Explanation Given information : f ' ( x ) exists throughout the interval [ x k − 1 , x k ] . Δ x k = x k − x k − 1 a n d Δ y k = f ( x k ) − f ( x k − 1 ) Theorem used: Mean value theorem: If f ( x ) is defined and continuous on [ a , b ] and differentiable on ( a , b ) , then there exist a number c in ( a , b ) such that f ' ( c ) = f ( b ) − f ( a ) b − a Proof: f ' ( x ) exists throughout the interval [ x k − 1 , x k ] . (Given) By Mean Value Theorem, there exist a number x * k in [ x k − 1 , x k ] such that f ' ( x * k ) = f ( x k ) − f ( x k − 1 ) x k − x k − 1 …..( 1) ( Δ x k ) 2 + ( Δ y k ) 2 = ( Δ x k ) 2 + ( Δ y k ) 2 ( Δ x k ) 2

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 6.4, Problem 55PS

To determine

To prove:There exist a number xk in [xk−1,xk] for which ( Δ x k )2+( Δ y k )2=1+[f'( x k )]2Δx

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Chapter 6.4, Problem 54PS

Chapter 6.4, Problem 56PS

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Chapter 6 Solutions

Calculus

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To prove: There exist a number x * k in [ x k − 1 , x k ] for which ( Δ x k ) 2 + ( Δ y k ) 2 = 1 + [ f ' ( x * k ) ] 2 Δ x

Solution Summary: The author proves that x*_k exists throughout the interval

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To prove:There exist a number xk in [xk−1,xk] for which ( Δ x k )2+( Δ y k )2=1+[f'( x k )]2Δx

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Chapter 6 Solutions

To prove: There exist a number x * k in [ x k − 1 , x k ] for which ( Δ x k ) 2 + ( Δ y k ) 2 = 1 + [ f ' ( x * k ) ] 2 Δ x

Solution Summary: The author proves that x*_k exists throughout the interval

Concept explainers

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To prove:There exist a number x*k in [xk−1,xk] for which ( Δ x k )2+( Δ y k )2=1+[f'( x * k )]2Δx

Want to see the full answer?

Chapter 6 Solutions

To prove:There exist a number xk in [xk−1,xk] for which ( Δ x k )2+( Δ y k )2=1+[f'( x k )]2Δx