University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.1 - Tangents and the Derivative at a Point - Exercises - Page 118: 8


The equation of the tangent line is $y=2x+3$.

Work Step by Step

The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$ $$y=f(x)=\frac{1}{x^2}\hspace{1cm}A(-1,1)$$ 1) Find the slope $m$ of the tangent: $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$ Here $a=-1$ and $f(a)=b=1$. $$m=\lim_{h\to0}\frac{\frac{1}{(h-1)^2}-1}{h}=\lim_{h\to0}\frac{\frac{1-(h-1)^2}{(h-1)^2}}{h}$$ $$m=\lim_{h\to0}\frac{1-(h^2-2h+1)}{h(h-1)^2}=\lim_{h\to0}\frac{2h-h^2}{h(h-1)^2}$$ $$m=\lim_{h\to0}\frac{2-h}{(h-1)^2}$$ $$m=\frac{2-0}{(0-1)^2}=\frac{2}{(-1)^2}=\frac{2}{1}=2$$ 2) Find the equation of the tangent line at $A(-1,1)$: The tangent line would have this form: $$y=2x+m$$ Substitute $A(-1,1)$ here to find $m$: $$2\times(-1)+m=1$$ $$-2+m=1$$ $$m=3$$ So the equation of the tangent line is $y=2x+3$.
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