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Question Number 30212 Answers: 0 Comments: 1
$${let}\:{f}\:\:\left[{a},{b}\right]\rightarrow{R}\:{continue}\:{let}\:{suppose}\:{f}\:{derivable}\:{on}\left[{a},{b}\right] \\ $$$${and}\:\forall\:{x}\:\in\left[{a},{b}\right]\:\:{f}\left({x}\right)>\mathrm{0}\:{prove}\:{that} \\ $$$$\left.\exists{c}\in\right]{a},{b}\left[\:/\:\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}=\:{e}^{\left({b}−{a}\right)\frac{{f}^{,} \left({c}\right)}{{f}\left({c}\right)}} .\right. \\ $$
Question Number 30206 Answers: 1 Comments: 1
$$\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}−{x}}}}} \\ $$$${x}=\frac{\sqrt{\mathrm{3}}−\mathrm{1}}{\mathrm{2}} \\ $$
Question Number 30194 Answers: 0 Comments: 1
$${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{{C}_{{n}} ^{{k}} }\:\:{with} \\ $$$${C}_{{n}} ^{{k}} \:\:=\frac{{n}!}{{k}!\left({n}−{k}\right)!}\:. \\ $$
Question Number 30192 Answers: 0 Comments: 0
$${let}\:\:{u}_{{n}} =\:\frac{\mathrm{1}}{{n}!}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}!\:\:\:\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} \:. \\ $$
Question Number 30191 Answers: 0 Comments: 0
$${let}\:{give}\:\:{A}\:=\:\begin{pmatrix}{\:\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$${calculate}\:{A}^{{n}} \:\:{and}\:\:{e}^{−{A}} \:\:. \\ $$$$ \\ $$
Question Number 30190 Answers: 0 Comments: 0
$${study}\:{the}\:{sequence}\:{u}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}\:+{u}_{{n}} ^{\mathrm{2}} }{\mathrm{2}}}\:\:\:{with}\:−\mathrm{1}<{u}_{\mathrm{0}} <\mathrm{1}\:. \\ $$
Question Number 30189 Answers: 0 Comments: 0
$${find}\:{lim}_{{n}\rightarrow\infty} \:\:{n}^{−\mathrm{4}} \:\prod_{{k}=\mathrm{1}} ^{\mathrm{2}{n}} \:\left({n}^{\mathrm{2}} \:+{k}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{{n}}} \:?. \\ $$
Question Number 30188 Answers: 0 Comments: 1
$$\:{solve}\:{x}^{\mathrm{3}} \left({x}^{\mathrm{2}} \:+\mathrm{1}\right){y}^{'} \:−\mathrm{2}{xy}\:=\mathrm{0} \\ $$
Question Number 30187 Answers: 1 Comments: 0
$${solve}\:{xy}^{'} \:−\mathrm{2}{y}\:=\:{x}^{\mathrm{4}} \:. \\ $$
Question Number 30186 Answers: 0 Comments: 0
$${solve}\:{x}^{\mathrm{2}} {y}^{''} \:−\mathrm{2}{y}\:={x} \\ $$
Question Number 30185 Answers: 0 Comments: 0
$${let}\:\:{I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{sinxcosx}}}{dx}\:{and} \\ $$$${J}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\sqrt{\mathrm{1}+{sinx}\:{cosx}}}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}\:+{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{I}\:{and}\:{J}. \\ $$
Question Number 30184 Answers: 0 Comments: 1
$${find}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{2}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx}\:.\:\left({arctan}={tan}^{−\mathrm{1}} \right). \\ $$
Question Number 30183 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{ax}^{\mathrm{2}} −{bx}+\mathrm{5}{c}=\mathrm{0} \\ $$$$\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{4}:\:\mathrm{5},\:\mathrm{then}\:\mathrm{4}{b}^{\mathrm{2}} =\mathrm{81}{ac}. \\ $$
Question Number 30182 Answers: 0 Comments: 2
$${find}\:\int_{\mathrm{2}} ^{\mathrm{3}} \:\:\:\frac{\sqrt{{x}+\mathrm{1}}}{{x}\sqrt{\mathrm{1}−{x}}}{dx}\:. \\ $$
Question Number 30181 Answers: 0 Comments: 0
$${find}\:\:\int\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} \:+{x}^{\mathrm{6}} }\:. \\ $$
Question Number 30180 Answers: 0 Comments: 1
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}\:{sinx}\:{cosx}}{{tan}^{\mathrm{2}} {x}\:+{cotan}^{\mathrm{2}} {x}}{dx}\:.\left({use}\:{the}\:{ch}.{x}=\frac{\pi}{\mathrm{2}}\:−{t}\right). \\ $$
Question Number 30179 Answers: 0 Comments: 1
$${find}\:\:\int\:\:\:\:\frac{{dt}}{\mathrm{1}+{cost}\:+{sint}}\:\:. \\ $$
Question Number 30178 Answers: 0 Comments: 1
$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cosx}\:{cos}\theta}\:\:{with}\:−\pi<\theta<\pi\:. \\ $$
Question Number 30177 Answers: 1 Comments: 0
$${let}\:{x}\in{R}\:\:{and}\:{u}_{{n}} =\:\prod_{{k}=\mathrm{0}} ^{{n}} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${u}_{{n}} . \\ $$
Question Number 30176 Answers: 0 Comments: 0
$${prove}\:{that}\:\:{v}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{2}{k}\:+\mathrm{1}}\:{is}\:{convergente}. \\ $$
Question Number 30175 Answers: 0 Comments: 2
$${prove}\:{that}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{n}+{k}}\:{is}\:{convergente}\:. \\ $$
Question Number 30174 Answers: 0 Comments: 0
$${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$$\mathrm{1}.\:{prove}\:{that}\:{ln}\left({n}+\mathrm{1}\right)\leqslant{u}_{{n}} \leqslant{ln}\left({n}\right)\:+\mathrm{1} \\ $$$$\mathrm{2}.\:{show}\:{that}\:{u}_{{n}} \:\:_{{n}\rightarrow\infty} \sim\:{ln}\left({n}\right)\:\:. \\ $$
Question Number 30193 Answers: 0 Comments: 0
$${let}\:{p}_{{n}} \left({x}\right)=−\mathrm{1}\:+\sum_{{k}=\mathrm{1}} ^{{k}} \:{x}^{{k}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{equation}\:{p}_{{n}} \left({x}\right)=\mathrm{0}\:{have}\:{only}\:{one}\: \\ $$$${solution}\:{x}_{{n}} \in\left[\mathrm{0},\mathrm{1}\right]\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{decreasing}\:{and}\:{minored}\:{by}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:{x}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$
Question Number 30166 Answers: 0 Comments: 3
Question Number 30165 Answers: 0 Comments: 3
Question Number 30160 Answers: 0 Comments: 4
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