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Question Number 56706 Answers: 0 Comments: 1

f : [0, 1) → R lim_(x→0) f(x)=0 lim_(x→0) ((f(x)−f((x/2)))/x)=0 lim_(x→0) ((f(x))/x) =...

$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right)\:\rightarrow\:\mathbb{R} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)−{f}\left(\frac{{x}}{\mathrm{2}}\right)}{{x}}=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{x}}\:=... \\ $$

Question Number 56705 Answers: 1 Comments: 0

the smallest value of S={^3 (√n)−^3 (√m) ∣ n, m ∈N} is...

$$\mathrm{the}\:\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{S}=\left\{\:^{\mathrm{3}} \sqrt{{n}}−^{\mathrm{3}} \sqrt{{m}}\:\mid\:{n},\:{m}\:\in\mathbb{N}\right\}\:\:\mathrm{is}... \\ $$

Question Number 56700 Answers: 1 Comments: 2

find ∫ (√(x−2(√x)+3))dx

$$\:{find}\:\int\:\sqrt{{x}−\mathrm{2}\sqrt{{x}}+\mathrm{3}}{dx} \\ $$

Question Number 56699 Answers: 0 Comments: 2

let f_n (a)=∫_(−∞) ^∞ ((sin(x^n ))/(x^2 +a^2 )) dx with a positif real not 0 and n from N 1) find a explicit form of f(a) 2) calculate g_n (a) =∫_(−∞) ^(+∞) ((sin(x^n ))/((x^2 +a^2 )^2 ))dx 3) calculate ∫_(−∞) ^(+∞) ((sin(x^3 ))/(x^2 +4)) dx and ∫_(−∞) ^(+∞) ((sin(x^2 ))/(x^2 +9))dx 4) calculate ∫_(−∞) ^(+∞) ((sin(x^3 ))/((x^2 +4)^2 ))dx .

$${let}\:{f}_{{n}} \left({a}\right)=\int_{−\infty} ^{\infty} \:\:\frac{{sin}\left({x}^{{n}} \right)}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }\:{dx}\:\:\:{with}\:{a}\:{positif}\:{real}\:{not}\:\mathrm{0}\:\:{and}\:{n}\:{from}\:{N} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:{g}_{{n}} \left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}\:{dx}\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{3}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 56698 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ ((sin(x^2 ))/(x^4 +4))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} \:+\mathrm{4}}{dx} \\ $$

Question Number 56697 Answers: 2 Comments: 0

Find the shortest distance between the lines L = (1, 4, 2) + N(1, 3, 2) and r = (−1, 1, −1) + λ(1, 2, −1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\:\:\:\mathrm{L}\:\:=\:\:\left(\mathrm{1},\:\mathrm{4},\:\mathrm{2}\right)\:+\:\mathrm{N}\left(\mathrm{1},\:\mathrm{3},\:\mathrm{2}\right)\:\:\:\mathrm{and} \\ $$$$\:\:\:\mathrm{r}\:\:=\:\:\left(−\mathrm{1},\:\mathrm{1},\:−\mathrm{1}\right)\:+\:\lambda\left(\mathrm{1},\:\mathrm{2},\:−\mathrm{1}\right) \\ $$

Question Number 56696 Answers: 0 Comments: 2

Find the perpendicular distance from (1, 7, 1) to 3x − 2y + 2z = 6

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{distance}\:\mathrm{from}\:\:\left(\mathrm{1},\:\mathrm{7},\:\mathrm{1}\right)\:\:\mathrm{to}\:\:\mathrm{3x}\:−\:\mathrm{2y}\:+\:\mathrm{2z}\:\:=\:\:\mathrm{6} \\ $$

Question Number 56719 Answers: 1 Comments: 6

The possible value of x that satisfy : x^2 + ⌊x^2 − 2x⌋ + ⌈x^2 + 2x + 1⌉ = 2017 , x ∈ R^+ and ⌊x^2 ⌋ − ⌈2x⌉ = ...

$${The}\:\:{possible}\:\:{value}\:\:{of}\:\:{x}\:\:{that}\:\:{satisfy}\:\:: \\ $$$$\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} \:+\:\lfloor{x}^{\mathrm{2}} \:−\:\mathrm{2}{x}\rfloor\:+\:\lceil{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1}\rceil\:\:=\:\:\mathrm{2017}\:\:,\:\:{x}\:\in\:\:\mathbb{R}^{+} \\ $$$${and}\:\:\:\lfloor{x}^{\mathrm{2}} \rfloor\:−\:\lceil\mathrm{2}{x}\rceil\:\:=\:\:... \\ $$

Question Number 56694 Answers: 1 Comments: 0

A tv set was marked for sale at 760.00 cedis in order to make a profit of 20%. The tv was actually sold at a discount of 5%. Calculate to 2 s.f the actual percentage profit

$$\mathrm{A}\:\mathrm{tv}\:\mathrm{set}\:\mathrm{was}\:\mathrm{marked}\:\mathrm{for}\:\mathrm{sale}\:\mathrm{at}\:\mathrm{760}.\mathrm{00} \\ $$$$\mathrm{cedis}\:\mathrm{in}\:\mathrm{order}\:\mathrm{to}\:\mathrm{make}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\mathrm{20\%}. \\ $$$$\mathrm{The}\:\mathrm{tv}\:\mathrm{was}\:\mathrm{actually}\:\mathrm{sold}\:\mathrm{at}\:\mathrm{a}\:\mathrm{discount}\: \\ $$$$\mathrm{of}\:\mathrm{5\%}.\:\mathrm{Calculate}\:\mathrm{to}\:\mathrm{2}\:\mathrm{s}.\mathrm{f}\:\mathrm{the}\:\mathrm{actual}\: \\ $$$$\mathrm{percentage}\:\mathrm{profit} \\ $$

Question Number 56688 Answers: 1 Comments: 0

How many odd numbers, greater than 60000, can be made from the digits 5,6,7,8,9,0, if no number contains any digit more than ones?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{odd}\:\mathrm{numbers},\:\mathrm{greater}\:\mathrm{than} \\ $$$$\mathrm{60000},\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},\mathrm{0},\:\mathrm{if}\:\mathrm{no}\:\mathrm{number}\:\mathrm{contains}\:\mathrm{any} \\ $$$$\mathrm{digit}\:\mathrm{more}\:\mathrm{than}\:\mathrm{ones}? \\ $$

Question Number 56685 Answers: 1 Comments: 1

show that α^4 +β^4 = (α^2 +β^2 )^2 −2α^2 β^2

$$\mathrm{show}\:\mathrm{that}\:\alpha^{\mathrm{4}} +\beta^{\mathrm{4}} \:=\:\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} \right)^{\mathrm{2}} \:−\mathrm{2}\alpha^{\mathrm{2}} \beta^{\mathrm{2}} \\ $$

Question Number 56681 Answers: 2 Comments: 1

Prove that sin ∣x∣ ≤ ∣x∣ ≤ tan ∣x∣ for ∣x∣ < (π/2)

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{sin}\:\mid{x}\mid\:\leqslant\:\mid{x}\mid\:\leqslant\:\mathrm{tan}\:\mid{x}\mid\:\:\:\:\mathrm{for}\:\:\:\:\mid{x}\mid\:<\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 56678 Answers: 1 Comments: 0

A box contains 9 whites, 7 red and 4 blue balls. Three balls are picked at random, one after the other without replacement. a) find the probability that: i. they are all of the same color ii. there is one of each colour iii. two of them are red b) if it is known that the first one picked is blue, find the probability that the rest are white.

$$\mathrm{A}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{9}\:\mathrm{whites},\:\mathrm{7}\:\mathrm{red}\:\mathrm{and}\:\mathrm{4}\:\mathrm{blue} \\ $$$$\mathrm{balls}.\:\mathrm{Three}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{picked}\:\mathrm{at}\:\mathrm{random}, \\ $$$$\mathrm{one}\:\mathrm{after}\:\mathrm{the}\:\mathrm{other}\:\mathrm{without}\:\mathrm{replacement}. \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}: \\ $$$$\mathrm{i}.\:\mathrm{they}\:\mathrm{are}\:\mathrm{all}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{color} \\ $$$$\mathrm{ii}.\:\mathrm{there}\:\mathrm{is}\:\mathrm{one}\:\mathrm{of}\:\mathrm{each}\:\mathrm{colour} \\ $$$$\mathrm{iii}.\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{red} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\:\mathrm{first}\:\mathrm{one}\:\mathrm{picked} \\ $$$$\mathrm{is}\:\mathrm{blue},\:\mathrm{find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{rest} \\ $$$$\mathrm{are}\:\mathrm{white}. \\ $$

Question Number 56668 Answers: 0 Comments: 0

I congratulate Nowruz celebration to all member of this forum

$${I}\:{congratulate}\:{Nowruz}\:{celebration} \\ $$$${to}\:{all}\:{member}\:{of}\:{this}\:{forum} \\ $$

Question Number 56664 Answers: 0 Comments: 2

Question Number 56663 Answers: 0 Comments: 0

i think everybody are in deep thought...

$${i}\:{think}\:{everybody}\:{are}\:{in}\:{deep}\:{thought}... \\ $$

Question Number 56661 Answers: 0 Comments: 0

Question Number 56660 Answers: 0 Comments: 5

Question Number 56659 Answers: 0 Comments: 1

Question Number 56650 Answers: 0 Comments: 0

ABCDEF is prime number which consist of six digits . CD, DE, EF, DEF are prime numbers . A + B + C + D + E + F = 19 ABC − DEF is a prime number . if ABC + DEF = x ∙ y ( x, y are prime numbers ) . ∣ x − y ∣ = ...

$${ABCDEF}\:\:{is}\:\:{prime}\:\:{number}\:\:{which}\:\:{consist}\:\:{of}\:\:{six}\:\:{digits}\:. \\ $$$${CD},\:{DE},\:{EF},\:{DEF}\:\:{are}\:\:{prime}\:\:{numbers}\:. \\ $$$${A}\:+\:{B}\:+\:{C}\:+\:{D}\:+\:{E}\:+\:{F}\:\:=\:\:\mathrm{19} \\ $$$${ABC}\:−\:{DEF}\:\:\:{is}\:\:{a}\:\:{prime}\:\:{number}\:. \\ $$$${if}\:\:{ABC}\:+\:{DEF}\:\:=\:\:{x}\:\centerdot\:{y}\:\:\:\:\left(\:\:{x},\:{y}\:\:\:{are}\:\:{prime}\:\:{numbers}\:\:\right)\:. \\ $$$$\mid\:{x}\:−\:{y}\:\mid\:\:=\:\:...\: \\ $$

Question Number 56647 Answers: 0 Comments: 1

sin((𝛑/x))=1 x∈[0.05;0.1]

$$\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\pi}}{\boldsymbol{\mathrm{x}}}\right)=\mathrm{1} \\ $$$$\boldsymbol{\mathrm{x}}\in\left[\mathrm{0}.\mathrm{05};\mathrm{0}.\mathrm{1}\right] \\ $$

Question Number 56643 Answers: 4 Comments: 2

Question Number 56641 Answers: 1 Comments: 2

x^5 +ax^4 +cx^2 +dx+e=0 let x=((rt+s)/(t+p)) . Find r,s,p such that equation gets transformed to λt^5 +Dt+E=0.

$$\mathrm{x}^{\mathrm{5}} +\mathrm{ax}^{\mathrm{4}} +\mathrm{cx}^{\mathrm{2}} +\mathrm{dx}+\mathrm{e}=\mathrm{0} \\ $$$$\mathrm{let}\:\mathrm{x}=\frac{\mathrm{rt}+\mathrm{s}}{\mathrm{t}+\mathrm{p}}\:\:.\:\mathrm{Find}\:\mathrm{r},\mathrm{s},\mathrm{p}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{equation}\:\mathrm{gets}\:\mathrm{transformed} \\ $$$$\mathrm{to}\:\:\:\:\:\lambda\mathrm{t}^{\mathrm{5}} +\mathrm{Dt}+\mathrm{E}=\mathrm{0}. \\ $$

Question Number 56639 Answers: 0 Comments: 1

Question Number 56635 Answers: 0 Comments: 1

∫((√(tanx))/(sinx))dx

$$\int\frac{\sqrt{\mathrm{tan}{x}}}{\mathrm{sin}{x}}\mathrm{d}{x} \\ $$

Question Number 56624 Answers: 0 Comments: 2

How many numbers greater than 2000 can be formed using the digits 1,3,4,5,6? how many of these will be even? repetition not allowed.

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2000} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{using}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{1},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}? \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{of}\:\mathrm{these}\:\mathrm{will}\:\mathrm{be}\:\mathrm{even}? \\ $$$$\mathrm{repetition}\:\mathrm{not}\:\mathrm{allowed}. \\ $$

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