Question Number 229012 by Lara2440 last updated on 04/May/26 Commented by Lara2440 last updated on 05/May/26 $$\: \\ $$$$\mathrm{What}\:\mathrm{do}\:\mathrm{you}\:\mathrm{think}\:\mathrm{of}\:\mathrm{this}??\:\mathrm{unlike}\:\mathrm{those}\:\mathrm{strange} \\ $$$$\mathrm{problems}\:\mathrm{from}\:\mathrm{the}\:\mathrm{web},\:\mathrm{these}\:\mathrm{are}\:\mathrm{actual}\:\mathrm{problems}\: \\ $$$$\mathrm{i}'\mathrm{m}\:\mathrm{solvng}\:\mathrm{right}\:\mathrm{now}.\:\mathrm{Care}\:\mathrm{to}\:\mathrm{try}\:\mathrm{Haha}. \\ $$$$#\mathrm{real}\:\mathrm{analysis}\:#\mathrm{SNU}\:#{L}^{{p}}…
Question Number 228981 by fantastic2 last updated on 03/May/26 $$\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2tan}^{−\mathrm{1}} \frac{{R}}{\left({a}−{R}\right)\sqrt{\mathrm{2}}}} \sqrt{\left(\mathrm{1}\left(\mathrm{sin}\:\alpha+\sqrt{\mathrm{2}}\right)\mathrm{tan}\:\alpha\right)^{\mathrm{2}} +\left(\mathrm{1}+\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha\left(\mathrm{5}−\mathrm{2}\right)−\mathrm{1}\right)^{\mathrm{2}} }\mathrm{cot}\:\left(\frac{\frac{\mathrm{3}\pi}{\mathrm{4}}−\mathrm{sin}^{−\mathrm{1}} \frac{\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha\left(\mathrm{5}−\mathrm{1}\right)−\mathrm{1}\right)^{\mathrm{2}} }}{\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}}}{\mathrm{2}}\right)−\left(\mathrm{1}−\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha\left(\mathrm{5}−\mathrm{1}\right)−\mathrm{1}\right)^{\mathrm{2}} }\right)\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−\mathrm{sin}^{−\mathrm{1}} \frac{\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha\left(\mathrm{5}−\mathrm{1}\right)−\mathrm{1}\right)^{\mathrm{2}} }}{\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}}\right)\right)^{\mathrm{2}} }×\frac{\mathrm{5}\sqrt{\mathrm{2}}}{\left(\mathrm{cos}\:\alpha\right)^{\mathrm{2}}…
Question Number 228908 by fantastic2 last updated on 03/May/26 $$\int_{\mathrm{0}} ^{\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}×\mathrm{1}\left(\mathrm{5}\sqrt{\mathrm{2}}−\mathrm{1}\right)}{\:\mathrm{5}\sqrt{\mathrm{2}}\left(\mathrm{5}\sqrt{\mathrm{2}}−\mathrm{2}\right)}\right)} \mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha−\mathrm{1}\right)^{\mathrm{2}} }\mathrm{cot}\:\left(\frac{\frac{\mathrm{3}\pi}{\mathrm{4}}−\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha−\mathrm{1}\right)^{\mathrm{2}} }}{\:\sqrt{\mathrm{2}}\left(\mathrm{5}−\mathrm{1}\right)}\right)}{\mathrm{2}}\right)−\left(\mathrm{1}−\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha−\mathrm{1}\right)^{\mathrm{2}} }\right)\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}^{\mathrm{2}} −\left(\left(\mathrm{5}−\mathrm{1}\right)\sqrt{\mathrm{2}}\mathrm{tan}\:\alpha−\mathrm{1}\right)^{\mathrm{2}} }}{\:\sqrt{\mathrm{2}}\left(\mathrm{5}−\mathrm{1}\right)}\right)\right)\right)\frac{\mathrm{5}\sqrt{\mathrm{2}}}{\mathrm{cos}\:^{\mathrm{2}} \alpha}{d}\alpha \\…
Question Number 228905 by fantastic2 last updated on 02/May/26 $${mu}={mv}\mathrm{cos}\:\phi+{MV}\mathrm{cos}\:\theta..{i} \\ $$$${mv}\mathrm{sin}\:\phi={MV}\mathrm{sin}\:\theta..{ii} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{mu}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}{mv}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{MV}^{\mathrm{2}} …{iii} \\ $$$${u},\mathrm{sin}\:\theta,{m\&M}\:{are}\:{given}.{find}\:{V},{v},\phi\:{in}\:{terms}\:{of} \\ $$$${given}\:{things} \\ $$ Terms of…
Question Number 228744 by Lara2440 last updated on 29/Apr/26 $$\mathrm{Solve}\:\mathrm{partial}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\: \\ $$$$\mathrm{1}.\:\:{k}\centerdot\frac{\partial\:\:}{\partial{x}}\:\frac{\partial{u}}{\partial{x}}=\frac{\partial{u}}{\partial{t}}\:,\:\mathrm{0}<{x}<{L}\:,\:{k}>\mathrm{0}\:,\:{t}>\mathrm{0} \\ $$$$\mathrm{condition} \\ $$$${u}\left(\mathrm{0},{t}\right)=\mathrm{0}\:,\:{u}\left({L},{t}\right)=\mathrm{0}\:,\:{t}>\mathrm{0} \\ $$$${u}\left({x},\mathrm{0}\right)={f}\left({x}\right)\:,\:\mathrm{0}<{x}<{L} \\ $$$$\: \\ $$$$\mathrm{2}.\:\frac{\partial{r}}{\partial{t}}=−\alpha\:^{\mathrm{2}} {r}…
Question Number 228684 by fantastic2 last updated on 27/Apr/26 Answered by taguim01 last updated on 29/Apr/26 $${r}^{\mathrm{2}} =\left(\mathrm{12}+\mathrm{18}\right)^{\mathrm{2}} +\left({r}−\mathrm{18}\right)^{\mathrm{2}} \\ $$$$\rightarrow{r}^{\mathrm{2}} =\mathrm{900}+{r}^{\mathrm{2}} −\mathrm{36}{r}+\mathrm{324} \\ $$$$\rightarrow{r}=\frac{−\mathrm{1224}}{−\mathrm{36}}=\mathrm{34}…
Question Number 228687 by fantastic2 last updated on 27/Apr/26 Answered by mr W last updated on 29/Apr/26 $$\pi{r}^{\mathrm{2}} =\pi\:\Rightarrow{r}=\mathrm{1} \\ $$$${R}+{r}+\sqrt{\left({R}+{r}\right)^{\mathrm{2}} −\left({R}−{r}\right)^{\mathrm{2}} }=\mathrm{9} \\ $$$${R}+{r}+\mathrm{2}\sqrt{{Rr}}=\mathrm{9}…
Question Number 228673 by Lara2440 last updated on 27/Apr/26 $$\mathrm{Let}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\pi−\frac{\mathrm{1}}{\mathrm{2}}{x}\:\left(\mathrm{mod}\:\mathrm{2}\pi\right)\:\mathrm{for}\:\mathrm{all}\:\mid{x}\mid<\mathrm{2}\pi \\ $$$$\mathrm{g}\left({x}\right)=\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{sin}\left({kx}\right)}{{k}}\:,\:\mathrm{for}\:\mathrm{all}\:\mid{x}\mid<\mathrm{2}\pi \\ $$$${f}\left({x}\right)\:\mathrm{and}\:\mathrm{g}\left({x}\right)\:\:\mathrm{are}\:\mathrm{identically}\:\mathrm{equal}\:\mathrm{function}. \\ $$$$\: \\ $$$$\left.\mathrm{1}\right)\mathrm{Even}\:\mathrm{though}\:{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{are}\:\mathrm{equivalent}\:\mathrm{functions}\: \\ $$$$\mathrm{why}\:\mathrm{is}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\:\:{f}\:\:\mathrm{well}-\mathrm{defined} \\ $$$$\mathrm{while}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{g}\:\mathrm{is}\:\mathrm{not}\:?? \\…
Question Number 228661 by fantastic2 last updated on 26/Apr/26 Commented by vnm last updated on 27/Apr/26 $${better}\:{use}\:{something}\:{else},\:{for}\:{instance}\:{Desmos}. \\ $$ Commented by mr W last updated…
Question Number 228659 by fantastic2 last updated on 26/Apr/26 Commented by mr W last updated on 27/Apr/26 $${say}\:{the}\:{friction}\:{coefficient}\:{is}\:\boldsymbol{\mu}. \\ $$$${if}\:{the}\:{block}\:\boldsymbol{{M}}\:{stops}\:{instantaneously} \\ $$$${after}\:{sliding}\:{a}\:{distance}\:\boldsymbol{{l}}\:{along}\:{the} \\ $$$${slope}\:{and}\:{then}\:{begins}\:{to}\:{move}\: \\…