Question Number 5121 by Yozzii last updated on 15/Apr/16 $${What}\:{solutions}\:{x}\in\mathbb{R}\:{exist}\:{for}\:{the}\: \\ $$$${equation}\:\left({tan}^{−\mathrm{1}} {x}\right)\left({cot}^{−\mathrm{1}} {x}\right)={n} \\ $$$${where}\:{n}\in\mathbb{Z}? \\ $$ Answered by prakash jain last updated on…
Question Number 5119 by sanusihammed last updated on 14/Apr/16 Commented by 123456 last updated on 15/Apr/16 $${x}=\sqrt{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}} \\ $$$${x}^{\mathrm{2}} =\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}} \\ $$$${x}=\sqrt{{a}}+\sqrt{{b}}\wedge{a}>\mathrm{0}\wedge{b}>\mathrm{0} \\ $$$${x}^{\mathrm{2}} ={a}+{b}+\mathrm{2}\sqrt{{ab}}…
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Question Number 5117 by FilupSmith last updated on 14/Apr/16 $$\mathrm{I}\:\mathrm{have}\:\mathrm{a}\:\mathrm{question}.\:\mathrm{I}\:\mathrm{am}\:\mathrm{unsure}\:\mathrm{how}\:\mathrm{this} \\ $$$$\mathrm{is}\:\mathrm{done}\:\mathrm{because}\:\mathrm{I}\:\mathrm{have}\:\mathrm{never}\:\mathrm{learnt}\:\mathrm{it}. \\ $$$$ \\ $$$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{line}\:\mathrm{of}\:\mathrm{best}\:\mathrm{fit}? \\ $$ Commented by Yozzii last updated on 14/Apr/16…
Question Number 70653 by naka3546 last updated on 06/Oct/19 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\:\left(\mathrm{sin}\:{x}\:−\:\mathrm{cos}\:{x}\right)^{\mathrm{tan}\:{x}} \:\:=\:\:… \\ $$ Commented by kaivan.ahmadi last updated on 06/Oct/19 $${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \left({sinx}−{cosx}−\mathrm{1}\right){tanx}= \\ $$$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}}…
Question Number 5115 by Yozzii last updated on 14/Apr/16 $${Given}\:{the}\:{space}\:{curve}\:\boldsymbol{{r}}=\boldsymbol{{r}}\left({t}\right),\:{show} \\ $$$${that}\:{its}\:{torsion}\:\tau\:{is}\:{given}\:{by} \\ $$$$\tau=\frac{\overset{.} {\boldsymbol{{r}}}\bullet\overset{..} {\boldsymbol{{r}}}×\overset{…} {\boldsymbol{{r}}}}{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid^{\mathrm{2}} }.\:{It}\:{may}\:{help}\:{to}\:{know}\:{that}\:{its} \\ $$$${curvature}\:{is}\:{numerically}\:{given}\:{by}\:\kappa=\frac{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid}{\mid\overset{.} {\boldsymbol{{r}}}\mid^{\mathrm{3}}…
Question Number 70651 by sadimuhmud 136 last updated on 06/Oct/19 Commented by Prithwish sen last updated on 06/Oct/19 $$\mathrm{Let}\:\sqrt{\mathrm{x}}\:=\:\mathrm{u}\:\:\Rightarrow\:\mathrm{d}\left(\sqrt{\mathrm{x}}\right)=\mathrm{du} \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{2}}{\:\sqrt{\mathrm{a}}}} \mathrm{e}^{\sqrt{\mathrm{a}}\mathrm{u}} \mathrm{du}\:\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{a}}}\:\left[\boldsymbol{\mathrm{e}}^{\sqrt{\boldsymbol{\mathrm{a}}}\boldsymbol{\mathrm{u}}} \right]_{\mathrm{0}}…
Question Number 5113 by FilupSmith last updated on 14/Apr/16 $$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}: \\ $$$$\left(\mathrm{1}\right)\:\:\mathrm{Focal}\:\mathrm{point} \\ $$$$\left(\mathrm{2}\right)\:\:\mathrm{Directrix} \\ $$$$\mathrm{for}\:{y}={ax}^{\mathrm{2}} +{bx}+{c} \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{simplicity},\:\mathrm{lets}\:\mathrm{assume}\:\mathrm{it}\:\mathrm{goes}\:\mathrm{throigh} \\ $$$$\mathrm{point}\:\left(\mathrm{0},\:\mathrm{0}\right). \\ $$…
Question Number 136186 by ZiYangLee last updated on 19/Mar/21 $$\mathrm{Let}\:{f}\left({x}\right)=\mathrm{cos}^{−\mathrm{1}} {x}+\mathrm{cos}^{−\mathrm{1}} \left({x}−\mathrm{1}\right),\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{f}. \\ $$ Answered by ZiYangLee last updated on 19/Mar/21 $$\mathrm{Ans}:\:\left[\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}}\right] \\…
Question Number 5111 by LMTV last updated on 13/Apr/16 $$\int\frac{{dy}}{{e}^{{y}} }=? \\ $$ Answered by FilupSmith last updated on 13/Apr/16 $$=\int{e}^{−{y}} {dy} \\ $$$$\int{e}^{{ax}+{b}} {dx}={ae}^{{ax}+{b}}…