Calculus EXERCISES 4.4 The Mean Value Theorem

Question

The Intermediate Value Theorem can be used to approximate a root. The following is an example of binary search in computer science. Suppose you want to approximate . You know that it is between 2 and 3. If you consider the function , then note that  and . Therefore by the Intermediate Value Theorem, there is a value,  such that . Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the actual root.  will either be less than 0 or greater than 0. You can use the Intermediate Value Theorem again replacing 2.5 with the previous endpoint that has the same sign as 2.5. Continuing this process gives a sequence of approximations  with . How many iterations must you do in order to be within 0.0009765625 of the root?

Answer 1

We can think of our maximum allowable error as the nth term of a geometric sequence with common ratio, r, = 1/2 (since with each iteration we halve the width of the interval we know the root lies in) and a₁ = 1/2, since our first guess is guaranteed to be within 1/2 of the actual root.

a_n = 1/2·1/2^(n-1) = .0009765625

1/2ⁿ = .0009765625

2ⁿ = 1,024

n = 10