1.
Part a)
Let f(x)= x3 + 3x2 - 1, where x
is a real variable. (This is the same function that was considered in
the supplementary class exercise.)
For each of the intervals on which a real zero was found in the supplementary class exercise, determine whether there is a zero on the left-hand half of the interval or on the right-hand half of the interval. (Conceiveably, the zero also could occur exactly at the midpoint . . . )
Sample: Suppose you have found that there is a zero
between -3 and -2 because
f(-3) < 0
and f(-2) > 0.
Then calculate f(-2.5), and
use that information (along with the Intermediate Value Theorem) to determine
which of the two halves of the interval could contain the zero.
Part b) Using your answer from part a) and a similar method (involving the Intermediate Value Theorem), determine which of the four quarters of the interval -3 < x < -2 could contain the zero of f(x). In other words, determine which of the following intervals has to contain the zero:
Part c) What would need to do to find an interval of length = 1/8 that contains a zero of f(x)? [You don't have to do the calculation; just write a couple of sentences to say what has to be done.]
Click here to view the related class exercise.
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