5.
Explanation
The bisection method is a numerical technique for finding the root of a function within a given interval. To approximate the value of √3 within the interval [1, 2], you can follow these steps:
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Start with the given interval [,]=[1,2][a,b]=[1,2].
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Calculate the midpoint c of the interval: =2=1+22=1.5c=2a+b=21+2=1.5
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Calculate the function value at c: −3=1.5−3f(c)=c−3=1.5−3
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Check the sign of ()f(c):
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If f(c) is very close to zero (within a predefined tolerance), consider c as the approximation for √3.
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If ()f(c) is positive, update =b=c and repeat from step 2 with the new interval [,][a,b].
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If f(c) is negative, update =a=c and repeat from step 2 with the new interval [,][a,b].
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Repeat the process until f(c) is very close to zero (within the desired tolerance).
Using this method, you can iteratively narrow down the interval until you find an approximation for √3. The accuracy of the approximation depends on the number of iterations and the chosen tolerance.