   Chapter 13.7, Problem 37E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Capital value Suppose that a continuous income stream has an annual rate of flow at time t given by f ( t ) =   A , where A is a constant. If the interest rate is r ( as a dec­imal, r > 0 ) compounded continuously, show that the capital value of the stream is A/r.

To determine

To prove: The capital value of the stream is Ar if annual flow rate at time t is given by f(t)=A and the interest rate is r.

Explanation

Given Information:

The annual flow rate at time t is given as, f(t)=A and the interest rate is r.

Formula used:

The capital value for a continuous income stream f(t) and rate r compounded continuously is,

C=0f(t)ertdt

The improper integral in form of a limit of integral is,

af(x)dx=limbabf(x)dx

The exponential rule of integrals,

exdx=ex+C

Proof:

Consider the provided that annual flow rate at time t is, f(t)=A

Now, the capital value for a continuous income stream f(t) and rate r compounded continuously is given as,

C=0f(t)ertdt

Substitute A for f(t) in above formula to get,

C="

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Exercises 7-28, perform the indicated operations and simplify each expression. 14. xexx+1+ex

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### The harmonic series is: 1 + 2 + 3 + 4 + …

Study Guide for Stewart's Multivariable Calculus, 8th

#### True or False: f(x) = tan x is differentiable at x=2.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 