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Question Number 98544    Answers: 1   Comments: 1

Question Number 98539    Answers: 2   Comments: 0

Given the function f(x) = ((ln x)/(x−1)) (a) State the domain D_f of f. (b) Find lim_(x→∞) ((ln x)/(x−1)). State its asymptotes. (c) Draw up a variation table for the curve y = f(x).

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{ln}\:{x}}{{x}−\mathrm{1}} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{State}\:\mathrm{the}\:\mathrm{domain}\:{D}_{{f}} \:\mathrm{of}\:{f}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:{x}}{{x}−\mathrm{1}}.\:\mathrm{State}\:\mathrm{its}\:\mathrm{asymptotes}. \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{up}\:\mathrm{a}\:\mathrm{variation}\:\mathrm{table}\:\mathrm{for}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right).\: \\ $$

Question Number 98537    Answers: 1   Comments: 1

Question Number 98535    Answers: 0   Comments: 0

f(x) = log _5 (x) + 5e^(3x) f^(−1) (x) = ?

$${f}\left({x}\right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left({x}\right)\:+\:\mathrm{5}{e}^{\mathrm{3}{x}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:=\:? \\ $$

Question Number 98531    Answers: 0   Comments: 0

Question Number 98528    Answers: 0   Comments: 2

Question Number 98521    Answers: 1   Comments: 0

Question Number 98520    Answers: 1   Comments: 0

Integrate the function f(x,y) = xy(x^2 +y^2 ) over the domain R:{−3≤x^2 −y^2 ≤3, 1≤xy≤4}

$$\mathrm{Integrate}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{xy}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{over}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{R}:\left\{−\mathrm{3}\leqslant\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \leqslant\mathrm{3},\:\mathrm{1}\leqslant\mathrm{xy}\leqslant\mathrm{4}\right\} \\ $$

Question Number 98516    Answers: 2   Comments: 2

Question Number 98506    Answers: 0   Comments: 1

Question Number 98470    Answers: 0   Comments: 0

prove that asymtotes y=mx−((∂xφ_K )/φ_n ) k=n−1 cuts the curve Σ_(r=0) ^n φ_r ((y/x))x^r in n(n−1) points

$${prove}\:{that}\:{asymtotes}\: \\ $$$${y}={mx}−\frac{\partial{x}\phi_{{K}} }{\phi_{{n}} } \\ $$$${k}={n}−\mathrm{1} \\ $$$$ \\ $$$${cuts}\:{the}\:{curve}\: \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\phi_{{r}} \left(\frac{{y}}{{x}}\right){x}^{{r}} \\ $$$${in}\:{n}\left({n}−\mathrm{1}\right)\:{points} \\ $$$$ \\ $$

Question Number 98468    Answers: 2   Comments: 10

Question Number 98464    Answers: 1   Comments: 0

Working individually, A, B and C can finish a piece of work in 16 days, 20 days and 30 days respectively. In how many days can A, B and C together complete a work which is 3(1/2) times the previous work?

$$\mathrm{Working}\:\mathrm{individually},\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{can} \\ $$$$\mathrm{finish}\:\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{work}\:\mathrm{in}\:\mathrm{16}\:\mathrm{days},\:\mathrm{20}\:\mathrm{days} \\ $$$$\mathrm{and}\:\mathrm{30}\:\mathrm{days}\:\mathrm{respectively}.\:\mathrm{In}\:\mathrm{how}\:\mathrm{many} \\ $$$$\mathrm{days}\:\mathrm{can}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{together}\:\mathrm{complete} \\ $$$$\mathrm{a}\:\mathrm{work}\:\mathrm{which}\:\mathrm{is}\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{times}\:\mathrm{the}\:\mathrm{previous} \\ $$$$\mathrm{work}? \\ $$

Question Number 98463    Answers: 1   Comments: 0

Question Number 98461    Answers: 3   Comments: 1

Question Number 98453    Answers: 0   Comments: 2

(m/(1×2))×(m/(2×3))×(m/(3×4))×...×(m/(99×100))=99 m?

$$\frac{{m}}{\mathrm{1}×\mathrm{2}}×\frac{{m}}{\mathrm{2}×\mathrm{3}}×\frac{{m}}{\mathrm{3}×\mathrm{4}}×...×\frac{{m}}{\mathrm{99}×\mathrm{100}}=\mathrm{99} \\ $$$$ \\ $$$${m}? \\ $$

Question Number 98613    Answers: 0   Comments: 0

Question Number 98450    Answers: 0   Comments: 1

Find the supremum and the infimum of f(x) = (x/(sin x)) ,x∈ (0,(π/2) ]

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{supremum}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{infimum}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}}{\mathrm{sin}\:\mathrm{x}}\:,\mathrm{x}\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\:\right] \\ $$

Question Number 98448    Answers: 1   Comments: 5

6^(273) +8^(273) :49 prove the division

$$\:\:\:\:\:\:\:\mathrm{6}^{\mathrm{273}} +\mathrm{8}^{\mathrm{273}} \:\::\mathrm{49}\:\:\:\boldsymbol{{prove}}\:\:\boldsymbol{{the}}\:\:\boldsymbol{{divi}\mathrm{s}{ion}} \\ $$

Question Number 98445    Answers: 2   Comments: 0

give at form of serie U_n =∫_0 ^1 ((x^n ln(x))/((1+x)^2 ))dx

$$\mathrm{give}\:\mathrm{at}\:\mathrm{form}\:\mathrm{of}\:\mathrm{serie}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{n}} \mathrm{ln}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 98444    Answers: 1   Comments: 0

calculate ∫_0 ^1 ((lnx)/((1+x)^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 98443    Answers: 1   Comments: 0

Given the sequence (U_n )_(n∈N) defined by U_0 =1 and U_(n+1) =f(U_n ) where f(x)=(x/((x+1)^2 )) Show by mathematical induction that ∀n∈N^∗ 0<U_n ≤(1/n)

$$\mathcal{G}\mathrm{iven}\:\mathrm{the}\:\mathrm{sequence}\:\left(\mathrm{U}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{defined}\:\mathrm{by}\:\mathrm{U}_{\mathrm{0}} =\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{U}_{\mathrm{n}+\mathrm{1}} =\mathrm{f}\left(\mathrm{U}_{\mathrm{n}} \right)\:\mathrm{where}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$$$\mathcal{S}\mathrm{how}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{0}<\mathrm{U}_{\mathrm{n}} \leqslant\frac{\mathrm{1}}{\mathrm{n}} \\ $$

Question Number 98434    Answers: 0   Comments: 5

prove that Ω=Σ_(n=0) ^(+∞) Σ_(m=0 ) ^(+∞) ((Γ(n+(1/2)).Γ(m+(1/2)))/(Γ(n+1).Γ(m+1))).((((2/3))^n .((1/2))^m )/((n+m+(1/2)))) =((√3)/(2(√2))).G_(2,2) ^(2,2) ((1/4)∣_(0 , 0) ^((1/2),(1/2)) ) =(((√3)π)/(√2))K((3/4))=(((√3)π^2 )/(2(√2)AGM(1,(1/2))))

$${prove}\:{that} \\ $$$$\Omega=\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\underset{{m}=\mathrm{0}\:} {\overset{+\infty} {\sum}}\frac{\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right).\Gamma\left({m}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left({n}+\mathrm{1}\right).\Gamma\left({m}+\mathrm{1}\right)}.\frac{\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{n}} .\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{m}} }{\left({n}+{m}+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$$=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}\sqrt{\mathrm{2}}}.{G}_{\mathrm{2},\mathrm{2}} ^{\mathrm{2},\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\mid_{\mathrm{0}\:\:,\:\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}} \right) \\ $$$$=\frac{\sqrt{\mathrm{3}}\pi}{\sqrt{\mathrm{2}}}{K}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)=\frac{\sqrt{\mathrm{3}}\pi^{\mathrm{2}} }{\mathrm{2}\sqrt{\mathrm{2}}{AGM}\left(\mathrm{1},\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$

Question Number 98430    Answers: 1   Comments: 0

let g(x) =(2/(cos(πx))) developp g at fourier serie

$$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)\:=\frac{\mathrm{2}}{\mathrm{cos}\left(\pi\mathrm{x}\right)}\:\:\mathrm{developp}\:\mathrm{g}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 98429    Answers: 1   Comments: 0

let f(x) =cos(αx) developp f at fourier serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{cos}\left(\alpha\mathrm{x}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 98416    Answers: 3   Comments: 0

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