Calculus (3rd Edition)

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

Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.4 Taylor Polynomials - Exercises - Page 492: 2


\begin{aligned} T_{2}(x) &= 1-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2}\\ T_3(x)&=1-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2} \end{aligned}

Work Step by Step

Since \begin{array}{ll} {f(x)=\sin x,} & {f\left(\frac{\pi}{2}\right)=1} \\ {f^{\prime}(x)=\cos x,} & {f^{\prime}\left(\frac{\pi}{2}\right)=0} \\ {f^{\prime \prime}(x)=-\sin x,} & {f^{\prime \prime}\left(\frac{\pi}{2}\right)=-1} \\ {f^{\prime \prime \prime}(x)=-\cos x,} & {f^{\prime \prime \prime}\left(\frac{\pi}{2}\right)=0} \end{array} Then \begin{aligned} T_{2}(x) &=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2} \\ &=1+(0)\left(x-\frac{\pi}{2}\right)-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2} \\ &=1-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2} \end{aligned} and \begin{aligned} T_{3}(x) &=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\frac{f^{\prime \prime \prime}(a)}{3 !}(x-a)^{3} \\ &=1+(0)\left(x-\frac{\pi}{2}\right)-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2}+(0)\left(x-\frac{\pi}{2}\right)^{3} \\ &=1-\frac{1}{2}\left(x-\frac{\pi}{2}\right)^{2} \end{aligned}
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