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A continuous money flow begins at $\$ 12,000$ and increases exponentially at 6\% per year for 12 years. Find the (a) present value and (b) final value, if money is worth $r=5 \%$ compounded continuously.

(a) $\$ 152,996.22$(b) $\$ 278,777.29$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 8

Applications of the Definite Integral

Integrals

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

02:21

Money Flow The rate of a c…

01:50

The rate of a continuous m…

06:27

Assuming an interest rate …

03:14

An amount of invested mone…

01:49

Find the present value of …

02:14

03:41

Money is deposited steadil…

02:37

Find the future value of $…

10:09

this problem. We're looking for the present value and the final amount or the accumulated value off. Uh, this problem. So we'll start with the present Value P is equal to the integral from 0 to 8 because we're looking for the 1st 8 years. Um, so 5000 was the rate of continuous flow of money. Uh, but also that's decreasing exponentially at 1% per year. So that's telling us e to the negative 0.1 tea. So this is the flow function times E to the negative 0.8 t. That's given as the interest rate of 8% peachy so we can use our exponents laws here group the ease so we'll have 5000 e to the negative 0.9 t d t. And, uh, taking the anti derivative, that's gonna be negative. 5000 over a 0.9 e to the negative 0.0 90 evaluated from 0 to 8. So when we put eight into the expression will get negative. 27,000 and 41 0.792 We put zero in there. We'll get 55,555 00.56 and that gives us a present value of $28,513.76. Part B is looking for the accumulated value, so this is really just a equals E to the 0.8 times eight years, and normally we would just write the entire integral. But really, we know that that's the value of P. So when we multiply E to the 0.64 with the value that we got in part a 20,513 0.76 that's gonna give us our accumulated value of 54,000 and $75 and 81 cents.

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