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Category: Differentiation

If-x-and-y-satisfy-the-equation-x-2-2-y-2-3-what-is-the-maximum-value-of-y-x-

Question Number 129344 by bramlexs22 last updated on 15/Jan/21 $$\:\mathrm{If}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\left(\mathrm{x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{3}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{y}}{\mathrm{x}}\:? \\ $$ Answered by liberty last updated on 15/Jan/21…

Prove-that-if-f-is-such-as-f-x-f-x-x-1-x-f-x-and-f-1-0-but-f-Then-f-is-the-unique-bijection-from-R-to-R-and-lim-x-0-f-x-and-lim-x-0-xf-x-0-0-f-1-

Question Number 129319 by snipers237 last updated on 14/Jan/21 $${Prove}\:{that}\: \\ $$$$\:{if}\:{f}\:{is}\:{such}\:{as}\:{f}\:'\left({x}\right)=\frac{{f}\left({x}\right)}{{x}\left(\mathrm{1}−{x}−{f}\left({x}\right)\right)} \\ $$$${and}\:{f}\left(\mathrm{1}\right)=\mathrm{0}\:{but}\:{f}\:\ncong\Theta\:.\:{Then} \\ $$$$\:\bigstar\:{f}\:{is}\:{the}\:{unique}\:{bijection}\:{from}\:\mathbb{R}^{\ast} \:{to}\:\mathbb{R}\:{and}\: \\ $$$$\:\bigstar\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)=+\infty\:\:{and}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{xf}\left({x}\right)=\mathrm{0} \\ $$$$\:\bigstar\:\int_{\mathrm{0}} ^{+\infty} {f}^{−\mathrm{1}}…

nice-calculus-evsluate-n-0-1-n-n-3-2-2-n-2n-2-

Question Number 129272 by mnjuly1970 last updated on 14/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:{evsluate}:: \\ $$$$\:\:\:\:\:\Omega=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \:\left(\frac{\Gamma\left({n}+\frac{\mathrm{3}}{\mathrm{2}}\right)}{\mathrm{2}^{{n}} \:\Gamma\left(\:\mathrm{2}{n}\:+\mathrm{2}\right)}\right)=??? \\ $$$$ \\ $$ Answered by Dwaipayan…

1-y-2-dx-1-x-2-dy-0-y-0-3-2-

Question Number 129171 by benjo_mathlover last updated on 13/Jan/21 $$\:\sqrt{\mathrm{1}−\mathrm{y}^{\mathrm{2}} }\:\mathrm{dx}\:−\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\mathrm{dy}\:=\:\mathrm{0} \\ $$$$\:\mathrm{y}\left(\mathrm{0}\right)\:=\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\: \\ $$ Answered by bramlexs22 last updated on 13/Jan/21 $$\:\Rightarrow\:\sqrt{\mathrm{1}−\mathrm{y}^{\mathrm{2}} }\:\mathrm{dx}\:=\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}}…

nice-calculus-prove-that-2-3-1-4-1-20-1-56-1-120-1-220-pi-3-

Question Number 129084 by mnjuly1970 last updated on 12/Jan/21 $$\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{20}}+\frac{\mathrm{1}}{\mathrm{56}}−\frac{\mathrm{1}}{\mathrm{120}}+\frac{\mathrm{1}}{\mathrm{220}}−…\right) \\ $$$$\:\:\:\:\:\:\:\overset{???} {=}\pi\:−\mathrm{3}\:\: \\ $$ Answered by Dwaipayan Shikari last updated…

What-is-the-F-0-so-that-the-function-F-x-4-x-1-3-sin-x-4-ln-1-x-2-3-becomes-continous-at-x-0-

Question Number 129072 by liberty last updated on 12/Jan/21 $$\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{F}\left(\mathrm{0}\right)\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\: \\ $$$$\:\mathrm{F}\left(\mathrm{x}\right)=\:\frac{\left(\mathrm{4}^{{x}} −\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{4}}\right)\:\mathrm{ln}\:\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\right)}\:\mathrm{becomes}\:\mathrm{continous} \\ $$$$\mathrm{at}\:{x}\:=\:\mathrm{0}? \\ $$ Answered by MJS_new last updated on…