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Thread: Please, help

  1. #1
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    Please, help

    24.) Let P (a,b) be a point on the curve X^1/2 + y ^1/2 = 1. Show that the slope of the tangent at P is - b/a^1/2
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    Quote Originally Posted by madman1611 View Post
    24.) Let P (a,b) be a point on the curve X^1/2 + y ^1/2 = 1. Show that the slope of the tangent at P is - b/a^1/2
    Use implicit differentiation:
    $\displaystyle \sqrt{x} + \sqrt{y} = 1$

    So taking the derivative:
    $\displaystyle \frac{1}{2\sqrt{x}} + \frac{y^{\prime}}{2 \sqrt{y}} = 0$

    $\displaystyle \frac{y^{\prime}}{2 \sqrt{y}} = -\frac{1}{2\sqrt{x}}$

    $\displaystyle y^{\prime} = -\frac{\sqrt{y}}{\sqrt{x}}$

    So at the point (x, y) = (a, b):

    $\displaystyle y^{\prime} = -\frac{\sqrt{b}}{\sqrt{a}}$

    -Dan
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  3. #3
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    Hello, madman!

    24) Let $\displaystyle P(a,b)$ be a point on the curve: .$\displaystyle x^{\frac{1}{2}}+ y^{\frac{1}{2}}\;=\;1$

    Show that the slope of the tangent at $\displaystyle P$ is: .$\displaystyle -\left(\frac{b}{a}\right)^{\frac{1}{2}}$
    Differentiate implicitly: .$\displaystyle \frac{1}{2}\!\cdot\!x^{-\frac{1}{2}} + \frac{1}{2}\!\cdot\!y^{-\frac{1}{2}}\left(\frac{dy}{dx}\right) \;=\;0\quad\Rightarrow\quad\frac{dy}{dx}\;=\;-\frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}} \;=\;-\left(\frac{y}{x}\right)^{\frac{1}{2}}$

    At $\displaystyle P(a,b)\!:\;\;\frac{dy}{dx}\;=\;\left(\frac{b}{a}\r ight)^{\frac{1}{2}} $

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