Question Number 131369 by liberty last updated on 04/Feb/21 $$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{m}}{\mathrm{n}}\:\mathrm{where}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n} \\ $$$$\mathrm{are}\:\mathrm{relative}\:\mathrm{prime}\:\mathrm{positive}\:\mathrm{integer}\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\left(\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)\:\mathrm{equals} \\ $$ Answered by EDWIN88 last updated…
Question Number 65834 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:\:\:\forall\:\:{x},\:{y}\:\:>\mathrm{0}\:\:\:\:{B}\left({x},{y}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt}\:\:\:\:\:\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{show}\:{that}\:\:\forall\:{x}>\mathrm{0}\:\:\:\:\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right)\:\:\:\:{and}\:\:{lim}_{{n}−>\infty}…
Question Number 298 by 123456 last updated on 25/Jan/15 $${a}\left(\mathrm{n},{m}\right)=\begin{cases}{{n}+{m}}&{{n}\leqslant\mathrm{0}}\\{{b}\left({n}\right)+{m}}&{{n}>\mathrm{0}\wedge{m}\leqslant\mathrm{0}}\\{{b}\left({n}\right)+{b}\left({m}\right)}&{{n}>\mathrm{0}\wedge{m}>\mathrm{0}}\end{cases} \\ $$$${b}\left({n}\right)=\begin{cases}{\mathrm{0}}&{{n}\leqslant\mathrm{0}}\\{{b}\left({n}−\mathrm{1}\right)}&{\mathrm{0}<{n}<\mathrm{5}}\\{{b}\left({n}−\mathrm{1}\right)+{b}\left({n}+\mathrm{1}\right)}&{{n}=\mathrm{5}}\\{{b}\left({n}+\mathrm{1}\right)}&{\mathrm{5}<{n}\leqslant\mathrm{10}}\\{\mathrm{1}}&{{n}>\mathrm{10}}\end{cases} \\ $$$$\mathrm{find} \\ $$$${a}\left(\mathrm{5},\mathrm{5}\right)+{a}\left(\mathrm{4},\mathrm{6}\right) \\ $$ Answered by prakash jain last updated on…
Question Number 131371 by EDWIN88 last updated on 04/Feb/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}\left(\mathrm{tan}\:{x}−\mathrm{sin}\:{x}\right)−{x}^{\mathrm{3}} }{{x}^{\mathrm{5}} }\:=? \\ $$ Answered by liberty last updated on 04/Feb/21 $$\mathrm{let}\:\mathrm{x}\:=\:\mathrm{2t}\: \\ $$$$\mathrm{L}=\underset{\mathrm{t}\rightarrow\mathrm{0}}…
Question Number 297 by 123456 last updated on 25/Jan/15 $${u}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${v}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\begin{cases}{{u}=\frac{{dv}}{{dx}}}\\{{v}=\frac{{du}}{{dx}}}\end{cases} \\ $$ Answered by prakash jain last updated on 19/Dec/14 $${u}=\frac{{dv}}{{dx}}=\frac{{d}^{\mathrm{2}}…
Question Number 131370 by EDWIN88 last updated on 10/Feb/21 $$\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\sqrt{\mathrm{4}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{2}} }\:+{x}\sqrt{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}}\:=? \\ $$ Answered by JDamian last updated on 04/Feb/21 $$\infty \\…
Question Number 65830 by ajfour last updated on 04/Aug/19 Commented by ajfour last updated on 04/Aug/19 $${x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$${let}\:{roots}\:{be}\:−{p},−{q},−{r},−{s}. \\ $$$${Find}\:{p},{q},{r},{s}\:{in}\:{terms}\:{of}\:{a},{b},{c},{d}. \\…
Question Number 295 by defg last updated on 25/Jan/15 $$\mathrm{If}\:{V}=\mathrm{log}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\:\mathrm{then}\:{V}_{{xx}} +{V}_{{yy}} =? \\ $$ Answered by 123456 last updated on 19/Dec/14 $${V}_{{x}} =\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}}…
Question Number 131364 by EDWIN88 last updated on 04/Feb/21 $$ \\ $$$$\mathrm{How}\:\mathrm{can}\:\mathrm{I}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{volume}\: \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{region}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} +\mathrm{3}\:;\mathrm{x}=\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{x}=\mathrm{2}\:\mathrm{rotating}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}=\mathrm{7}\:\mathrm{using} \\ $$$$\mathrm{the}\:\mathrm{shell}\:\mathrm{method}. \\ $$ Answered by bramlexs22 last…
Question Number 65828 by ~ À ® @ 237 ~ last updated on 04/Aug/19 $$\:{Let}\:{go}\:{toward}\:{a}\:{rational}\:{order}\:{of}\:{derivation} \\ $$$$ \\ $$$${Part}\:\mathrm{1}\::\:\:{What}'{s}\:{that}\:{special}\:{factor}\:\: \\ $$$${Let}\:{n}\:,\:{p}\:{and}\:{k}\:{three}\:{integer}\:\:{different}\:{of}\:{zero} \\ $$$${We}\:\:{state}\:{J}_{{n},{k}} \left({p}\right)=\int_{\mathrm{0}} ^{\mathrm{1}}…