Find interval and root of polynomial with absolute error less than 1/8

songoku
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Homework Statement
Find an interval of length ##\frac{1}{4}## which contains root of ##x^3+2x+1=0##. Then using that interval, give approximate value of the root with absolute error less than ##\frac{1}{8}##
Relevant Equations
Intermediate Value Theorem (IVT)

Limit (maybe)
By IVT and trial and error, I get the interval to be ##(-\frac{1}{2},-\frac{1}{4})##

I don't know how to do the next part.

Let the actual root of the polynomial be ##x_{0}## and the approximate value is ##p##, we have ##|p-x_{0}|<\frac{1}{8}##

I am not sure how to continue.

Thanks
 
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https://www.wolframalpha.com/input?i=plot+y=x^3+2x+1&lang=en
1729769732212.png


f(x)=x^3+2x+1
Say [a,b] is the range we choose, linear line between, i.e.,
y-f(a)=\frac{f(b)-f(a)}{b-a}(x-a)
cross the x axis at
x=-\frac{(b-a)f(a)}{f(b)-f(a)}+a
This is a candidate of approximate solution which should be investigated to be within the cited error.

[EDIT]
f(a)<0, f(b)>0
If f(##\frac{a+b}{2}##)<0 we can halve the size of the section [##\frac{a+b}{2}##,b].
If f(##\frac{a+b}{2}##)>0 we can halve the size of the section [a,##\frac{a+b}{2}##].
 
Last edited:
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anuttarasammyak said:
https://www.wolframalpha.com/input?i=plot+y=x^3+2x+1&lang=en
View attachment 352641

f(x)=x^3+2x+1
Say [a,b] is the range we choose, linear line between, i.e.,
y-f(a)=\frac{f(b)-f(a)}{b-a}(x-a)
cross the x axis at
x=-\frac{(b-a)f(a)}{f(b)-f(a)}+a
This is a candidate of approximate solution which should be investigated to be within the cited error.

[EDIT]
f(a)<0, f(b)>0
If f(##\frac{a+b}{2}##)<0 we can halve the size of the section [##\frac{a+b}{2}##,b].
If f(##\frac{a+b}{2}##)>0 we can halve the size of the section [a,##\frac{a+b}{2}##].
So the idea is to use linearization and then change the interval based on the value of ##x## obtained from linearization until the interval length is < ##\frac{1}{8}##?

Thanks
 
anuttarasammyak said:
f(a)<0, f(b)>0
If f((a+b)/2)<0 we can halve the size of the section [(a+b)/2,b].
If f((a+b)/2)>0 we can halve the size of the section [a,(a+b)/2].
Please forget the part before [EDIT]. Bisection method https://en.wikipedia.org/wiki/Bisection_method applies here. We can get better convergence by Newton method.
 
Last edited:
Thank you very much for the help and explanation anuttarasammyak
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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