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let give f(x) = e^(−(1/x)) with f(0)=0 1) is f derivable in point 0? 2)prove that f^((n)) = F_n (x) e^(−(1/x)) with F_(n ) is rational function 3) calculate f^((6)) (x) and f^((9)) (x) .

$${let}\:{give}\:\:{f}\left({x}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:\:{with}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{is}\:{f}\:{derivable}\:{in}\:{point}\:\mathrm{0}? \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:{f}^{\left({n}\right)} =\:{F}_{{n}} \left({x}\right)\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:{with}\:{F}_{{n}\:} {is}\:{rational}\:{function} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\:{f}^{\left(\mathrm{6}\right)} \:\left({x}\right)\:{and}\:\:{f}^{\left(\mathrm{9}\right)} \left({x}\right)\:\:. \\ $$

Question Number 27782 Answers: 0 Comments: 1

solve the e.d. xy^′ +αy = xe^(−x) .

$${solve}\:{the}\:{e}.{d}.\:\:\:\:\:{xy}^{'} \:+\alpha{y}\:\:=\:{xe}^{−{x}} \:\:. \\ $$

Question Number 27781 Answers: 0 Comments: 1

find the value of F(x)=∫_0 ^(π/2) ((ln(1+x sin^2 t))/(sin^2 t)) dt knowing that −1<x<1 .

$${find}\:{the}\:{value}\:{of}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right)}{{sin}^{\mathrm{2}} {t}}\:{dt}\:{knowing}\:{that} \\ $$$$−\mathrm{1}<{x}<\mathrm{1}\:. \\ $$

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Question Number 27771 Answers: 2 Comments: 1

Question Number 27769 Answers: 1 Comments: 0

A glass bottle full of mercury has mass 500g. On being heated through 35°C, 2.43g of mercury are expelled. calculate the mass of mercury remaining in the bottle (Cubic expansivity of mercury is 1.8 × 10^(−4) per K. linear expansivity of glass is 8.0 × 10^(−6) per K.

$$\mathrm{A}\:\mathrm{glass}\:\mathrm{bottle}\:\mathrm{full}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{has}\:\mathrm{mass}\:\mathrm{500g}.\:\mathrm{On}\:\mathrm{being}\:\mathrm{heated}\:\mathrm{through}\:\mathrm{35}°\mathrm{C}, \\ $$$$\mathrm{2}.\mathrm{43g}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{are}\:\mathrm{expelled}.\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{remaining}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{bottle}\:\:\left(\mathrm{Cubic}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{is}\:\mathrm{1}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{4}} \:\mathrm{per}\:\mathrm{K}.\right. \\ $$$$\mathrm{linear}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\mathrm{8}.\mathrm{0}\:×\:\mathrm{10}^{−\mathrm{6}} \:\mathrm{per}\:\mathrm{K}. \\ $$

Question Number 27767 Answers: 1 Comments: 0

If the number of divisors of a number is odd,prove that the number is perfect square and vice versa.

$$\mathrm{If}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{odd}},\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{square}}\:\mathrm{and} \\ $$$$\mathrm{vice}\:\mathrm{versa}. \\ $$

Question Number 27764 Answers: 1 Comments: 0

∫(√(tan x))dx

$$\int\sqrt{\mathrm{tan}\:{x}}{dx} \\ $$

Question Number 27761 Answers: 1 Comments: 0

4(2a+b)^2 −(a−b)^2

$$\mathrm{4}\left(\mathrm{2a}+\mathrm{b}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} \\ $$

Question Number 27729 Answers: 0 Comments: 0