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Unformatted text preview: xactly twice the exponent on x. How could we go about approximating
this zero without resorting to the ‘Zero’ command of a graphing calculator? We use the Bisection
Method. The ﬁrst step in the Bisection Method is to ﬁnd an interval on which f changes sign.
We know f (1) = −1 and we ﬁnd f (2) = 29. By the Intermediate Value Theorem, we know the
zero of f lies in the interval [1, 2]. Next, we ‘bisect’ this interval and ﬁnd the midpoint, 1.5. We
ﬁnd f (1.5) ≈ 5.09. This means our zero is between 1 and 1.5, since f changes sign on this interval.
Now, we ‘bisect’ the interval [1, 1.5] and ﬁnd f (1.25) ≈ 0.80, so now we have the zero between 1 and
1.25. Bisecting [1, 1.25], we ﬁnd f (1.125) ≈ −0.32, which means the zero of f is between 1.125 and
1.25. We continue in this fashion until we have ‘sandwiched’ the zero between two numbers which
diﬀer by no more than a desired accuracy. You can think of the Bisection Method as reversing
the sign diagram process: instead of...
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