Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 964: 41


The solution is $$R_t(0,1)=1.$$

Work Step by Step

First, find the derivative: $$R_t=(t e^{s/t})'_t=(t)'_te^{s/t}+t(e^{s/t})'_t=e^{s/t}+te^{s/t}(s/t)'_t=e^{s/t}+te^{s/t}\left(-\frac{s}{t^2}\right)=e^{s/t}-\frac{s}{t}e^{s/t}=e^{s/t}\left(1-\frac{s}{t}\right).$$ Now calculate by direct substitution $s=0$ and $t=1$: $$R_t(0,1)=e^{0/1}\left(1-\frac{0}{1}\right)=1\cdot1=1.$$
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