Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.5 Compound Interest and Present Value - Exercises - Page 355: 17


$$PV=\frac{R}{r}\left(1-e^{-r T}\right)$$

Work Step by Step

Since $$ R(t) =R $$ Then \begin{align*} P V&=\int_{0}^{T} R(t) e^{-r t} d t\\ &=\int_{0}^{T} R e^{-r t} d t\\ &=\left.\frac{R}{-r} e^{-r t}\right|_{0} ^{T}\\ &=\frac{R}{-r}\left(e^{-r T}-e^{0}\right)\\ &=\frac{R}{r}\left(1-e^{-r T}\right) \end{align*}
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