Explanation Definition used: Improper integral : “If f is continuous on [ a , ∞ ) , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . If the limit exists, then the improper integral converges; otherwise diverges”. Calculation: The rate of change of the voting population of a city with respect to time t in years is given by, N ′ ( t ) = 100 t ( 1 + t 2 ) 2 . Obtain the value of indefinite integral ∫ 100 t ( 1 + t 2 ) 2 d t by substitution method. Let the variable v be 1 + t 2 . Differentiate v with respect to x . d d t ( v ) = d d t ( 1 + t 2 ) d v d t = 2 t d v = 2 t d t t d t = d v 2 Substitute v = 1 + t 2 and t d t = d v 2 in the integral ∫ 100 t ( 1 + t 2 ) 2 d t , ∫ 100 t ( 1 + t 2 ) 2 d t = ∫ 100 d v 2 v 2 = 50 ∫ ( 1 v 2 ) d v = 50 ∫ v − 2 d v = 50 ( v − 2 + 1 − 2 + 1 ) + C [ ∵ ∫ x n d x = x n + 1 n + 1 + C ] Simplify further as, ∫ 100 t ( 1 + t 2 ) 2 d t = 50 ( v − 1 − 1 ) + C = − 50 v + C = − 50 ( 1 + t 2 ) + C [ ∵ v = 1 + t 2 ] Thus, ∫ 100 t ( 1 + t 2 ) 2 d t = − 50 ( 1 + t 2 ) + C (1) Thus, N ( t ) = − 50 ( 1 + t 2 ) + C

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 11, Problem 48RE

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To find: The increase in the population during the next 3 years and the total increase in the voting population.

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Chapter 11, Problem 47RE

Chapter A.1, Problem 1MP

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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The increase in the population during the next 3 years and the total increase in the voting population.

Solution Summary: The author calculates the rate of change of the voting population of a city with respect to time t in years.

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To find: The increase in the population during the next 3 years and the total increase in the voting population.

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Chapter 11 Solutions