Question Number 201502 by necx122 last updated on 07/Dec/23 $${A}\:{generation}\:{is}\:{about}\:{one}-{third}\:{of}\:{a} \\ $$$${lifetime}.{Approximately}\:{about}\:{how} \\ $$$${many}\:{generations}\:{have}\:{passed}\:{since} \\ $$$${the}\:{year}\:\mathrm{0}{AD}? \\ $$ Commented by AST last updated on 07/Dec/23…
Question Number 201495 by Rodier97 last updated on 07/Dec/23 $$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\left({Un}\right)_{{n}\geqslant\mathrm{1}\:;} \:\:\:\:\frac{\mathrm{1}}{{nC}_{\mathrm{2}{n}} ^{{n}} \:} \\ $$$$ \\ $$$$\:\:\:\:{study}\:\:{convergence}…
Question Number 201491 by tri26112004 last updated on 07/Dec/23 $$\mathrm{1}.\:{x}^{\mathrm{2}} −\mathrm{1}+\sqrt[{\mathrm{3}}]{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} }=\mathrm{10}{x} \\ $$$$\mathrm{2}.\:\sqrt{{x}−\frac{\mathrm{1}}{{x}}}+\sqrt{\mathrm{1}−\frac{\mathrm{1}}{{x}}}={x} \\ $$$$\mathrm{3}.\:\sqrt{\mathrm{2}{x}−\frac{\mathrm{8}}{{x}}}+\mathrm{2}\sqrt{\mathrm{1}−\frac{\mathrm{2}}{{x}}}\geqslant{x} \\ $$$$\mathrm{4}.\:\sqrt{{x}^{\mathrm{2}} +{x}}+\sqrt{{x}+\mathrm{2}}\geqslant\sqrt{\mathrm{3}\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right)} \\ $$$$\mathrm{5}.\:\left(\sqrt{{x}+\mathrm{5}}−\sqrt{{x}−\mathrm{3}}\right)\left(\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{15}}\right)\geqslant\mathrm{8} \\…
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Question Number 201509 by Calculusboy last updated on 07/Dec/23 Answered by witcher3 last updated on 09/Dec/23 $$\mathrm{tack}\:\mathrm{principal}\:\:\mathrm{definition}\:\mathrm{of}\:\mathrm{Log}\left(\mathrm{z}\right)=\mathrm{ln}\mid\mathrm{z}\mid+\mathrm{iarg}\left(\mathrm{z}\right) \\ $$$$\left.\mathrm{z}\in\right]−\frac{\pi}{\mathrm{2}},\frac{\mathrm{3}\pi}{\mathrm{2}}\left[\:\right. \\ $$$$\mathrm{ln}\left(\mathrm{ix}+\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{x}\right)\right)=\mathrm{ln}\mid\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{ln}^{\mathrm{2}} \left(\mathrm{cos}\left(\mathrm{x}\right)\right.}\mid+\mathrm{i}\:\mathrm{arg}\left(\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{x}\right)+\mathrm{ix}\right)\right.\right. \\ $$$$\mathrm{arg}\left(\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{x}\right)+\mathrm{ix}\right)\in\left[\frac{\pi}{\mathrm{2}},\pi\left[\:\right.\right.\right.…
Question Number 201477 by York12 last updated on 07/Dec/23 $$\mathrm{how}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{3d}_{\mathrm{3}} +\mathrm{4d}_{\mathrm{2}} +\mathrm{3d}_{\mathrm{1}} \right)^{\mathrm{2}} \leqslant\mathrm{5}\left(\mathrm{d}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{d}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{d}_{\mathrm{3}} ^{\mathrm{2}} +\left(\mathrm{d}_{\mathrm{2}} +\mathrm{d}_{\mathrm{1}} \right)^{\mathrm{2}} +\left(\mathrm{d}_{\mathrm{3}}…
Question Number 201439 by Red1ight last updated on 06/Dec/23 $$ \\ $$$$ \\ $$There is a field. Everyday kids throw some balls on the field.…
Question Number 201464 by hardmath last updated on 06/Dec/23 $$\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{2}\right)\:=\:\mathrm{3}\:\:\:\mathrm{and}\:\:\:\mathrm{f}\left(\mathrm{4}\right)\:=\:\mathrm{5} \\ $$$$\mathrm{find}\:\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \:\mathrm{f}\left(\mathrm{x}\right)\:\centerdot\:\mathrm{f}\:^{'} \left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by mahdipoor last updated on 06/Dec/23 $$=\left[\frac{\left({f}\left({x}\right)\right)^{\mathrm{2}}…
Question Number 201433 by necx122 last updated on 06/Dec/23 $${A}\:{truck},\:{P},\:{travelling}\:{at}\:\mathrm{54}{km}/{h}\:{passes} \\ $$$${a}\:{point}\:{at}\:\mathrm{10}:\mathrm{30}\:{am}\:{while}\:{another}\:{truck}, \\ $$$${Q}\:{travelling}\:{at}\:\mathrm{90}{km}/{h}\:{passes}\:{through} \\ $$$${this}\:{same}\:{point}\:\mathrm{30}\:{minutes}\:{later}.\:{At} \\ $$$${what}\:{time}\:{will}\:{truck}\:{Q}\:{overtake}\:{P}? \\ $$ Answered by Rasheed.Sindhi last updated…
Question Number 201460 by hardmath last updated on 06/Dec/23 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{period}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{function}: \\ $$$$\mathrm{y}\:=\:\mid\:\mathrm{tan}\:\mathrm{2x}\:\mid\:\:+\:\:\mid\:\mathrm{cot}\:\mathrm{2x}\:\mid \\ $$ Answered by Mathspace last updated on 07/Dec/23 $${y}\left({x}\right)=\mid{tan}\left(\mathrm{2}{x}\right)\mid+\frac{\mathrm{1}}{\mid{tan}\left(\mathrm{2}{x}\right)\mid} \\…