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Question Number 116323 Answers: 2 Comments: 0

(1)Let a,b and c real number such that ((ab)/(a+b)) = (1/3), ((bc)/(b+c)) = (1/4) and ((ac)/(a+c)) = (1/5). Find the value of ((24abc)/(ab+ac+bc)) ? (2) Let p and q be two real number that satisfy p.q=2013. What is the minimum value of (p+q)^2 ?

$$\left(\mathrm{1}\right)\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{ab}}{\mathrm{a}+\mathrm{b}}\:=\:\frac{\mathrm{1}}{\mathrm{3}},\:\frac{\mathrm{bc}}{\mathrm{b}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:\mathrm{and}\:\frac{\mathrm{ac}}{\mathrm{a}+\mathrm{c}}\:=\:\frac{\mathrm{1}}{\mathrm{5}}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{24abc}}{\mathrm{ab}+\mathrm{ac}+\mathrm{bc}}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Let}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{be}\:\mathrm{two}\:\mathrm{real}\:\mathrm{number}\:\mathrm{that} \\ $$$$\mathrm{satisfy}\:\mathrm{p}.\mathrm{q}=\mathrm{2013}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\left(\mathrm{p}+\mathrm{q}\right)^{\mathrm{2}} \:? \\ $$

Question Number 116319 Answers: 1 Comments: 0

If z = x^2 tan^(−1) ((y/x)), find (∂^2 z/(∂x∂y)) at (1,1)

$$\mathrm{If}\:\mathrm{z}\:=\:\mathrm{x}^{\mathrm{2}} \:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{y}}{\mathrm{x}}\right),\:\mathrm{find}\:\frac{\partial^{\mathrm{2}} \mathrm{z}}{\partial\mathrm{x}\partial\mathrm{y}}\: \\ $$$$\mathrm{at}\:\left(\mathrm{1},\mathrm{1}\right) \\ $$

Question Number 116318 Answers: 1 Comments: 0

1) explicite f(a) =∫_(−∞) ^(+∞) ((arctan(a+x))/(x^2 +4))dx 1) 1)calculate ∫_(−∞) ^(+∞) ((arctan(1+x))/(x^2 +4))dx and ∫_(−∞) ^(+∞) ((arctan(3+x))/(x^2 +4))dx

$$\left.\mathrm{1}\right)\:{explicite}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left({a}+{x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$$$\left.\mathrm{1}\left.\right)\:\mathrm{1}\right){calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{1}+{x}\right)}{{x}^{\mathrm{2}} +\mathrm{4}}{dx} \\ $$$${and}\:\int_{−\infty} ^{+\infty} \:\frac{{arctan}\left(\mathrm{3}+{x}\right)}{{x}^{\mathrm{2}} +\mathrm{4}}{dx} \\ $$

Question Number 116317 Answers: 2 Comments: 0

lim_(n→∞) ((((√(n+1))+(√(n+2))+(√(n+3))+...+(√(2n−1)))/n^(3/2) ) ) =

$$\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\sqrt{\mathrm{n}+\mathrm{1}}+\sqrt{\mathrm{n}+\mathrm{2}}+\sqrt{\mathrm{n}+\mathrm{3}}+...+\sqrt{\mathrm{2n}−\mathrm{1}}}{\mathrm{n}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\right)\:= \\ $$

Question Number 116311 Answers: 3 Comments: 0

∫_0 ^(π/3) ((sin 2x)/((sin x)^(4/3) )) dx

$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{3}}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2x}}{\left(\mathrm{sin}\:\mathrm{x}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:\mathrm{dx}\: \\ $$$$ \\ $$

Question Number 116310 Answers: 0 Comments: 2