Calculus: Early Transcendentals 8th Edition

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

Chapter 2 - Section 2.7 - Derivatives and Rates of Change - 2.7 Exercises: 22

Answer

$f(4)=3$ and $f'(4)=\frac{1}{4}$

Work Step by Step

The tangent line $(l):y=f(x)$ is at point $A(4,3)$, so point $A(4,3)$ also lies in $(l)$. Therefore, $f(4)=3$ The equation of the tangent line $l$ would have the following form: $$(l): y=ax+b$$ Since $l$ passes through point $A(4,3)$, we apply the equation of $l$ to $A$, which means $$4a+b=3\hspace{1cm}(1)$$ $l$ also passes through point $B(0,2)$, we also can apply the equation of $l$ to $B$, which means $$0a+b=2$$$$b=2\hspace{1cm}(2)$$ Apply (2) to (1), we have $$4a+2=3$$$$a=\frac{1}{4}$$ Since $a$ is the slope of the tangent line $l$ at point $A(4,3)$, $f'(4)=a=\frac{1}{4}$
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