Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.5 Higher Derivatives - Exercises - Page 135: 27

Answer

\begin{align*} y(0)&=d \\ y'(0)&= c\\ y''(0)&= 2b\\ y'''(0)&= 6a\\ y^{(4)}(0)&= 24\\ y^{(5)}(0)&= 0 \end{align*}

Work Step by Step

Given $$y = x^4 + ax^3 + bx^2 + cx + d $$ We compute the derivatives \begin{align*} y'&= 4x^3+3ax^2+2bx+c\\ y''&= 12x^2+6ax+2b\\ y'''&= 24x+6a\\ y^{(4)}&= 24\\ y^{(5)}&= 0 \end{align*} Now, we can plug in $x=0$ \begin{align*} y(0)&=d \\ y'(0)&= c\\ y''(0)&= 2b\\ y'''(0)&= 6a\\ y^{(4)}(0)&= 24\\ y^{(5)}(0)&= 0 \end{align*}
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