Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises - Page 853: 32


$\approx 70.53^\circ$ at $x=\frac{\pi}{4}$

Work Step by Step

$f(x)=sinx$ and $g(x)=cosx$ ; $ 0\leq x\leq \pi/2$ $sinx=cosx$ when $x=\frac{\pi}{4}$ $f'(\frac{\pi}{4})=cos\frac{\pi}{4}=\frac{1}{\sqrt 2}$ and $g'(\frac{\pi}{4})=-sin\frac{\pi}{4}=-\frac{1}{\sqrt 2}$ Thus, $a= \lt \sqrt 2,1\gt$ and $b =\lt \sqrt 2,-1 \gt$ $ \theta = cos^{-1}\dfrac{a \cdot b}{|a||b|}=cos^{-1}\dfrac{\sqrt 2\sqrt 2+1(-1)}{\sqrt {3}\sqrt {3}}$ $ \theta = cos^{-1}\dfrac{a \cdot b}{|a||b|}=cos^{-1}\dfrac{1}{3}\approx 70.53^\circ$ at $x=\frac{\pi}{4}$
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