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Question Number 4911    Answers: 1   Comments: 2

Prove that: lim_(n→∞) (∅^n −⌊φ^n ⌋)=0 Where: φ =((1+(√5))/2) ⌊x⌋ is the floor function

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\emptyset^{{n}} −\lfloor\phi^{{n}} \rfloor\right)=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Where}:\:\:\:\:\:\:\:\phi\:=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{function} \\ $$

Question Number 4905    Answers: 0   Comments: 1

f(x)=0 is a even function or a odd function?

$${f}\left({x}\right)=\mathrm{0}\:\mathrm{is}\:\mathrm{a}\:\mathrm{even}\:\mathrm{function}\:\mathrm{or}\:\mathrm{a}\:\mathrm{odd}\:\mathrm{function}? \\ $$

Question Number 4904    Answers: 0   Comments: 2

Determine the smallest natural value of n, so the function y= 5x sin 5nx will be even.

$${Determine}\:{the}\:{smallest}\:{natural} \\ $$$${value}\:{of}\:{n},\:{so}\:{the}\:{function}\: \\ $$$${y}=\:\mathrm{5}{x}\:{sin}\:\mathrm{5}{nx}\:{will}\:{be}\:{even}. \\ $$

Question Number 4900    Answers: 0   Comments: 1

∫_n ^( n+1) f(x)dx, n∈N where f(x)=Π_(i=0) ^(n+1) (x−i) Can this be integrated with evaluating the product?

$$\int_{{n}} ^{\:{n}+\mathrm{1}} {f}\left({x}\right)\mathrm{d}{x},\:{n}\in\mathbb{N} \\ $$$${where}\:{f}\left({x}\right)=\underset{{i}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\prod}}\left({x}−{i}\right) \\ $$$$\mathrm{Can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{integrated}\:\mathrm{with}\:\mathrm{evaluating}\:\mathrm{the} \\ $$$$\mathrm{product}? \\ $$

Question Number 4897    Answers: 1   Comments: 2

f(x)=10sin x g(x)=x^2 −5x+1 h(x)=3x^2 −2x+1 h(g(f(x)))=0 x=???

$${f}\left({x}\right)=\mathrm{10sin}\:{x} \\ $$$${g}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{1} \\ $$$${h}\left({x}\right)=\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1} \\ $$$${h}\left({g}\left({f}\left({x}\right)\right)\right)=\mathrm{0} \\ $$$${x}=??? \\ $$

Question Number 4891    Answers: 1   Comments: 1

A train 150 metres long completely passes a boy walking in the opposite direction at 6 kmph in 9 seconds and a car travelling in the opposite direction in 6 sec. Find the speed of the car.

$$\mathrm{A}\:\mathrm{train}\:\mathrm{150}\:\mathrm{metres}\:\mathrm{long}\:\mathrm{completely} \\ $$$$\mathrm{passes}\:\mathrm{a}\:\mathrm{boy}\:\mathrm{walking}\:\mathrm{in}\:\mathrm{the}\:\mathrm{opposite} \\ $$$$\mathrm{direction}\:\mathrm{at}\:\mathrm{6}\:\mathrm{kmph}\:\mathrm{in}\:\mathrm{9}\:\mathrm{seconds}\:\mathrm{and} \\ $$$$\mathrm{a}\:\mathrm{car}\:\:\mathrm{travelling}\:\:\mathrm{in}\:\mathrm{the}\:\:\mathrm{opposite}\: \\ $$$$\mathrm{direction}\:\mathrm{in}\:\mathrm{6}\:\mathrm{sec}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{car}. \\ $$

Question Number 4889    Answers: 1   Comments: 0

f(x)=x^2 −2x+3 g(x)=2x^2 +7x−1 h(x)=x^2 +10x−7 f(g(h(x)))=0,x=?

$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{3} \\ $$$${g}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}{x}−\mathrm{1} \\ $$$${h}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{10}{x}−\mathrm{7} \\ $$$${f}\left({g}\left({h}\left({x}\right)\right)\right)=\mathrm{0},{x}=? \\ $$

Question Number 4883    Answers: 1   Comments: 3

f(x)= { (x,(x<0)),((xf(x−1)+1),(x≥0)) :} ∫_(−5) ^5 f(x)dx=?

$${f}\left({x}\right)=\begin{cases}{{x}}&{{x}<\mathrm{0}}\\{{xf}\left({x}−\mathrm{1}\right)+\mathrm{1}}&{{x}\geqslant\mathrm{0}}\end{cases} \\ $$$$\underset{−\mathrm{5}} {\overset{\mathrm{5}} {\int}}{f}\left({x}\right){dx}=? \\ $$

Question Number 4882    Answers: 1   Comments: 0

Roots of the equation 9x^2 −18∣x∣+5=0 belonging to the domain of definition of the function f(x)=log (x^2 −x−2) is/ are

$$\mathrm{Roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{9}{x}^{\mathrm{2}} −\mathrm{18}\mid{x}\mid+\mathrm{5}=\mathrm{0} \\ $$$$\mathrm{belonging}\:\mathrm{to}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{definition} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)=\mathrm{log}\:\left({x}^{\mathrm{2}} −{x}−\mathrm{2}\right)\:\mathrm{is}/ \\ $$$$\mathrm{are} \\ $$

Question Number 4873    Answers: 0   Comments: 5

If f(x) is a Continous and odd function. Then is f(0)=0?

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{Continous}\:\mathrm{and}\:\mathrm{odd}\:\mathrm{function}. \\ $$$$\mathrm{Then}\:\mathrm{is}\:{f}\left(\mathrm{0}\right)=\mathrm{0}? \\ $$

Question Number 4855    Answers: 0   Comments: 8

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Question Number 4851    Answers: 0   Comments: 5

The distance moved by particle in the sixth and eight seconds of the motion are 45m and 53m repectively. Determine the acceleration and the initial speed.

$${The}\:{distance}\:{moved}\:{by}\:{particle}\:{in}\:{the}\:{sixth}\:{and}\:{eight}\:{seconds} \\ $$$${of}\:{the}\:{motion}\:{are}\:\mathrm{45}{m}\:{and}\:\mathrm{53}{m}\:{repectively}. \\ $$$${Determine}\:{the}\:{acceleration}\:{and}\:{the}\:{initial}\:{speed}. \\ $$

Question Number 4848    Answers: 1   Comments: 0

please help .... thanks in advance ... Find the value of x ... (√x^x^x ) = 729

$${please}\:{help}\:....\:{thanks}\:{in}\:{advance}\:... \\ $$$$ \\ $$$${Find}\:{the}\:{value}\:{of}\:{x}\:... \\ $$$$ \\ $$$$\sqrt{{x}^{{x}^{{x}} } }\:\:=\:\:\mathrm{729} \\ $$

Question Number 4847    Answers: 0   Comments: 0

(√(6+(√(6+(√(6+(√(6+(√6))))))))) SOLUTION let x = (√(6+(√(6+(√(6+(√(6+(√6))))))))) therefore.. x^(2 ) = 6+(√(6+(√(6+(√(6+(√(6 )))))))) the equation is a continuos funtion Thus x^2 = 6+(√(6+(√(6+(√(6+(√(6+(√6) ))))))))...... since x = (√(6+(√(6+(√(6+(√(6+(√6))))))))) Therdfore x^2 = 6 + x x^2 − x − 6 = 0 x^2 − 3x + 2x − 6 = 0 (x^2 − 3x) + (2x − 6) = 0 x(x − 3) + 2(x − 3) = 0 (x − 3)(x + 2) = 0 x − 3 = 0 or x − 2 = 0 x = 3 or x = −2 since negative is not allowed Thus x = 6 Meaning that (√(6+(√(6+(√(6+(√(6+(√(6 )))))))))) = 3 DONE THANK YOU SO MUCH. I UNDERSTAND THE SOLUTION.

$$\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}}}}}} \\ $$$$ \\ $$$${SOLUTION} \\ $$$$ \\ $$$${let}\:{x}\:=\:\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}}}}}} \\ $$$$ \\ $$$${therefore}.. \\ $$$$ \\ $$$${x}^{\mathrm{2}\:} =\:\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}\:\:}}}} \\ $$$$ \\ $$$${the}\:{equation}\:{is}\:{a}\:{continuos}\:{funtion} \\ $$$${Thus} \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:=\:\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}}\:}}}}...... \\ $$$$ \\ $$$${since}\:\:\:{x}\:=\:\:\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}}}}}}\: \\ $$$$ \\ $$$${Therdfore} \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:=\:\mathrm{6}\:+\:{x} \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:−\:{x}\:−\:\mathrm{6}\:=\:\mathrm{0} \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{3}{x}\:+\:\mathrm{2}{x}\:−\:\mathrm{6}\:=\:\mathrm{0} \\ $$$$ \\ $$$$\left({x}^{\mathrm{2}} \:−\:\mathrm{3}{x}\right)\:+\:\left(\mathrm{2}{x}\:−\:\mathrm{6}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$${x}\left({x}\:−\:\mathrm{3}\right)\:+\:\mathrm{2}\left({x}\:−\:\mathrm{3}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$$\left({x}\:−\:\mathrm{3}\right)\left({x}\:+\:\mathrm{2}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$${x}\:−\:\mathrm{3}\:=\:\mathrm{0}\:{or}\:{x}\:−\:\mathrm{2}\:=\:\mathrm{0} \\ $$$$ \\ $$$${x}\:=\:\mathrm{3}\:{or}\:{x}\:=\:−\mathrm{2} \\ $$$$ \\ $$$${since}\:{negative}\:{is}\:{not}\:{allowed} \\ $$$${Thus} \\ $$$$ \\ $$$${x}\:=\:\mathrm{6} \\ $$$$ \\ $$$${Meaning}\:{that} \\ $$$$ \\ $$$$\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}\:}}}}}\:\:\:=\:\:\mathrm{3} \\ $$$$ \\ $$$${DONE} \\ $$$$ \\ $$$${THANK}\:{YOU}\:{SO}\:{MUCH}.\:{I}\:{UNDERSTAND}\:{THE}\:{SOLUTION}. \\ $$

Question Number 4844    Answers: 1   Comments: 1

y(x)=f(x)+g(x)+h(x) y(x)=x^2 +sin x+x(1−x) f(x) is even g(x) is odd f(0)g(0)−h(0)=?

$${y}\left({x}\right)={f}\left({x}\right)+{g}\left({x}\right)+{h}\left({x}\right) \\ $$$${y}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{sin}\:{x}+{x}\left(\mathrm{1}−{x}\right) \\ $$$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{even} \\ $$$${g}\left({x}\right)\:\mathrm{is}\:\mathrm{odd} \\ $$$${f}\left(\mathrm{0}\right){g}\left(\mathrm{0}\right)−{h}\left(\mathrm{0}\right)=? \\ $$

Question Number 4841    Answers: 0   Comments: 1

what mean for symbol Π?

$$ \\ $$$${what}\:{mean}\:{for}\:{symbol}\:\Pi? \\ $$

Question Number 4862    Answers: 1   Comments: 0

Solve for x x=(√(.1+(√(.1+(√(.1+(√(.1+...))))))))

$$\mathrm{Solve}\:\mathrm{for}\:{x} \\ $$$${x}=\sqrt{.\mathrm{1}+\sqrt{.\mathrm{1}+\sqrt{.\mathrm{1}+\sqrt{.\mathrm{1}+...}}}} \\ $$

Question Number 4839    Answers: 3   Comments: 1

(1) S_1 =(√(1−4x)) (2) S_2 =(√(1+4x)) For S_1 ,S_2 ∈Z, x=?

$$\left(\mathrm{1}\right)\:\:\:{S}_{\mathrm{1}} =\sqrt{\mathrm{1}−\mathrm{4}{x}} \\ $$$$\left(\mathrm{2}\right)\:\:\:{S}_{\mathrm{2}} =\sqrt{\mathrm{1}+\mathrm{4}{x}} \\ $$$$\mathrm{For}\:{S}_{\mathrm{1}} ,{S}_{\mathrm{2}} \in\mathbb{Z},\:{x}=? \\ $$

Question Number 4825    Answers: 1   Comments: 2

Find the value of (√(6+(√(6+(√(6+(√(6+(√6)))))))))

$${Find}\:{the}\:{value}\:{of}\: \\ $$$$ \\ $$$$\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}+\sqrt{\mathrm{6}}}}}} \\ $$

Question Number 4822    Answers: 0   Comments: 0

Can you please mathematically explain how some infinities can be bigger than others? Thank you!

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{mathematically}\:\mathrm{explain} \\ $$$$\mathrm{how}\:\mathrm{some}\:\mathrm{infinities}\:\mathrm{can}\:\mathrm{be}\:\mathrm{bigger}\:\mathrm{than} \\ $$$$\mathrm{others}?\:\mathrm{Thank}\:\mathrm{you}! \\ $$

Question Number 4820    Answers: 1   Comments: 0

y=f(x)+g(x) f(x) − odd function g(x) − even function find f(0), if y= 2x^2 +((sin x)/3)+1

$${y}={f}\left({x}\right)+{g}\left({x}\right) \\ $$$$ \\ $$$${f}\left({x}\right)\:−\:{odd}\:{function} \\ $$$${g}\left({x}\right)\:−\:{even}\:{function} \\ $$$$ \\ $$$${find}\:{f}\left(\mathrm{0}\right),\:{if}\:{y}=\:\mathrm{2}{x}^{\mathrm{2}} +\frac{{sin}\:{x}}{\mathrm{3}}+\mathrm{1} \\ $$

Question Number 4817    Answers: 0   Comments: 1

f(αx)=αf(x−α) f(x)=?

$${f}\left(\alpha{x}\right)=\alpha{f}\left({x}−\alpha\right) \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 4816    Answers: 0   Comments: 1

Question Number 4812    Answers: 0   Comments: 6

Question Number 4809    Answers: 1   Comments: 0

Show that ((x^2 +a^2 )/(x^2 −a^2 )) > ((x+a)/(x−a)).

$${Show}\:{that}\:\frac{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\:>\:\frac{{x}+{a}}{{x}−{a}}. \\ $$

Question Number 4807    Answers: 1   Comments: 2

Find X if... ∫_0 ^e x≠0 and ((xr)/r^e )=(√(e+n)) or Σ_(n!) e=0 and ((℧x≠y)/(℧y≠x))=−1

$${Find}\:\mathbb{X}\:{if}... \\ $$$$\underset{\mathrm{0}} {\overset{{e}} {\int}}{x}\neq\mathrm{0}\:{and}\:\frac{{xr}}{{r}^{{e}} }=\sqrt{{e}+{n}} \\ $$$${or} \\ $$$$\underset{{n}!} {\sum}{e}=\mathrm{0}\:{and}\:\frac{\mho{x}\neq{y}}{\mho{y}\neq{x}}=−\mathrm{1} \\ $$

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