Explanation Given Information: The value of income, c , is 1500 t dollar. The time period, t 1 , is 10 years . The rate of the annual inflation, r , is 7 % . Formula used: The expression for the present value is written as, Present value = ∫ 0 t 1 c ( t ) e − r t d t Here, r is the rate of inflation and c ( t ) represents the income which is a function of time t . The method of integration by parts: If v and u are two differentiable function of x . Then, ∫ u d v = u v − ∫ v d u Steps to solve the integral problems: Step1: At first find the most complicated portion of the integrand and try to letter it as d v so that it can fit a fundamental integration rule. Then, the remaining factor or factors of the integrand will be u . Step2: First find the factor whose derivative is simple and consider it as u and then the remaining factor or factors of the integrand will be d v and d v should always include the term d x of the original integrand. Calculation: Recall the expression of the present value. Present value = ∫ 0 t 1 c ( t ) e − r t d t Substitute 10 for t 1 , 1500 t for c ( t ) and 0.07 for r . Present value = ∫ 0 10 ( 1500 t ) e − 0.07 t d t = ∫ 0 10 1500 t e − 0.07 t d t = 1500 ∫ 0 10 t e − 0.07 t d t In the above integrand, the simplest portion of the integrand is t . So, consider, u = t and the remaining factors as d v = e − 0.07 t d t . Therefore, d u = d t And, d v = e − 0.07 t d t Integrate the above expression for v . d v = ∫ e − 0

Calculus: An Applied Approach (Providence College: MTH 109)

9th Edition

ISBN: 9781285142616

Author: Ron Larson

Publisher: CENGAGE C

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 6, Problem 14RE

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To calculate: The present value of the income (c, in dollar) of 1500t (c, in dollar) over a time period (t1) of 10 years at an annual inflation rate (r) of 7%.

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Chapter 6, Problem 13RE

Chapter 6, Problem 15RE

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Chapter 6 Solutions

Calculus: An Applied Approach (Providence College: MTH 109)

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The present value of the income ( c , in dollar) of 1500 t ( c , in dollar) over a time period ( t 1 ) of 10 years at an annual inflation rate ( r ) of 7 % .

Solution Summary: The author explains how to calculate the present value of the income of 1500t dollar over a time period.

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To calculate: The present value of the income (c, in dollar) of 1500t (c, in dollar) over a time period (t1) of 10 years at an annual inflation rate (r) of 7%.

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Chapter 6 Solutions