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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Present Value In Exercises 13-16, find the present value of the income c (in dollars) over t 1 years at the given annual inflation rate r.

c = 20 , 000 + 100 t e t / 2 , r = 5 % , t 1 = 5  years

To determine

To calculate: The present value of the income (c, in dollar) of 20,000+100tet/2 (c, in dollar) over a time period (t1) of 5 years at an annual inflation rate (r) of 5%.

Explanation

Given Information:

The income, c, is 20,000+100tet/2 in dollar and the time period, t1, is 5 years.

Also the rate of the annual inflation, r, is 5%.

Formula used:

The expression for the present value is written as,

Present value=0t1c(t)ertdt

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,

udv=uvvdu

Following is the guideline that should be followed while solving a problem by integration by parts:

1. Find the most complicated portion of the integrand and try to letter it as dv so that it can fit a fundamental integration rule. Then, the remaining factor or factors of the integrand will be u.

2. Find the factor whose derivative is simple and consider it as u. Then, the remaining factor or factors of the integrand will be dv.

Note: dv should always include the term dx of the original integrand.

Calculation:

Consider the expression of the present value.

Present value=0t1c(t)ertdt

Substitute 5 for t1, 20,000+100tet/2 for c(t) and 0.05 for r.

Present value=05(20,000+100tet/2)e0.05tdt

Substitute u=e0.05t then du=0.05e0.05tdt.

Then

05(20,000+100tet/2)e0.05tdt=20000u102000ln(u)0.05u10du=10.0520000u102000ln(u)u10du=20000u102000ln(u)u10=20000u10u102000ln(u)u10

Further calculate it,

05(20,000+100tet/2)e0

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