Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.1 Definition of the Derivative - Exercises - Page 104: 67

Answer

\begin{aligned} P^{\prime}(303)& \approx 0.00265 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(313)& \approx 0.004145 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(323)& \approx 0.006295 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(333)& \approx 0.00931 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(343)& \approx 0.013435 \mathrm{atm} / \mathrm{K} \end{aligned}

Work Step by Step

We use the symmetric difference quotient $$ f'(a) =\frac{f(a+h)-f(a-h)}{2h}$$ and from the given table, we have \begin{aligned} P^{\prime}(303)& \approx \frac{P(313)-P(293)}{20}=\frac{0.0808-0.0278}{20}=0.00265 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(313)& \approx \frac{P(323)-P(303)}{20}=\frac{0.1311-0.0482}{20}=0.004145 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(323)& \approx \frac{P(333)-P(313)}{20}=\frac{0.2067-0.0808}{20}=0.006295 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(333)& \approx \frac{P(343)-P(323)}{20}=\frac{0.3173-0.1311}{20}=0.00931 \mathrm{atm} / \mathrm{K}\\ P^{\prime}(343)& \approx \frac{P(353)-P(333)}{20}=\frac{0.4754-0.2067}{20}=0.013435 \mathrm{atm} / \mathrm{K} \end{aligned}
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