3. In this question we look at the bisection method of computing a square root. (a) Suppose a, b > 0 and a? < b². Prove that a < b. (Hint: Factor 62 - a².) (b) Prove that if x > 0 and x² E [4,9] then xE [2,3]. (c) Define the sequence of closed intervals [a1, b1], [a2, b2], . ... as follows: antba] if (nbn) 2 7 [an [41, b1] = [2, 3), [an+1; bn+1] = for all n E N. [antbn, bn] if (an tbn)² < 7 2 Compute the first 4 intervals. (d) Prove that (an) and (bn) are bounded monotone sequences.
3. In this question we look at the bisection method of computing a square root. (a) Suppose a, b > 0 and a? < b². Prove that a < b. (Hint: Factor 62 - a².) (b) Prove that if x > 0 and x² E [4,9] then xE [2,3]. (c) Define the sequence of closed intervals [a1, b1], [a2, b2], . ... as follows: antba] if (nbn) 2 7 [an [41, b1] = [2, 3), [an+1; bn+1] = for all n E N. [antbn, bn] if (an tbn)² < 7 2 Compute the first 4 intervals. (d) Prove that (an) and (bn) are bounded monotone sequences.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
3
![3. In this question we look at the bisection method of computing a square root.
(a) Suppose a, b > 0 and a? < b². Prove that a < b. (Hint: Factor b2 – a2.)
(b) Prove that if x > 0 and x² E [4, 9] then xE [2,3].
(c) Define the sequence of closed intervals [a1, b1], [a2, b2], . .. as follows:
S [an, an tbn] if (antbn)² > 7
, bn] if ()
2
[a1, b1] = [2, 3],
[an+1, bn+1]
for all n e N.
2
[an+bn
´an+bn
< 7
Compute the first 4 intervals.
(d) Prove that (an) and (b,n) are bounded monotone sequences.
(e) Consider a1, b1, a2, b2, . . ., i.e., the sequence (Cn) defined by
Ja(n+1)/2 if n is odd
Cn =
bn/2
if n is even
Prove that (cn) is a Cauchy sequence.
(f) Based on your proof, find a positive integer N such that |Cm – Cn| < 0.01 for all m, n > N.
(g) Prove that (Cn) → c where c= sup{a: n e N} = inf{b, : n E N}.
(h) Prove that c> 0 and c? = 7. (Hence c =
V7.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab68ef6f-9760-4c1f-88a8-6e5ceb2168b2%2F986fbbf9-946b-4e4a-bae3-bac801c38f03%2Fkly0ffg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. In this question we look at the bisection method of computing a square root.
(a) Suppose a, b > 0 and a? < b². Prove that a < b. (Hint: Factor b2 – a2.)
(b) Prove that if x > 0 and x² E [4, 9] then xE [2,3].
(c) Define the sequence of closed intervals [a1, b1], [a2, b2], . .. as follows:
S [an, an tbn] if (antbn)² > 7
, bn] if ()
2
[a1, b1] = [2, 3],
[an+1, bn+1]
for all n e N.
2
[an+bn
´an+bn
< 7
Compute the first 4 intervals.
(d) Prove that (an) and (b,n) are bounded monotone sequences.
(e) Consider a1, b1, a2, b2, . . ., i.e., the sequence (Cn) defined by
Ja(n+1)/2 if n is odd
Cn =
bn/2
if n is even
Prove that (cn) is a Cauchy sequence.
(f) Based on your proof, find a positive integer N such that |Cm – Cn| < 0.01 for all m, n > N.
(g) Prove that (Cn) → c where c= sup{a: n e N} = inf{b, : n E N}.
(h) Prove that c> 0 and c? = 7. (Hence c =
V7.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning