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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 √5. You know that it is between 2 and 3. If you consider the function f(x) = x² – 5, then note that f(2) < 0 and f(3) > 0. Therefore by the Intermediate Value Theorem, there is a value, 2 ≤ c ≤ 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) 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 on with x₁ = 2.5. How many iterations must you do in order to be within 0.015625 of the root?

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 √5. You know that it is between 2 and 3. If you consider the function f(x) = x² – 5, then note that f(2) < 0 and f(3) > 0. Therefore by the Intermediate Value Theorem, there is a value, 2 ≤ c ≤ 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) 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 on with x₁ = 2.5. How many iterations must you do in order to be within 0.015625 of the root?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 √5. You know that it is between 2 and 3. If you consider the function f(x) = x² – 5, then note that f(2) < 0 and f(3) > 0. Therefore by the Intermediate Value Theorem, there is a value, 2 ≤ c ≤ 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) 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 on with x₁ = 2.5. How many iterations must you do in order to be within 0.015625 of the root?
Transcribed Image Text: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 √5. You know that it is between 2 and 3. If you consider the function f(x) = x² – 5, then note that f(2) < 0 and f(3) > 0. Therefore by the Intermediate Value Theorem, there is a value, 2 ≤ c ≤ 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) 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 on with x₁ = 2.5. How many iterations must you do in order to be within 0.015625 of the root?
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