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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.

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

Question Number 97779 Answers: 0 Comments: 1