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let-x-x-3-x-1-1-prove-that-have-one-real-root-2-determine-a-approximate-value-for-by-use-of-newton-method-3-factorise-inside-R-x-f-x-4-calculste-dx-x-




Question Number 104772 by mathmax by abdo last updated on 23/Jul/20
let ϕ(x) = x^3  +x+1  1) prove that ϕ have one real root α  2)determine a approximate value for α  by use of newton method  3)factorise inside R(x) f(x)  4) calculste ∫ (dx/(ϕ(x)))
letφ(x)=x3+x+11)provethatφhaveonerealrootα2)determineaapproximatevalueforαbyuseofnewtonmethod3)factoriseinsideR(x)f(x)4)calculstedxφ(x)
Answered by MAB last updated on 23/Jul/20
1) ϕ′(x)=3x^2 +1>0   lim_(x→−∞) ϕ(x)=−∞  lim_(x→+∞) ϕ(x)=+∞  hence ϕ is a bijection of ]−∞,+∞[ to  itself, ϕ has a unique real root  2)x_(n+1) =x_n −((ϕ(x_n ))/(ϕ′(x_n )))  x_(n+1) =x_n −((x_n ^3 +x_n +1)/(3x_n ^2 +1))  x_(n+1) =((2x_n ^3 −1)/(3x_n ^2 −1))  let x_0 =0  using python x_5 =−0.6823278039465127  to be continued...
1)φ(x)=3x2+1>0limxφ(x)=limx+φ(x)=+henceφisabijectionof],+[toitself,φhasauniquerealroot2)xn+1=xnφ(xn)φ(xn)xn+1=xnxn3+xn+13xn2+1xn+1=2xn313xn21letx0=0usingpythonx5=0.6823278039465127tobecontinued
Commented by abdomsup last updated on 23/Jul/20
thank you sir.
thankyousir.
Commented by MAB last updated on 23/Jul/20
you are welcome sir
youarewelcomesir
Answered by MAB last updated on 23/Jul/20
3) ϕ(x)=(x−α)(x^2 +αx+1+α^2 )  (easy to check)  4)∫(dx/(ϕ(x)))=∫((1/((α^2 +1)))((1/(x−α))−(x/(x^2 +αx+α^2 +1)))dx  =(1/(α^2 +1))(ln(x−α)−(1/2)ln(x^2 +αx+α^2 +1)+2α((arctan(((α+2x)/( (√(2α^2 +4))))))/( (√(3α^2 +4)))))+C
3)φ(x)=(xα)(x2+αx+1+α2)(easytocheck)4)dxφ(x)=(1(α2+1)(1xαxx2+αx+α2+1)dx=1α2+1(ln(xα)12ln(x2+αx+α2+1)+2αarctan(α+2x2α2+4)3α2+4)+C

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