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goalltheway4768
02.11.2020 •
Mathematics
Are the fractions 1/2 and 479 proportional to each other?
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Ответ:
a) The equation has a root in the interval (-1,0)
b) The solution of
by using the Bisection Method is x=0.9977 accurate within ![10^{-3}](/tpl/images/1402/7688/229cf.png)
a) The intermediate zero theorem (Bolzano's Theorem) tells us that whenever you have a continuous function in a given interval and the extremes of the functions on this interval have oposite signs, then there must be a zero in between those extreme values.
A formal definition of this theorem is written like this:"If a function f on the closed interval [a,b] is a continuous function and it holds that f(a)>0 and f(b)<0 or f(a)<0 and f(b)>0, then there is at least one x-value such that f(x)=0"
So basically we need to evaluate the given equation for both extremes of the interval x=-1 and x=0, if they return results opposite in sign, then there must be a zero in that interval, so let's evaluate the function for x=-1:
Let's now test for x=0
So notice we ended up with two values -0.0112 and 4. One is positive and the other is negative, therefore there must be a zero in that interval.
b) The zero is located at x=-0.9977
The idea of the bisection method is to find values for x in the middle of two x-values that return opposite sign answers when evaluated on the given function. So we can start with the extremes of the given interval:
x=-1 and x=0
so we find the value in the middle by using the following formula:
so we get:
mid-value=-0.5
Next, we evaluate the given function for that value:
Since we got a positive answer, we now find the midpoint between -0.5 and -1 (which was the last x-value that returned a negative answer) so we get:
mid-value=-0.75
Next, we evaluate the given function for that value:
and we repeat the process until que get to the desired accuracy. I uploaded a table that has the corresponding iterations and its answers. There were 14 iterations done until we got to the final answer x=-0.9977.
Learn more about the intermediate zero theorem here: