Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.1 Definition of the Derivative - Exercises - Page 103: 40


$$\frac{3}{2 \sqrt{2}}$$$$y=\frac{3}{2 \sqrt{2}}t +\frac{7}{2 \sqrt{2}}$$

Work Step by Step

Let $f(x)=\sqrt{3t+5}$. Then \begin{aligned} f^{\prime}(-1)&=\lim _{h \rightarrow 0} \frac{f(-1+h)-f(-1)}{h}\\ &=\lim _{h \rightarrow 0} \frac{\sqrt{3 h+2}-\sqrt{2}}{h} \\ &=\lim _{h \rightarrow 0} \frac{\sqrt{3 h+2}-\sqrt{2}}{h} \cdot \frac{\sqrt{3 h+2}+\sqrt{2}}{\sqrt{3 h+2}+\sqrt{2}}\\ &=\lim _{h \rightarrow 0} \frac{3 h}{h(\sqrt{3 h+2}+\sqrt{2})}\\ &=\lim _{h \rightarrow 0} \frac{3}{\sqrt{3 h+2}+\sqrt{2}}=\frac{3}{2 \sqrt{2}} \end{aligned} The tangent line at $a=-1$ is \begin{aligned} y&=f^{\prime}(-1)(t+1)+f(1)\\ &=\frac{3}{2 \sqrt{2}}(t+1)+\sqrt{2}\\ &=\frac{3}{2 \sqrt{2}}t +\frac{7}{2 \sqrt{2}} \end{aligned}
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