Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - 12.3 Taylor Polynomials at 0 - 12.3 Exercises - Page 632: 40



Work Step by Step

The profit when x thousand tons of apples are sold is $P(x)=\frac{20+x^{2}}{50+x}$ $P'(x)=\frac{2x(50+x)-1(20+x^{2})}{(50+x)^{2}}=\frac{x^{2}+100x-20}{x^{2}+100x+2500}$ $P''(x)=\frac{(2x+100)(x^{2}+100x+2500)-(2x+100)(x^{2}+100x-20)}{(x^{2}+100x+2500)^{2}}=\frac{2x^{3}+15000x+300x^{2}+250000-(2x^{3}+300x^{2}+9960x-2000)}{(x^{2}+100x+2500)^{2}}=\frac{5040x+27000}{(x^{2}+100x+2500)^{2}}$ $P(0)=\frac{2}{5}$ $P'(0)=\frac{-1}{125}$ $P''(0)=\frac{27}{6250}$ $P_2(x)=\frac{2}{5}-\frac{1}{125}+\frac{\frac{27}{6250}}{2!}x^{2}$ $P_{2}(x)=\frac{2}{5}+\frac{27}{3125}x^{2}$ $P_{2}(0.3)=\frac{2}{5}+\frac{27}{3125}x^{2} \approx 0.4$
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