Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.1 - Derivatives of Polynomials and Exponential Functions - 3.1 Exercises: 67


The equation of the second-order polynomial $P$ is $$P=x^2-x+3$$

Work Step by Step

A second-order polynomial $P$ would have the following equation form: $$P=ax^2+bx+c$$ Therefore, - First derivative: $P'=a\frac{d}{dx}(x^2)+b\frac{d}{dx}(x)+\frac{d}{dx}(c)=2ax+b$ - Second derivative: $P''=2a\frac{d}{dx}(x)+\frac{d}{dx}(b)=2a$ We know that $P''(2)=2$. So, $$2a=2$$ $$a=1$$ We also know that $P'(2)=3$. So, $$2a\times2+b=3$$ $$2\times1\times2+b=3$$ $$b=-1$$ Finally, we know that $P(2)=5$. So, $$a\times(2^2)+b\times2+c=5$$ $$1\times4+(-1)\times2+c=5$$ $$c=3$$ Therefore, the equation of the second-order polynomial $P$ is $$P=x^2-x+3$$
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