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Suppose that a company expects its annual profits t years from now to be f(t) dollars and that...

Question:

Suppose that a company expects its annual profits t years from now to be f(t) dollars and that interest is considered to be compounded continuously at an annual rate r. Then, the present value of all future profits can be shown to be {eq}FP= \int_{0}^{\infty}e^{-rt}f(t)\ dt. {/eq}

Find FP if r = 0.08 and f(t) = 100,000 + 1000t. The present value is _____ dollars.

Integration:

Definite integrals represent the area under the curve between the lower limit and the upper limit.

Unlike indefinite integrals, definite integrals do not contain a constant in the final answer.

Answer and Explanation:

Given r=0.08 and f(t) = 100000+1000t .

The present value will be :

{eq}FP= \int_{0}^{\infty}e^{-0.08t}(100000+100t)\ dt. {/eq}

The indefinite integral will be :

{eq}I = -12500e^{-0.08x} x - 1406250e^{-0.08x} + c {/eq}

Finite integral will be obtained by applying limits

{eq}FP = 0-(-1406250) {/eq}


Learn more about this topic:

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Definite Integrals: Definition

from Math 104: Calculus

Chapter 12 / Lesson 6
8.4K

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