Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous functions differentiable on (a,b). Then there exists a point c E (a, b) such that (F(b) – f(a)) o'(c) =f'(c)(@(b) – 9(a)).
Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous functions differentiable on (a,b). Then there exists a point c E (a, b) such that (F(b) – f(a)) o'(c) =f'(c)(@(b) – 9(a)).
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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Prove the theorem
![Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous
functions differentiable on (a,b). Then there exists a point c E (a,b) such that
(S(b) – f(a)) o'(c) = f'(c)(@(b) – p(a)).
The mean value theorem has the distinction of being one of the few theorems commonly cited
in court. That is, when police measure the speed of cars by aircraft, or via cameras reading license
plates, they measure the time the car takes to go between two points. The mean value theorem then
says that the car must have somewhere attained the speed you get by dividing the difference in
distance by the difference in time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2Fbbbb9f40-11c6-421d-a019-586e600cc8d3%2Fpf2vb8w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous
functions differentiable on (a,b). Then there exists a point c E (a,b) such that
(S(b) – f(a)) o'(c) = f'(c)(@(b) – p(a)).
The mean value theorem has the distinction of being one of the few theorems commonly cited
in court. That is, when police measure the speed of cars by aircraft, or via cameras reading license
plates, they measure the time the car takes to go between two points. The mean value theorem then
says that the car must have somewhere attained the speed you get by dividing the difference in
distance by the difference in time.
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