Category: Differentiation

Question Number 132459 by mnjuly1970 last updated on 14/Feb/21 $$\dots .nicecalculus\dots .$$$$\mathit{\varphi }={\int }_0^{\mathrm{\infty }}\frac{sin\left(x^2\right)ln\left(x\right)}{x^{\frac{3}{2}}}dx=$$$$solution:$$$$\boldsymbol{\phi}\overset{{x}^{\mathrm{2}} ={t}} {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({t}\right){ln}\left(\sqrt{{t}}\:\right)}{{t}^{\frac{\mathrm{3}}{\mathrm{4}}} }\:\frac{{dt}}{{t}^{\frac{\mathrm{1}}{\mathrm{2}}}…

Question Number 132440 by liberty last updated on 14/Feb/21 $$\mathrm{A}\mathrm{vessel}\mathrm{containing}\mathrm{water}\mathrm{has}\mathrm{the}$$$$\mathrm{shape}\mathrm{of}\mathrm{and}\mathrm{inverted}\mathrm{right}\mathrm{circular}$$$$\mathrm{cone}\mathrm{with}\mathrm{base}\mathrm{radius}2\mathrm{m}\mathrm{and}\mathrm{height}5\mathrm{m}$$$$\mathrm{The}\mathrm{water}\mathrm{flows}\mathrm{from}\mathrm{the}\mathrm{apex}$$$$\mathrm{of}\mathrm{the}\mathrm{cone}\mathrm{at}\mathrm{a}\mathrm{constant}\mathrm{rate}$$$$\mathrm{of}0.2{\mathrm{m}}^3/\min .\mathrm{Find}\mathrm{the}\mathrm{rate}\mathrm{at}$$$$\mathrm{which}\mathrm{the}\mathrm{water}\mathrm{level}\mathrm{is}\mathrm{dropping}$$$$\mathrm{when}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{water}\:\mathrm{is}…

Question Number 66866 by rajesh4661kumar@gmail.com last updated on 20/Aug/19 Commented by mathmax by abdo last updated on 20/Aug/19 $$firstx+\sqrt{1+x^2}>0forallxsoy=ln(x+\sqrt{1+x^2})=argsh\left(x\right)\Rightarrow$$$${y}^{'} \left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:=\left(\mathrm{1}+{x}^{\mathrm{2}}…

Question Number 132211 by benjo_mathlover last updated on 12/Feb/21 $$$$$$\mathrm{how}\mathrm{fast}\mathrm{is}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{a}$$$$\mathrm{rectangle}\mathrm{changing}\mathrm{if}\mathrm{one}\mathrm{side}\mathrm{is}10$$$$\mathrm{cm}\mathrm{long}\mathrm{and}\mathrm{increasing}\mathrm{at}\mathrm{a}$$$$\mathrm{rate}\mathrm{of}2\mathrm{cm}/\mathrm{s}\mathrm{and}\mathrm{the}\mathrm{other}\mathrm{side}\mathrm{is}$$$$8\mathrm{cm}\mathrm{long}\mathrm{and}\mathrm{is}\mathrm{decreasing}\mathrm{at}$$$$\mathrm{a}\mathrm{rate}\mathrm{of}3\mathrm{cm}/\mathrm{s}$$ Answered…

Question Number 132104 by mnjuly1970 last updated on 11/Feb/21 $$\dots calculus\left(1\right)\dots$$$$ify=\sqrt[3]{\frac{1}{sin\left(x\right)+cos\left(x\right)}}then:$$$$3y^{″}-12{\left(y^{\prime }\right)}^2-y^2=???$$$$\dots \dots \dots$$ Answered by mnjuly1970 last updated…