Explanation Let f be a real function which is continuous on the closed interval [ a , b ] . Let ∫ a b f ( x ) d x be the definite integral of f ( x ) over [ a , b ] . Then m ( b − a ) ≤ ∫ a b f ( x ) d x ≤ M ( b − a ) where, M is the maximum value of f on [ a , b ] and m is the minimum value of f on [ a , b ] Given: To prove that a) 2 ≤ ∫ 1 2 2 x d x ≤ 4 b) 1 9 ≤ ∫ 1 2 3 − x d x ≤ 1 3 Formula used: Let f be a real function which is continuous on the closed interval [ a , b ] . Let ∫ a b f ( x ) d x be the definite integral of f ( x ) over [ a , b ] . Then m ( b − a ) ≤ ∫ a b f ( x ) d x ≤ M ( b − a ) where, M is the maximum value of f on [ a , b ] and m is the minimum value of f on [ a , b ] Proof: a) The graph of 2 x in the interval [1,2] is: From the graph, it is clear that the maximum value of 2 x in the interval [ 1 , 2 ] is 4 and the minimum value of 2 x in the interval [ 1 , 2 ] is 2...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 5, Problem 83CRE

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To Prove:

a) $2 \leq \int_{1}^{2} 2^{x} d x \leq 4$ b) $\frac{1}{9} \leq \int_{1}^{2} 3^{- x} d x \leq \frac{1}{3}$

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Chapter 5, Problem 82CRE

Chapter 5, Problem 84CRE

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Chapter 5 Solutions

Calculus - Standalone book

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

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To Prove: a) 2 ≤ ∫ 1 2 2 x d x ≤ 4 b) 1 9 ≤ ∫ 1 2 3 − x d x ≤ 1 3

Solution Summary: The author explains how to prove that f is a real function which is continuous on the closed interval.

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To Prove:

a) $2 \leq \int_{1}^{2} 2^{x} d x \leq 4$ b) $\frac{1}{9} \leq \int_{1}^{2} 3^{- x} d x \leq \frac{1}{3}$

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To Prove: a) 2 ≤ ∫ 1 2 2 x d x ≤ 4 b) 1 9 ≤ ∫ 1 2 3 − x d x ≤ 1 3

Solution Summary: The author explains how to prove that f is a real function which is continuous on the closed interval.

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To Prove: a) 2≤∫122xdx≤4 b) 19≤∫123−xdx≤13

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Chapter 5 Solutions

To Prove:

a) $2 \leq \int_{1}^{2} 2^{x} d x \leq 4$ b) $\frac{1}{9} \leq \int_{1}^{2} 3^{- x} d x \leq \frac{1}{3}$