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Question Number 97901 Answers: 2 Comments: 2

Question Number 97888 Answers: 1 Comments: 2

how to prove ((the volumn of dimensional sphare)) formula V_N (R)=(π^(N/2) /(Γ((N/2)+1))) R^N

$${how}\:{to}\:{prove}\:\left(\left({the}\:{volumn}\:{of}\right.\right. \\ $$$$\left.{d}\left.{imensional}\:{sphare}\right)\right)\:{formula} \\ $$$${V}_{{N}} \left({R}\right)=\frac{\pi^{{N}/\mathrm{2}} }{\Gamma\left(\frac{{N}}{\mathrm{2}}+\mathrm{1}\right)}\:{R}^{{N}} \\ $$$$ \\ $$

Question Number 97891 Answers: 0 Comments: 6

Question Number 97885 Answers: 2 Comments: 2

hello every one how do they calculated the universe old wich is 13.8 billion years

$${hello}\:{every}\:{one} \\ $$$${how}\:{do}\:{they}\:{calculated}\:{the}\:{universe}\:{old} \\ $$$${wich}\:{is}\:\mathrm{13}.\mathrm{8}\:{billion}\:{years} \\ $$

Question Number 97868 Answers: 2 Comments: 0

3y′ = 2x+y−1

$$\mathrm{3y}'\:=\:\mathrm{2x}+\mathrm{y}−\mathrm{1}\: \\ $$

Question Number 97866 Answers: 5 Comments: 4

Question Number 97858 Answers: 1 Comments: 1

Question Number 97847 Answers: 1 Comments: 0

(x^2 +x) (d^2 y/dx^2 ) + (1−2x^2 ) (dy/dx) + (x^2 −x−1)y = x^2 −x−1

$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:+\:\left(\mathrm{1}−\mathrm{2x}^{\mathrm{2}} \right)\:\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{1}\right)\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{1} \\ $$

Question Number 97840 Answers: 0 Comments: 1

lim_(γ→0) (arc sin ((k cos γ)/(√(1+k^2 ))) − arc sin (k/(√(1+k^2 )))) =?

$$\underset{\gamma\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{arc}\:\mathrm{sin}\:\frac{\mathrm{k}\:\mathrm{cos}\:\gamma}{\sqrt{\mathrm{1}+\mathrm{k}^{\mathrm{2}} }}\:−\:\mathrm{arc}\:\mathrm{sin}\:\frac{\mathrm{k}}{\sqrt{\mathrm{1}+\mathrm{k}^{\mathrm{2}} }}\right)\:=? \\ $$

Question Number 97839 Answers: 3 Comments: 1

calculate A_n =∫_0 ^((nπ)/4) (dx/(3cos^4 x +3sin^4 x−1))

$$\mathrm{calculate}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\mathrm{n}\pi}{\mathrm{4}}} \:\frac{\mathrm{dx}}{\mathrm{3cos}^{\mathrm{4}} \mathrm{x}\:+\mathrm{3sin}^{\mathrm{4}} \mathrm{x}−\mathrm{1}} \\ $$

Question Number 97833 Answers: 0 Comments: 10

Hello members of maths editor i want to say something concerning this forum. Well it is my own point of view: This forum is a place where people from all over the world come to interact together. A place where people from different back grounds, class and qualification speak a common language − mathematics. I in particular thank Tinkutara′s team for such a great platform. But i notice some people literally don′t show respect to others, like persuading others to answer thier questions, others show no appreciation for the given answers while others give rude comments with no reason behind them. please i just want to urge the Maths editor users to show more respect for others since we don′t know the identity or qualification of people who post and solve questions here. we remain one family as God guides us through our endervous.

$$\:\mathrm{Hello}\:\mathrm{members}\:\mathrm{of}\:\mathrm{maths}\:\mathrm{editor}\:\mathrm{i}\:\mathrm{want}\:\mathrm{to}\:\mathrm{say} \\ $$$$\mathrm{something}\:\mathrm{concerning}\:\mathrm{this}\:\mathrm{forum}.\:\mathrm{Well}\:\mathrm{it} \\ $$$$\mathrm{is}\:\mathrm{my}\:\mathrm{own}\:\mathrm{point}\:\mathrm{of}\:\mathrm{view}:\:\mathrm{This}\:\mathrm{forum}\:\mathrm{is}\:\mathrm{a}\:\mathrm{place}\:\mathrm{where} \\ $$$$\mathrm{people}\:\mathrm{from}\:\mathrm{all}\:\mathrm{over}\:\mathrm{the}\:\mathrm{world}\:\mathrm{come}\:\mathrm{to}\:\mathrm{interact}\:\mathrm{together}. \\ $$$$\mathrm{A}\:\mathrm{place}\:\mathrm{where}\:\mathrm{people}\:\mathrm{from}\:\mathrm{different}\:\mathrm{back}\:\mathrm{grounds}, \\ $$$$\mathrm{class}\:\mathrm{and}\:\mathrm{qualification}\:\mathrm{speak}\:\mathrm{a}\:\mathrm{common}\:\mathrm{language}\:−\: \\ $$$$\mathrm{mathematics}.\:\mathrm{I}\:\mathrm{in}\:\mathrm{particular}\:\mathrm{thank}\:\mathrm{Tinkutara}'\mathrm{s}\:\mathrm{team}\:\mathrm{for} \\ $$$$\mathrm{such}\:\mathrm{a}\:\mathrm{great}\:\mathrm{platform}.\:\mathrm{But}\:\mathrm{i}\:\mathrm{notice}\:\mathrm{some}\:\mathrm{people}\:\mathrm{literally} \\ $$$$\mathrm{don}'\mathrm{t}\:\mathrm{show}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{others},\:\mathrm{like}\:\mathrm{persuading}\:\mathrm{others} \\ $$$$\mathrm{to}\:\mathrm{answer}\:\mathrm{thier}\:\mathrm{questions},\:\mathrm{others}\:\mathrm{show}\:\mathrm{no}\: \\ $$$$\mathrm{appreciation}\:\mathrm{for}\:\mathrm{the}\:\mathrm{given}\:\mathrm{answers}\:\mathrm{while}\:\mathrm{others}\:\mathrm{give}\:\mathrm{rude} \\ $$$$\mathrm{comments}\:\mathrm{with}\:\mathrm{no}\:\mathrm{reason}\:\mathrm{behind}\:\mathrm{them}. \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{just}\:\mathrm{want}\:\mathrm{to}\:\mathrm{urge}\:\mathrm{the}\:\mathrm{Maths}\:\mathrm{editor}\:\mathrm{users}\:\mathrm{to}\:\mathrm{show} \\ $$$$\mathrm{more}\:\mathrm{respect}\:\mathrm{for}\:\mathrm{others}\:\mathrm{since}\:\mathrm{we}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{or} \\ $$$$\mathrm{qualification}\:\mathrm{of}\:\mathrm{people}\:\mathrm{who}\:\mathrm{post}\:\mathrm{and}\:\mathrm{solve}\:\mathrm{questions}\:\mathrm{here}. \\ $$$$\mathrm{we}\:\mathrm{remain}\:\mathrm{one}\:\mathrm{family}\:\mathrm{as}\:\mathrm{God}\:\mathrm{guides}\:\mathrm{us}\:\mathrm{through}\:\mathrm{our}\:\mathrm{endervous}. \\ $$

Question Number 97827 Answers: 0 Comments: 1

Question Number 97823 Answers: 4 Comments: 2

If x and y are integers , prove that x^3 −7x divisible by 3

$$\mathrm{If}\:{x}\:\mathrm{and}\:{y}\:\mathrm{are}\:\mathrm{integers}\:,\:\mathrm{prove} \\ $$$$\mathrm{that}\:{x}^{\mathrm{3}} −\mathrm{7}{x}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}\: \\ $$

Question Number 97818 Answers: 1 Comments: 0

if y^2 = ax^2 + bx + c Show that: y (d^3 y/dx^3 ) + 3 (dy/dx) (d^2 y/dx^2 ) = 0

Question Number 97808 Answers: 0 Comments: 1

Question Number 97807 Answers: 0 Comments: 0

Given (u_n )_(n∈N) , suppose (u_(2n) )_(n∈N) and (u_(2n+1) )_(n∈N) converge towards the same limit, L. Show that (u_n )_(n∈N) equally converges to L.

$$\mathcal{G}\mathrm{iven}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} ,\:\mathrm{suppose}\:\left(\mathrm{u}_{\mathrm{2n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{and}\:\left(\mathrm{u}_{\mathrm{2n}+\mathrm{1}} \right)_{\mathrm{n}\in\mathbb{N}} \\ $$$$\mathrm{converge}\:\mathrm{towards}\:\mathrm{the}\:\mathrm{same}\:\mathrm{limit},\:\mathrm{L}. \\ $$$$\mathcal{S}\mathrm{how}\:\mathrm{that}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{equally}\:\mathrm{converges}\:\mathrm{to}\:\mathrm{L}. \\ $$

Question Number 97803 Answers: 1 Comments: 0

The annual salaries of employees in a large company are approximately normally disributed with a mean of $50,000 and a standard deviation of $20,000. a. what percent of people earn less than $40,000? b. what percent of people earn between $45,000 and $65,000? c. what percent of people earn more than $70,000?

$$\mathrm{The}\:\mathrm{annual}\:\mathrm{salaries}\:\mathrm{of}\:\mathrm{employees}\:\mathrm{in}\:\mathrm{a}\:\mathrm{large} \\ $$$$\mathrm{company}\:\mathrm{are}\:\mathrm{approximately}\:\mathrm{normally}\: \\ $$$$\mathrm{disributed}\:\mathrm{with}\:\mathrm{a}\:\mathrm{mean}\:\mathrm{of}\:\$\mathrm{50},\mathrm{000}\:\mathrm{and}\:\mathrm{a}\:\mathrm{standard} \\ $$$$\mathrm{deviation}\:\mathrm{of}\:\$\mathrm{20},\mathrm{000}. \\ $$$$\mathrm{a}.\:\mathrm{what}\:\mathrm{percent}\:\mathrm{of}\:\mathrm{people}\:\mathrm{earn}\:\mathrm{less}\:\mathrm{than} \\ $$$$\$\mathrm{40},\mathrm{000}? \\ $$$$\mathrm{b}.\:\mathrm{what}\:\mathrm{percent}\:\mathrm{of}\:\mathrm{people}\:\mathrm{earn}\:\mathrm{between} \\ $$$$\$\mathrm{45},\mathrm{000}\:\mathrm{and}\:\$\mathrm{65},\mathrm{000}? \\ $$$$\mathrm{c}.\:\mathrm{what}\:\mathrm{percent}\:\mathrm{of}\:\mathrm{people}\:\mathrm{earn}\:\mathrm{more}\:\mathrm{than} \\ $$$$\$\mathrm{70},\mathrm{000}? \\ $$

Question Number 97800 Answers: 3 Comments: 0

1) findf(a)= ∫_0 ^1 (√(x^2 −x+a))dx with a>(1/2) 2)explicite g(a) =∫_0 ^1 (dx/(√(x^2 −x+a))) 3) calculate ∫_0 ^1 (dx/(√(x^2 −x +3)))

$$\left.\mathrm{1}\right)\:\mathrm{findf}\left(\mathrm{a}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{a}}\mathrm{dx}\:\:\:\:\:\mathrm{with}\:\mathrm{a}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\mathrm{explicite}\:\mathrm{g}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{a}}}\: \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}\:+\mathrm{3}}} \\ $$

Question Number 97799 Answers: 0 Comments: 1

solve y^′ cosx +y sinx =cosx +sinx

$$\mathrm{solve}\:\mathrm{y}^{'} \mathrm{cosx}\:+\mathrm{y}\:\mathrm{sinx}\:=\mathrm{cosx}\:+\mathrm{sinx} \\ $$

Question Number 97798 Answers: 0 Comments: 0

solve y′′−y =xsin(2x)

$$\mathrm{solve}\:\mathrm{y}''−\mathrm{y}\:=\mathrm{xsin}\left(\mathrm{2x}\right) \\ $$

Question Number 97797 Answers: 3 Comments: 0

solve y^(′′) −y = x

$$\mathrm{solve}\:\mathrm{y}^{''} \:−\mathrm{y}\:=\:\mathrm{x} \\ $$

Question Number 97795 Answers: 1 Comments: 0

calculate Σ_(n=1) ^∞ (((−1)^(n−1) )/([(√n)])) [..] meant the floor

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\left[\sqrt{\mathrm{n}}\right]} \\ $$$$\left[..\right]\:\mathrm{meant}\:\mathrm{the}\:\mathrm{floor} \\ $$

Question Number 97794 Answers: 1 Comments: 3

solve y^(′′) +y =(1/(cosx))

$$\mathrm{solve}\:\mathrm{y}^{''} \:+\mathrm{y}\:=\frac{\mathrm{1}}{\mathrm{cosx}} \\ $$

Question Number 97784 Answers: 1 Comments: 0

If f(x)=((( x)^(1/2) )^(1/2) )^⋰ find: (dy/dx)

$${If}\:{f}\left({x}\right)=\left(\left(\left(\:{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\iddots} \\ $$$${find}:\:\frac{{dy}}{{dx}} \\ $$

Question Number 97782 Answers: 2 Comments: 1

Evaluate: ∫ ((sinx)/(1 +sin^2 x))dx

$${Evaluate}: \\ $$$$\int\:\frac{{sinx}}{\mathrm{1}\:+{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 97781 Answers: 1 Comments: 0

Given the sequences (u_n )_(n∈N) and (v_n )_(n∈N) defined by u_n =Σ_(k=0) ^n (1/(k!)) and v_n =u_n +(1/(n(n!))) a\ Show that (u_n )_n is of Cauchy. Deduce that (u_n )_n converges. b\ Show that (u_n )_n and (v_n )_n are adjacent c\ Show that their common limit is not a rational number.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{sequences}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{and}\:\left(\mathrm{v}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \mathrm{defined} \\ $$$$\mathrm{by}\:\mathrm{u}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}!}\:\mathrm{and}\:\mathrm{v}_{\mathrm{n}} =\mathrm{u}_{\mathrm{n}} +\frac{\mathrm{1}}{\mathrm{n}\left(\mathrm{n}!\right)} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}} \:\mathrm{is}\:\mathrm{of}\:\mathrm{Cauchy}.\:\mathcal{D}\mathrm{educe}\:\mathrm{that} \\ $$$$\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}} \:\mathrm{converges}. \\ $$$$\mathrm{b}\backslash\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}} \:\mathrm{and}\:\left(\mathrm{v}_{\mathrm{n}} \right)_{\mathrm{n}} \:\mathrm{are}\:\mathrm{adjacent} \\ $$$$\mathrm{c}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{their}\:\mathrm{common}\:\mathrm{limit}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{rational} \\ $$$$\mathrm{number}. \\ $$

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