\}\; \backslash sum\_\{i=1\}^N\; (x\_i\; -\; \backslash mu)^2\}$

$s\; =\; \backslash sqrt\{\backslash frac\{1\}\{N-1\}\; \backslash sum\_\{i=1\}^N\; (x\_i\; -\; \backslash overline\{x\})^2\}$

$\backslash sum\_\{k=0\}^\{n-1\}\; (ar^k)\; =\; a\; \backslash left\; (\backslash frac\{1-r^n\}\{1-r\}\backslash right\; )$

$\backslash sum\_\{k=0\}^\{\backslash infty\}\; (ar^k)\; =\; a\; \backslash left\; (\backslash frac\{1\}\{1-r\}\backslash right\; )$

$\backslash sum\_\{k=0\}^\{\backslash infty\}\; \backslash left\; (\backslash frac\{1\}\{2\}\; \backslash cdot\; (\backslash frac\{1\}\{2\})^k\; \backslash right\; )\; =\; \backslash frac\{1\}\{2\}\; \backslash left\; (\backslash frac\{1\}\{1-\backslash frac\{1\}\{2\}\}\backslash right\; )$

$\backslash sum\_\{k=0\}^\{\backslash infty\}\; (\backslash tfrac\{1\}\{2\}\; \backslash cdot\; (\backslash tfrac\{1\}\{2\})^k\; )\; =\; \backslash tfrac\{1\}\{2\}\; \backslash left\; (\backslash frac\{1\}\{1-\backslash frac\{1\}\{2\}\}\backslash right\; )$

\begin{align}

0.999... & = 0.9 + 0.09 + 0.009 + ... \\

& = 0.9 \cdot 0.1^0 + 0.9 \cdot 0.1^1 + 0.9 \cdot 0.1^2 + ... \\

& = \sum_{k=0}^{\infty} 0.9 \cdot 0.1^k \\

\end{align}

$\backslash sum\_\{k=0\}^\{\backslash infty\}\; 0.9\; \backslash times\; 0.1^k\; =\; 0.9\; \backslash left\; (\backslash frac\{1\}\{1-0.1\}\backslash right\; )\; =\; 0.9\; \backslash left\; (\backslash frac\{1\}\{0.9\}\backslash right\; )\; =\; 1$

$\backslash frac\{\backslash sqrt\{a^2-b^2\}\}\{a\}$

$\backslash frac\{\backslash sqrt\{a^2+b^2\}\}\{a\}$

$\backslash frac\{a\}\{\backslash sin\; A\}\; =\; \backslash frac\{b\}\{\backslash sin\; B\}\; =\; \backslash frac\{c\}\{\backslash sin\; C\}$

$\backslash frac\{\backslash sin\; A\}\{a\}\; =\; \backslash frac\{\backslash sin\; B\}\{b\}\; =\; \backslash frac\{\backslash sin\; C\}\{c\}$

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

$a^2\; =\; b^2\; +\; c^2\; -\; 2bc\backslash cos(A)\; \backslash ,$

$b^2\; =\; a^2\; +\; c^2\; -\; 2ac\backslash cos(B)\; \backslash ,$

$\backslash tan\; \backslash theta\; =\; \backslash frac\{\backslash sin\; \backslash theta\}\{\backslash cos\; \backslash theta\}$

$\backslash frac\{\backslash sin\; \backslash theta\}\{\backslash cos\; \backslash theta\}\; =\; \backslash frac\{Opposite\; /\; Hypotenuse\}\{Adjacent\; /\; Hypotenuse\}\; =\; \backslash frac\{Opposite\}\{Adjacent\}\; =\; \backslash tan\; \backslash theta$

$\backslash cot\; \backslash theta\; =\; \backslash frac\{\backslash cos\; \backslash theta\}\{\backslash sin\; \backslash theta\}$

$\backslash sin\; \backslash frac\{\backslash theta\}\{2\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{\backslash frac\{1\; -\; \backslash cos\; \backslash theta\}\{2\}\}$

$\backslash cos\; \backslash frac\{\backslash theta\}\{2\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{\backslash frac\{1\; +\; \backslash cos\backslash theta\}\{2\}\}$

$\backslash tan\; \backslash frac\{\backslash theta\}\{2\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{1\; -\; \backslash cos\; \backslash theta\; \backslash over\; 1\; +\; \backslash cos\; \backslash theta\}\; =\; \backslash frac\{\backslash sin\; \backslash theta\}\{1\; +\; \backslash cos\; \backslash theta\}\; =\; \backslash frac\{1-\backslash cos\; \backslash theta\}\{\backslash sin\; \backslash theta\}\; =\; \backslash csc\; \backslash theta\; -\; \backslash cot\; \backslash theta$

$\backslash cot\; \backslash frac\{\backslash theta\}\{2\}\; =\; \backslash pm\backslash ,\; \backslash sqrt\{1\; +\; \backslash cos\; \backslash theta\; \backslash over\; 1\; -\; \backslash cos\; \backslash theta\}\; =\; \backslash frac\{\backslash sin\; \backslash theta\}\{1\; -\; \backslash cos\; \backslash theta\}\; =\; \backslash frac\{1\; +\; \backslash cos\; \backslash theta\}\{\backslash sin\; \backslash theta\}\; =\; \backslash csc\; \backslash theta\; +\; \backslash cot\; \backslash theta$

$c\; =\; \backslash sqrt\{x^2\; +\; y^2\}$

$c\; =\; \backslash sqrt\{(-2)^2\; +\; 3^2\}\; =\; \backslash sqrt\{4\; +\; 9\}\; =\; \backslash sqrt\{13\}$

$c\; =\; \backslash sqrt\{(3-9)^2\; +\; (2-7)^2\}$

$c\; =\; \backslash sqrt\{(-6)^2\; +\; (-5)^2\}\; =\; \backslash sqrt\{36+25\}=\; \backslash sqrt\{61\}\; =\; 7.81...$

$c\; =\; \backslash sqrt\{(-3-7)^2\; +\; (5-(-1))^2\}$

$c\; =\; \backslash sqrt\{(-10)^2\; +\; (6)^2\}\; =\; \backslash sqrt\{100+36\}=\; \backslash sqrt\{136\}\; =\; 11.66...$

$c\; =\; \backslash sqrt\{(9-3)^2\; +\; (7-2)^2\}$

$c\; =\; \backslash sqrt\{6^2\; +\; 5^2\}\; =\; \backslash sqrt\{36+25\}=\; \backslash sqrt\{61\}\; =\; 7.81...$

$c\; =\; \backslash sqrt\{(x\_\backslash text\{A\}-x\_\backslash text\{B\})^2\; +(y\_\backslash text\{A\}-y\_\backslash text\{B\})^2\}$

\begin{align}

c &= \sqrt{(9-4)^2 + (2-8)^2 + (7-10)^2} \\

&= \sqrt{25 + 36 + 9} = \sqrt{70} = 8.37... \\

\end{align}

$\backslash frac\{2x^2-5x-1\}\{x-3\}\; =\; 2x\; +\; 1\; +\; \backslash frac\{2\}\{x-3\}$

$\backslash frac\{1\}\{\backslash sqrt\{2\}\}$

$\backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash times\; \backslash frac\{\backslash sqrt\{2\}\}\{\backslash sqrt\{2\}\}\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}$

$\backslash sqrt[n]\{a^m\}\; =\; (\backslash sqrt[n]\{a\})^m$

$\backslash sqrt[3]\{27^2\}\; =\; (\backslash sqrt[3]\{27\})^2\; =\; 3^2\; =\; 9$

$\backslash sqrt[3]\{4^6\}\; =\; (4)^\backslash frac\{6\}\{3\}\; =\; 4^2\; =\; 16$

$\backslash sqrt[n]\{a^m\}\; =\; a^\backslash frac\{m\}\{n\}$

$\backslash sqrt[n]\{a\}\; =\; a^\backslash frac\{1\}\{n\}$

$\backslash sqrt[3]\{2^3\}\; =\; 2$

$\backslash sqrt[3]\{-2^3\}\; =\; -2$

$\backslash sqrt[4]\{-2^4\}\; =\; |-2|\; =\; 2$

$5^4=625\; \backslash \; \backslash \; so\; \backslash \; \backslash \; 5\; =\; \backslash sqrt[4]\{625\}$

$\backslash sqrt[n]\{ab\}\; =\; \backslash sqrt[n]\{a\}\backslash cdot\backslash sqrt[n]\{b\}$

$\backslash sqrt[3]\{128\}\; =\; \backslash sqrt[3]\{64\backslash cdot2\}\; =\; \backslash sqrt[3]\{64\}\backslash cdot\backslash sqrt[3]\{2\}\; =\; 4\backslash sqrt[3]\{2\}$

$\backslash sqrt[n]\{\backslash frac\{a\}\{b\}\}\; =\; \backslash frac\{\backslash sqrt[n]\{a\}\}\{\backslash sqrt[n]\{b\}\}$

$\backslash sqrt[3]\{\backslash frac\{1\}\{64\}\}\; =\; \backslash frac\{\backslash sqrt[3]\{1\}\}\{\backslash sqrt[3]\{64\}\}\; =\; \backslash frac\{1\}\{4\}$

$\backslash sqrt[n]\{a+b\}\; \backslash neq\; \backslash sqrt[n]\{a\}+\backslash sqrt[n]\{b\}$

$\backslash sqrt[n]\{a-b\}\; \backslash neq\; \backslash sqrt[n]\{a\}-\backslash sqrt[n]\{b\}$

$\backslash sqrt[n]\{a^n+b^n\}\; \backslash neq\; a+b$

$\backslash sqrt[n]\{a^n\}\; =\; a$

$\backslash sqrt\{a\}\; \backslash times\; \backslash sqrt\{a\}\; =\; a$

$\backslash sqrt[3]\{a\}\; \backslash times\; \backslash sqrt[3]\{a\}\; \backslash times\; \backslash sqrt[3]\{a\}\; =\; a$

$\backslash underbrace\{\backslash sqrt[n]\{a\}\; \backslash times\; \backslash sqrt[n]\{a\}\; \backslash times\; ...\; \backslash times\; \backslash sqrt[n]\{a\}\}\_\{n\backslash \; of\backslash \; them\}\; =\; a$

$h\; =\; \backslash frac\{(a-b)^2\}\{(a+b)^2\}$

$p\; =\; \backslash pi\; (a+b)\; \backslash left(\; 1\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \{0.5\; \backslash choose\; n\}^2\; \backslash cdot\; h^n\; \backslash right)\backslash !\backslash ,$

$p\; =\; \backslash pi\; (a+b)\; \backslash sum\_\{n=0\}^\backslash infty\; \{0.5\; \backslash choose\; n\}^2\; h^n\; \backslash !\backslash ,$

$p\; =\; \backslash pi\; (a+b)\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{4\}h\; +\; \backslash frac\{1\}\{64\}h^2\; +\; \backslash frac\{1\}\{256\}h^3\; +\; ...\backslash right)\backslash !\backslash ,$

Ellipse perimeter, simple formula:

$p\; \backslash approx\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{a^2\; +\; b^2\}\{2\}\}\backslash !\backslash ,$

A better approximation by Ramanujan is:

$p\; \backslash approx\; \backslash pi\; \backslash left[3(a+b)\; -\; \backslash sqrt\{(3a+b)(a+3b)\}\backslash right]\backslash !\backslash ,$

$p\backslash approx\backslash pi\backslash left(a+b\backslash right)\backslash left(1+\backslash frac\{3h\}\{10+\backslash sqrt\{4-3h\}\}\backslash right)$

$e\; =\; \backslash frac\{\backslash sqrt\{a^2-b^2\}\}\{a\}$

$p\; =\; 2\; a\; \backslash pi\; \backslash left(\; 1\; -\; \backslash sum\_\{i=1\}^\backslash infty\; \backslash frac\{(2i)!^2\}\{(2^i\; \backslash cdot\; i!)^4\; \}\; \backslash cdot\; \backslash frac\{e\; ^\{2i\}\}\{2i-1\}\; \backslash right)$

$p\; =\; 2\; a\; \backslash pi\; \backslash left[1\; -\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2\; e\; ^2\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\backslash right)^2\; \backslash frac\{e\; ^4\}\{3\}\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\; \backslash cdot\; 5\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\}\backslash right)^2\; \backslash frac\{e\; ^\{6\}\}\{5\}\; -\; \backslash dots\; \backslash right]$

$p\; =\; 2\; a\; \backslash pi\; \backslash left[1\; -\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2\; e\; ^2\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\backslash right)^2\; \backslash frac\{e\; ^4\}\{3\}\; -\; \backslash dots\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\; \backslash cdot\; 5\; \backslash cdot\; \backslash dots\; \backslash cdot\; (2n-1)\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\; \backslash cdot\; \backslash dots\; \backslash cdot\; 2n\}\backslash right)^2\; \backslash frac\{e\; ^\{2n\}\}\{2n-1\}\; -\; \backslash dots\; \backslash right]$

$Ax^2\; +\; Bxy\; +\; Cy^2\; +Dx\; +\; Ey\; +\; F\; =\; 0\backslash ;$

$h\; =\; \backslash frac\{(r-s)^2\}\{(r+s)^2\}$

$p\; =\; \backslash pi\; (r+s)\; \backslash left(\; 1\; +\; \backslash sum\_\{n=1\}^\backslash infty\; \{0.5\; \backslash choose\; n\}^2\; \backslash cdot\; h^n\; \backslash right)\backslash !\backslash ,$

$p\; =\; \backslash pi\; (r+s)\; \backslash sum\_\{n=0\}^\backslash infty\; \{0.5\; \backslash choose\; n\}^2\; h^n\; \backslash !\backslash ,$

$p\; =\; \backslash pi\; (r+s)\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{4\}h\; +\; \backslash frac\{1\}\{64\}h^2\; +\; \backslash frac\{1\}\{256\}h^3\; +\; ...\backslash right)\backslash !\backslash ,$

Ellipse perimeter, simple formula:

$p\; \backslash approx\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{r^2\; +\; s^2\}\{2\}\}\backslash !\backslash ,$

A better approximation by Ramanujan is:

$p\; \backslash approx\; \backslash pi\; \backslash left[3(r+s)\; -\; \backslash sqrt\{(3r+s)(r+3s)\}\backslash right]\backslash !\backslash ,$

$\backslash varepsilon=\; \backslash frac\{\backslash sqrt\{r^2-s^2\}\}\{r\}$

$p\; =\; 2\; r\; \backslash pi\; \backslash left(\; 1\; -\; \backslash sum\_\{i=1\}^\backslash infty\; \backslash frac\{(2i)!^2\}\{(2^i\; \backslash cdot\; i!)^4\; \}\; \backslash cdot\; \backslash frac\{\backslash varepsilon^\{2i\}\}\{2i-1\}\; \backslash right)$

$p\; =\; 2\; r\; \backslash pi\; \backslash left[1\; -\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2\; \backslash varepsilon^2\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\backslash right)^2\; \backslash frac\{\backslash varepsilon^4\}\{3\}\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\; \backslash cdot\; 5\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\}\backslash right)^2\; \backslash frac\{\backslash varepsilon^\{6\}\}\{5\}\; -\; \backslash dots\; \backslash right]$

$p\; =\; 2\; r\; \backslash pi\; \backslash left[1\; -\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^2\; \backslash varepsilon^2\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\backslash right)^2\; \backslash frac\{\backslash varepsilon^4\}\{3\}\; -\; \backslash dots\; -\; \backslash left(\backslash frac\{1\; \backslash cdot\; 3\; \backslash cdot\; 5\; \backslash cdot\; \backslash dots\; \backslash cdot\; (2n-1)\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\; \backslash cdot\; \backslash dots\; \backslash cdot\; 2n\}\backslash right)^2\; \backslash frac\{\backslash varepsilon^\{2n\}\}\{2n-1\}\; -\; \backslash dots\; \backslash right]$

$Ax^2\; +\; Bxy\; +\; Cy^2\; +Dx\; +\; Ey\; +\; F\; =\; 0\backslash ;$

$x^\backslash frac\{1\}\{n\}\; =\; \backslash sqrt[n]\{x\}$

$27^\backslash frac\{1\}\{3\}\; =\; \backslash sqrt[3]\{27\}\; =\; 3$

$x^\backslash frac\{m\}\{n\}\; =\; \backslash sqrt[n]\{x^m\}$

$\backslash sqrt[n]\{625\}\; =\; 5$

$\backslash sqrt[4]\{625\}\; =\; 5$

$\backslash sqrt[n]\{a^n\}\; =\; a$

\begin{align}

x^\frac{m}{n} & = \sqrt[n]{x^m} \\

& = (\sqrt[n]{x})^m \\

\end{align}

$x^\backslash frac\{m\}\{n\}\; =\; \backslash sqrt[n]\{x^m\}\; =\; (\backslash sqrt[n]\{x\})^m$

\begin{align}

x^\frac{2}{3} & = \sqrt[3]{x^2} \\

& = (\sqrt[3]{x})^2 \\

\end{align}

$x^\backslash frac\{m\}\{n\}\; =\; x^\{(m\; \backslash times\; \backslash frac\{1\}\{n\})\}\; =\; (x^m)^\{\backslash frac\{1\}\{n\}\}\; =\; \backslash sqrt[n]\{x^m\}$

$x^\backslash frac\{m\}\{n\}\; =\; x^\{(\backslash frac\{1\}\{n\}\; \backslash times\; m)\}\; =\; (x^\{\backslash frac\{1\}\{n\}\})^m\; =\; (\backslash sqrt[n]\{x\})^m$

$\backslash int\_a^b\; f(x)\backslash ,dx$

$2x^2+5x+3=0\backslash ,$

$x^2-3x=0\backslash ,$

$5x-3=0\backslash ,$

$\backslash lim\_\{x\backslash rightarrow\; 0\}\backslash frac\{\backslash sin(x)\}\{x\}\; =\; 1$

$\backslash sqrt\{1-e^2\}$

$\backslash frac\{\backslash sqrt\{2\}\}\{3\; \backslash sqrt\{3\}\}\; \backslash times\; \backslash frac\{\backslash sqrt\{3\}\}\{\backslash sqrt\{3\}\}\; =$$\backslash frac\{\backslash sqrt\{2\}\; \backslash times\; \backslash sqrt\{3\}\}\{3\; \backslash times\; 3\}\; =$$\backslash frac\{\backslash sqrt\{6\}\}\{9\}$

$\backslash sqrt\{\backslash tfrac\{1\}\{2\}\}\; =\; \backslash sqrt\{\backslash tfrac\{2\}\{4\}\}\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{\backslash sqrt\{4\}\}\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}$

$\backslash sqrt\{\backslash tfrac\{3\}\{4\}\}\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{\backslash sqrt\{4\}\}\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$

$x^\backslash frac\{2\}\{3\}\; =\; \backslash sqrt[3]\{x^2\}$

$10^\{\backslash ,\backslash !10^\{100\}\}$

$10^\{\backslash ,\backslash !10^\{10^\{1000\}\}\}$

$\{\{n!\}\; \backslash over\; \{(n\; -\; r)!\}\}\; \backslash times\; \{\{1\}\; \backslash over\; \{r!\}\}\; =\; \{\{n!\}\; \backslash over\; \{r!(n\; -\; r)!\}\}$

$\{\{n!\}\; \backslash over\; \{r!(n\; -\; r)!\}\}\; =\; \{n\; \backslash choose\; r\}$

$f(k;n,p)=\{n\backslash choose\; k\}p^k(1-p)^\{n-k\}$

for $k=0,1,2,\backslash dots,n$ and where

$\{n\backslash choose\; k\}=\backslash frac\{n!\}\{k!(n-k)!\}$

$f(3;10,0.5)=\{10\backslash choose\; 3\}0.5^3(1-0.5)^\{(10-3)\}=\{10\backslash choose\; 3\}0.5^30.5^7$

$\{10\backslash choose\; 3\}=\backslash frac\{10!\}\{3!(10-3)!\}=\backslash frac\{10!\}\{3!7!\}=120$

$\{4\backslash choose\; 2\}=\backslash frac\{4!\}\{2!(4-2)!\}=\backslash frac\{4!\}\{2!2!\}=\backslash frac\{4\backslash cdot3\backslash cdot2\backslash cdot1\}\{2\backslash cdot1\backslash cdot2\backslash cdot1\}=6$

$f(3;10,0.5)=120\backslash times0.5^30.5^7=0.1171875$

$P(n,r)\; =\; \{\}^n\backslash !P\_r\; =\; \{\}\_n\backslash !P\_r\; =\; \backslash frac\{n!\}\{(n-r)!\}$

$C(n,r)\; =\; \{\}^n\backslash !C\_r\; =\; \{\}\_n\backslash !C\_r\; =\; \{n\backslash choose\; r\}=\backslash frac\{n!\}\{r!(n-r)!\}$

$P(n,r)\; =\; \backslash frac\{n!\}\{(n-r)!\}.$

$\backslash vec\{i\}\; \backslash times\; \backslash vec\{j\}=\backslash vec\{k\}$

Image:Hexadecimal_multiplication_table.svg]]

$nC\_\{r\}=\backslash frac\{n!\}\{(n-r)!(r!)\}$

$\backslash sum\_\{k=1\}^\backslash infty\; 10^\{-k!\}$ = 0.110001000000000000000001000...

$A\; =\; \backslash sqrt\{s\backslash left(s-a\backslash right)\backslash left(s-b\backslash right)\backslash left(s-c\backslash right)\}\backslash ,$

$C\; =\; cos^\{-1\}(\backslash frac\{a^2+b^2-c^2\}\{2ab\})$

$\backslash varphi\; =\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{\backslash sqrt\{5\}\}\{2\}\; =\; \backslash frac\{1\; +\; \backslash sqrt\{5\}\}\{2\}$

$\backslash varphi\; =\; 1\; +\; \backslash frac\{1\}\{1\; +\; \backslash frac\{1\}\{1\; +\; \backslash frac\{1\}\{1\; +\; \backslash cdots\}\}\}\; =\; 1.618...$

$\backslash frac\{1\}\{3-\backslash sqrt\{2\}\}$

$\backslash frac\{1\}\{3-\backslash sqrt\{2\}\}\; \backslash times\; \backslash frac\{3+\backslash sqrt\{2\}\}\{3+\backslash sqrt\{2\}\}\; =\; \backslash frac\{3+\backslash sqrt\{2\}\}\{3^2-(\backslash sqrt\{2\})^2\}\; =\; \backslash frac\{3+\backslash sqrt\{2\}\}\{7\}$

$\backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}$

$\backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}\; \backslash times\; \backslash frac\{2+\backslash sqrt\{x\}\}\{2+\backslash sqrt\{x\}\}\; =\; \backslash frac\{2^2-(\backslash sqrt\{2\})^2\}\{(4-x)(2+\backslash sqrt\{x\})\}\; =\; \backslash frac\{(4-x)\}\{(4-x)(2+\backslash sqrt\{x\})\}\; =\; \backslash frac\{1\}\{2+\backslash sqrt\{x\}\}$

$n^r\; \backslash ,$

$\{n!\}\; \backslash over\; \{(n\; -\; r)!\}\; \backslash ,$

$\{n!\}\; \backslash over\; \{r!(n\; -\; r)!\}\; \backslash ,$

$\{\{(r\; +\; n\; -\; 1)!\}\; \backslash over\; \{r!(n\; -\; 1)!\}\}$

$\{\{r\; +\; n\; -\; 1\}\; \backslash choose\; \{r\}\}\; =\; \{\{(r\; +\; n\; -\; 1)!\}\; \backslash over\; \{r!(n\; -\; 1)!\}\}$

$\{\{r\; +\; n\; -\; 1\}\; \backslash choose\; \{r\}\}\; =\; \{\{r\; +\; n\; -\; 1\}\; \backslash choose\; \{n\; -\; 1\}\}\; =\; \{\{(r\; +\; n\; -\; 1)!\}\; \backslash over\; \{r!(n\; -\; 1)!\}\}$

f(t)=\begin{cases}

\$50&\text{if } t\leq6\\

\$80&\text{if } t>6 \text{ and } t\leq 15\\

\$80+\$5(t-15)&\text{if } t>15\end{cases}

f(x)=\begin{cases}

x^2&\text{if } x<2\\

6&\text{if } x=2\\

10-x&\text{if } x>2 \text{ and } x\leq 6 \text{ .}\end{cases}

$\backslash frac\{1\}\{x\}$ $\backslash frac\{1\}\{y\}$ $x^2\backslash ,$ $\backslash sqrt\{y\}\backslash ,$ $x^n\backslash ,$ $\backslash sqrt[n]\{y\}$ $y^\backslash frac\{1\}\{n\}$

$e^x\backslash ,$ $ln(y)\backslash ,$ $a^x\backslash ,$ $log\_a(y)\backslash ,$

$sin(x)\backslash ,$ $arcsin(y)\backslash ,$ $sin^\{-1\}(y)\backslash ,$

$cos(x)\backslash ,$ $arccos(y)\backslash ,$ $cos^\{-1\}(y)\backslash ,$

$tan(x)\backslash ,$ $arctan(y)\backslash ,$ $tan^\{-1\}(y)\backslash ,$

Help:Displaying_a_formula

$f\backslash colon\; \backslash mathbb\{N\}\backslash rightarrow\backslash mathbb\{N\}$

$f\backslash colon\; \backslash \{1,2,3,...\backslash \}\backslash rightarrow\backslash \{1,2,3,...\backslash \}$

$f\backslash colon\; \backslash mathbb\{R\}\backslash rightarrow\backslash mathbb\{R\}$

$f\backslash colon\backslash ,x\backslash mapsto\; x^2$

$f(x)\; =\; x^2\; \backslash ,$

From Set-builder notation

Examples:

• $\backslash \{x\; \backslash mid\; x\; =\; x^2\; \backslash \}$ is the set $\backslash \{0,\; 1\backslash \}$,
• $\backslash \{x\; :\; x\; \backslash in\; \backslash mathbb\{R\}\; \backslash land\; x\; >\; 0\backslash \}$ is the set of all positive real numbers,
• $\backslash \{k\; :\; n\; \backslash in\; \backslash mathbb\{N\}\; \backslash land\; k\; =\; 2n\backslash \}$ is the set of all even natural numbers,
• $\backslash \{a\; :\; \backslash exists\; \backslash \; p,\; q\; \backslash in\; \backslash mathbb\{Z\},\; q\; \backslash ne\; 0\; :\; a\; =\; p/q\backslash \}$ is the set of rational numbers, or numbers that can be written as the ratio of two integers.

$\backslash sum\_\{n=1\}^4\; (2n+1)$

-

$\backslash lim\_\{x\backslash to1\}\; \backslash frac\{x^2-1\}\{x-1\}\; =\; 2$

$\backslash lim\_\{x\backslash to1\}\; \backslash frac\{x^2-1\}\{x-1\}\; =\; \backslash lim\_\{x\backslash to1\}\; \backslash frac\{(x-1)(x+1)\}\{x-1\}\; =\; \backslash lim\_\{x\backslash to1\}\; (x+1)$

$\backslash lim\_\{x\backslash to1\}\; (x+1)\; =\; 1+1\; =\; 2$

$\backslash frac\{x^2-1\}\{x-1\}\; =\; \backslash frac\{(x-1)(x+1)\}\{x-1\}\; =\; x+1$

$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{n\}\; \backslash right)^n\; =\; e$

$\backslash lim\_\{x\backslash to10\}\; \backslash frac\{x\}\{2\}=\; 5$

$\backslash lim\_\{x\backslash to4\}\; \backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}$

$\backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}\; \backslash times\; \backslash frac\{2+\backslash sqrt\{x\}\}\{2+\backslash sqrt\{x\}\}\; =\; \backslash frac\{2^2-(\backslash sqrt\{x\})^2\}\{(4-x)(2+\backslash sqrt\{x\})\}\; =\; \backslash frac\{(4-x)\}\{(4-x)(2+\backslash sqrt\{x\})\}\; =\; \backslash frac\{1\}\{2+\backslash sqrt\{x\}\}$

$\backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}\; \backslash times\; \backslash frac\{2+\backslash sqrt\{x\}\}\{2+\backslash sqrt\{x\}\}$

$\backslash frac\{2^2-(\backslash sqrt\{x\})^2\}\{(4-x)(2+\backslash sqrt\{x\})\}$

$\backslash frac\{(4-x)\}\{(4-x)(2+\backslash sqrt\{x\})\}$

$\backslash frac\{1\}\{2+\backslash sqrt\{x\}\}$

$\backslash lim\_\{x\backslash to4\}\; \backslash frac\{2-\backslash sqrt\{x\}\}\{4-x\}\; =\; \backslash lim\_\{x\backslash to4\}\; \backslash frac\{1\}\{2+\backslash sqrt\{x\}\}\; =\; \backslash frac\{1\}\{2+\backslash sqrt\{4\}\}\; =\; \backslash frac\{1\}\{4\}$

$\backslash frac\{2x\backslash Delta\; x\; +\; \backslash Delta\; x^2\}\{\backslash Delta\; x\}$

$\backslash lim\_\{\backslash Delta\; x\backslash to0\}\; \backslash frac\{2x\backslash Delta\; x\; +\; \backslash Delta\; x^2\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\backslash to0\}\; 2x\; +\; \backslash Delta\; x$

$\backslash lim\_\{\backslash Delta\; x\backslash to0\}\; 2x\; +\; \backslash Delta\; x\; =\; 2x$

$\backslash lim\_\{\backslash Delta\; x\backslash to0\}\; \backslash frac\{2x\backslash Delta\; x\; +\; \backslash Delta\; x^2\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\backslash to0\}\; \backslash frac\{2x\backslash Delta\; x\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\backslash to0\}\; 2x\; =\; 2x$

$\backslash lim\_\{\backslash Delta\; x\backslash to0\}\; \backslash frac\{3x^2\backslash Delta\; x\; +\; 3x\backslash Delta\; x^2\; +\; \backslash Delta\; x^3\}\{\backslash Delta\; x\}\; =\; \backslash lim\_\{\backslash Delta\; x\backslash to0\}\; 3x^2\; +\; 3x\backslash Delta\; x\; +\; \backslash Delta\; x^2\; =\; 3x^2$

$f^\backslash prime(x)\backslash \; =\; \backslash lim\_\{\backslash Delta\; x\backslash to0\}\backslash frac\{f(x+\backslash Delta\; x)-f(x)\}\{\backslash Delta\; x\}$

$\backslash frac\{\backslash Delta\; y\}\{\backslash Delta\; x\}\; =\; \backslash frac\{f(x+\backslash Delta\; x)-f(x)\}\{\backslash Delta\; x\}$

$\backslash frac\{dy\}\{dx\}\; =\; \backslash frac\{f(x+dx)-f(x)\}\{dx\}\backslash ,$

$\backslash frac\{dy\}\{dx\}\backslash ,$

$=\; \backslash frac\{f(x+dx)-f(x)\}\{dx\}\backslash ,$

$=\; \backslash frac\{(x+dx)^2\; -\; x^2\}\{dx\}\backslash ,$

$=\; \backslash frac\{x^2\; +\; 2x\; \backslash cdot\; dx\; +\; dx^2\; -\; x^2\}\{dx\}\backslash ,$

$=\; \backslash frac\{2x\; \backslash cdot\; dx\; +\; dx^2\}\{dx\}\backslash ,$

$=\; 2x\; +\; dx\backslash ,$

$=\; 2x\backslash ,$

$\backslash int\_1^2\; 2x\backslash ,dx\; =\; 2^2\; -\; 1^2\; =\; 3$

$\backslash int\_\{0.5\}^1\; cos(x)\backslash ,dx\; =\; sin(1)\; -\; sin(0.5)\; =\; 0.841...\; -\; 0.479...\; =\; 0.362...$

$\backslash int\_1^3\; cos(x)\backslash ,dx\; =\; sin(3)\; -\; sin(1)\; =\; 0.141...\; -\; 0.841...\; =\; -0.700...$

$\backslash int\_0^1\; sin(x)\backslash ,dx\; =\; -cos(1)\; -\; (-cos(0))\; =\; -0.540...\; -\; (-1)\; =\; 0.460...$

$\backslash int\_\backslash frac\{\backslash pi\}\{4\}^\backslash frac\{\backslash pi\}\{2\}\; cos(x)\backslash ,dx$