To determine To find: The capital value at an interest rate of 5% for an investment of $6,000 and the income increases linearly at a rate of $200 per year. Explanation Formula used: The capital value is C V = ∫ 0 ∞ Re − r t d t , where r is annual rate of interest and t is time. 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 it exists and finite, then it is converges otherwise diverges.” Theorem used: Fundamental Theorem: “Let f be a continuous function on [ a , b ] . If F is anti-derivative of f on [ a , b ] , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) .” Calculation: The income investment is $6,000 per year. Since the income increases at a rate of $200 per year, the total in come become R ( t ) = 6000 + 200 t , at t years. Substitute the value of R ( t ) = 6000 + 200 t and r = 0.05 in the above formula and compute the capital value. C V = ∫ 0 ∞ ( 6 , 000 + 200 t ) e − 0.05 t d t = lim b → ∞ ∫ 0 b ( 6 , 000 + 200 t ) e − 0.05 t d t ( ∵ Definition used ) = ( 6 , 000 ) lim b → ∞ ∫ 0 b e − 0.05 t d t + ( 200 ) lim b → ∞ ∫ 0 b t e − 0

Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

11th Edition

ISBN: 9780133886849

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 8.4, Problem 46E

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To find: The capital value at an interest rate of 5% for an investment of $6,000 and the income increases linearly at a rate of $200 per year.

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Chapter 8.4, Problem 45E

Chapter 8.4, Problem 47E

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Calculus With Applications, Books a la Carte Plus MyLab Math Package (11th Edition)

Ch. 8.1 - YOUR TURN 1 Find Ch. 8.1 - YOUR TURN 2 Find Ch. 8.1 - YOUR TURN 3 Find Ch. 8.1 - YOUR TURN 4 Find . Ch. 8.1 - YOUR TURN 5 Find Ch. 8.1 - Prob. 1WE Ch. 8.1 - Find the following. W2. Ch. 8.1 - Find the following. W3. Ch. 8.1 - Find the following. W4. Ch. 8.1 - Use integration by parts to find the integrals in...

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