Single Variable Calculus

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Single Variable Calculus, Chapter 3, 3.6, Section 3.6, Problem 28

Expert Answers

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Find an equation of the tangent to the curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 4$ at the point $\displaystyle \left(-3\sqrt{3},1\right)$ using Implicit Differentiation.

If $y'= m\text{ (slope)}$ then,

$\displaystyle \frac{d}{dx} \left(x^{\frac{2}{3}}\right) + \frac{d}{dx} \left(y^{\frac{2}{3}}\right) = \frac{d}{dx} (4)$

$ \begin{equation} \begin{aligned} \frac{2}{3} (x)^{\frac{-1}{3}} + \frac{2}{3} (y)^{\frac{-1}{3}} \frac{dy}{dx} &= 0\\ \\ \frac{2}{3\sqrt[3]{x}} + \frac{2}{3 \sqrt[3]{y}} y' &= 0\\ \\ \frac{2}{3\sqrt[3]{y}} y' &= \frac{-2}{3 \sqrt[3]{x}}\\ \\ y' &= \frac{-2(3\sqrt[3]{y})}{2(3\sqrt[3]{x})}\\ \\ y' &= \frac{-\sqrt[3]{y}}{\sqrt[3]{x}} \end{aligned} \end{equation} $

For $x = -3\sqrt{3}$ and $y =1 $, we obtain

$ \begin{equation} \begin{aligned} y' = m &= - \frac{\sqrt[3]{y}}{\sqrt[3]{x}}\\ \\ m &= - \frac{\sqrt[3]{1}}{\sqrt[3]{3-\sqrt{3}}}\\ \\ m &= \frac{1}{\sqrt{3}} \end{aligned} \end{equation} $

Using the point slope form

$ \begin{equation} \begin{aligned} y - y_1 &= m(x-x_1)\\ \\ y - 1 & = \frac{1}{\sqrt{3}} [x - ( -3 \sqrt{3})]\\ \\ y - 1 & = \frac{1}{\sqrt{3}} (x + 3\sqrt{3})\\ \\ y &= \frac{x+3\sqrt{3}}{\sqrt{3}} +1 \\ \\ y &= \frac{x+3\sqrt{3}+\sqrt{3}}{\sqrt{3}}\\ \\ y &= \frac{x+4\sqrt{3}}{\sqrt{3}} && \text{Equation of the tangent line at } (-3\sqrt{3},1) \end{aligned} \end{equation} $

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