Question Number 1865 by Denbang last updated on 18/Oct/15 $${proof}\:{that}\:\sqrt{\mathrm{2}}\:{is}\:{an}\:{irrational}\:{number} \\ $$$$ \\ $$ Answered by 123456 last updated on 18/Oct/15 $$\mathrm{suppuse}\:\mathrm{by}\:\mathrm{absurf}\:\mathrm{that}\:\sqrt{\mathrm{2}}\in\mathbb{Q},\:\mathrm{then} \\ $$$$\exists\left({p},{q}\right)\in\mathbb{Z},{q}\neq\mathrm{0}\:\mathrm{such}\:\mathrm{that}\:\sqrt{\mathrm{2}}=\frac{{p}}{{q}},\left({p},{q}\right)=\mathrm{1} \\…
Question Number 67398 by mhmd last updated on 26/Aug/19 $${solve}\:{the}\:{defrintion}\:{eguation}\:\left({xp}^{\mathrm{2}} −{p}+\mathrm{2}{x}\right)=\mathrm{0} \\ $$$${when}\:{p}={dy}/{dx} \\ $$ Commented by mathmax by abdo last updated on 26/Aug/19 $${xy}^{''}…
Question Number 132932 by bobhans last updated on 17/Feb/21 $${Find}\:{f}\left({x}\right)\:{such}\:{that}\:{f}\left(\mathrm{2}{x}\right)={f}\left({x}\right) \\ $$ Answered by Olaf last updated on 17/Feb/21 $${f}\left(\mathrm{2}{x}\right)\:=\:{f}\left({x}\right) \\ $$$$\Rightarrow\:{f}\left({x}\right)\:=\:{f}\left(\frac{{x}}{\mathrm{2}}\right)\:=\:{f}\left(\frac{{x}}{\mathrm{4}}\right)\:=\:…{f}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)… \\ $$$${f}\left({x}\right)\:=\:\underset{{n}\rightarrow\infty}…
Question Number 1862 by 123456 last updated on 17/Oct/15 $${f}\left[{x}−{f}\left({y}\right)\right]={f}\left({x}\right)−{f}\left({y}\right),\forall\left({x},{y}\right)\in\mathbb{R} \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}\right)=? \\ $$ Answered by prakash jain last updated on 18/Oct/15 $${f}\left({x}\right)={x}\:\mathrm{is}\:\mathrm{one}\:\mathrm{solution}.…
Question Number 132935 by liberty last updated on 17/Feb/21 Answered by EDWIN88 last updated on 17/Feb/21 $$\:\mathrm{let}\:\mathrm{x}=\mathrm{2}\:\Rightarrow\:\mathrm{f}\left(\mathrm{2}\right).\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{f}\left(\mathrm{2}\right)+\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\:\mathrm{65}.\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{65}+\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\frac{\mathrm{65}}{\mathrm{64}}\:\Rightarrow\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{64}} \\ $$$$\Rightarrow\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{6}} }\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{1}+\mathrm{x}^{\mathrm{6}} \\…
Question Number 67396 by Cmr 237 last updated on 26/Aug/19 Commented by mathmax by abdo last updated on 26/Aug/19 $${let}\:{P}\left({z}\right)\:=\:{z}^{\mathrm{2}{n}} \:+\mathrm{1}\:{let}\:{determine}\:{the}\:{roots}\:{of}\:{P}\left({z}\right) \\ $$$${P}\left({z}\right)=\mathrm{0}\:\Leftrightarrow\:{z}^{\mathrm{2}{n}} \:=−\mathrm{1}\:\:\Leftrightarrow\:\:{z}^{\mathrm{2}{n}} ={e}^{{i}\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}…
Question Number 1861 by 123456 last updated on 17/Oct/15 $$\mathrm{V}\left(\xi\right)=\pi\rho^{\mathrm{2}} \underset{−\infty} {\overset{\xi} {\int}}{e}^{\mathrm{2}{z}} {dz} \\ $$$$\mathrm{S}\left(\xi\right)=\mathrm{2}\pi\rho\underset{−\infty} {\overset{\xi} {\int}}{e}^{{z}} \sqrt{\mathrm{1}+\rho^{\mathrm{2}} {e}^{\mathrm{2}{z}} }{dz} \\ $$$$\mathrm{V}\left(\xi\right)−\mathrm{S}\left(\xi\right)=? \\ $$…
Question Number 132928 by metamorfose last updated on 17/Feb/21 $$\underset{{k}=\mathrm{1}} {\overset{+\infty} {\sum}}\left(−\mathrm{1}\right)^{{k}} {ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{k}}\right) \\ $$ Commented by Olaf last updated on 17/Feb/21 $${sorry}\:{sir},\:{I}\:{deleted}\:{my}\:{answer}. \\ $$$${it}\:{was}\:{wrong}.…
Question Number 1858 by 123456 last updated on 16/Oct/15 $$\begin{bmatrix}{{x}\left(\rho,\theta,\xi\right)}\\{{y}\left(\rho,\theta,\xi\right)}\\{{z}\left(\rho,\theta,\xi\right)}\end{bmatrix}=\begin{bmatrix}{\rho{e}^{\xi} \mathrm{cos}\:\theta}\\{\rho{e}^{\xi} \mathrm{sin}\:\theta}\\{\xi}\end{bmatrix}\begin{cases}{\rho\in\left[\mathrm{0},+\infty\right)}\\{\theta\in\left[\mathrm{0},\mathrm{2}\pi\right)}\\{\xi\in\mathbb{R}}\end{cases} \\ $$$$\boldsymbol{{r}}\left(\rho,\theta,\xi\right)={x}\left(\rho,\theta,\xi\right)\boldsymbol{{i}}+{y}\left(\rho,\theta,\xi\right)\boldsymbol{{j}}+{z}\left(\rho,\theta,\xi\right)\boldsymbol{{k}} \\ $$$$\boldsymbol{{a}}_{\rho} =\frac{\partial\boldsymbol{{r}}}{\partial\rho} \\ $$$$\boldsymbol{{a}}_{\theta} =\frac{\partial\boldsymbol{{r}}}{\partial\theta} \\ $$$$\boldsymbol{{a}}_{\xi} =\frac{\partial\boldsymbol{{r}}}{\partial\xi} \\ $$$$\boldsymbol{{a}}_{\rho}…
Question Number 67392 by Prof 40 last updated on 26/Aug/19 $$\int\left(\mathrm{6}{x}\right. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com