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Category: Differentiation

Prove-that-for-all-n-1-1-There-exist-a-n-0-1-such-as-sin-1-n-1-n-1-6n-3-cos-1-n-a-n-2-Prove-that-lim-n-a-n-1-10-

Question Number 127455 by snipers237 last updated on 29/Dec/20 $${Prove}\:{that}\:{for}\:{all}\:{n}\geqslant\mathrm{1}\: \\ $$$$\left.\mathrm{1}\left.\right){There}\:{exist}\:\:{a}_{{n}} \in\right]\mathrm{0},\mathrm{1}\left[\:{such}\:{as}\:\:\right. \\ $$$${sin}\left(\frac{\mathrm{1}}{{n}}\right)=\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{\mathrm{6}{n}^{\mathrm{3}} }{cos}\left(\frac{\mathrm{1}}{{n}}{a}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right)\:{Prove}\:{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{10}}\: \\ $$ Terms of…

Question-192901

Question Number 192901 by cortano12 last updated on 30/May/23 Answered by Frix last updated on 30/May/23 $$\mathrm{Assuming}\:{a},\:{b}\:>\mathrm{0} \\ $$$${z}=\frac{{x}}{{a}+\frac{{x}}{{b}+{z}}}\:\Rightarrow\:{z}=\frac{−{b}+\sqrt{\left(\mathrm{4}{x}+{ab}\right){b}}}{\mathrm{2}} \\ $$$${y}={x}+{z}\:\Rightarrow\:{y}={x}+\frac{−{b}+\sqrt{\left(\mathrm{4}{x}+{ab}\right){b}}}{\mathrm{2}} \\ $$$$\frac{{d}\left[{x}+\frac{−{b}+\sqrt{\left(\mathrm{4}{x}+{ab}\right){b}}}{\mathrm{2}}\right]}{{dx}}=\mathrm{1}+\frac{\sqrt{{b}}}{\:\sqrt{\left(\mathrm{4}{x}+{ab}\right){a}}} \\ $$…

differential-equation-if-d-2-y-dx-2-y-x-amp-y-0-y-0-1-then-evaluate-0-y-x-y-1-x-dx-

Question Number 127152 by mnjuly1970 last updated on 27/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{differential}\:\:{equation}…\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:{if}\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:=\:{y}\left({x}\right)\:\:\&\:{y}\left(\mathrm{0}\right)=−{y}'\left(\mathrm{0}\right)=\mathrm{1} \\ $$$$\:\:{then}\:\:\:{evaluate}\:\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {y}\left({x}\right).{y}^{−\mathrm{1}} \left(−{x}\right){dx}=? \\ $$$$ \\ $$…

The-barometric-pressure-p-at-an-altitude-of-h-miles-above-sea-level-satisfies-the-differential-equation-dp-dh-0-2p-If-the-pressure-at-sea-level-is-29-92-inches-of-mercury-find-the-barometric-

Question Number 127119 by liberty last updated on 27/Dec/20 $${The}\:{barometric}\:{pressure}\:{p}\:{at}\:{an}\:{altitude} \\ $$$${of}\:{h}\:{miles}\:{above}\:{sea}\:{level}\:{satisfies}\:{the} \\ $$$${differential}\:{equation}\:\frac{{dp}}{{dh}}\:=\:−\mathrm{0}.\mathrm{2}{p}\:. \\ $$$${If}\:{the}\:{pressure}\:{at}\:{sea}\:{level}\:{is}\:\mathrm{29}.\mathrm{92}\:{inches} \\ $$$${of}\:{mercury},\:{find}\:{the}\:{barometric}\: \\ $$$${preassure}\:{at}\:\mathrm{17},\mathrm{000}\:{ft}\: \\ $$$$\left({A}\right)\:\mathrm{56}.\mathrm{97}\:{in}\:\:\:\:\:\:\left({B}\right)\:\mathrm{15}.\mathrm{71}\:{in} \\ $$$$\left({C}\right)\:\mathrm{7}.\mathrm{86}\:{in}\:\:\:\:\:\:\:\:\:\:\left({D}\right)\:\mathrm{1}\:{in} \\…