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Thread: equation of tangent line

  1. #1
    Junior Member shepherdm1270's Avatar
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    equation of tangent line

    Find the equation of the tangent line at the given value (in form Ax+By+C=0) Implicit Differentiation must be used:

    If you could show me the steps, as well, that would be great... yet again, i'm totally lost.

    a) x^2 y^3 = 8 ; (-1,2)

    b) y^3 + xy - y = 8x^4 ; x = 1



    thanks again, guys :-/
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  2. #2
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    Hello, shepherdm1270!

    Here's the first one . . .


    Find the equation of the tangent line at the given value
    (in the form $\displaystyle Ax+By+C\:=\:0$).
    Implicit Differentiation must be used:

    $\displaystyle a)\;\;x^2y^3 \:=\:8\qquad(\text{-}1,2)$

    Differentiate: .$\displaystyle x^2\!\cdot\!3y^2y' + 2x\!\cdot\!y^3 \:=\:0\quad\Rightarrow\quad y' \:=\:-\frac{2y}{3x}$

    At $\displaystyle (\text{-}1,2)\!:\;\;y' \:=\:-\frac{2(2)}{3(\text{-}1)} \:=\:\frac{4}{3}\quad\hdots$ The slope of the tangent is $\displaystyle \frac{4}{3}$


    The line through $\displaystyle (\text{-}1,2)$ with slope $\displaystyle \frac{4}{3}$ is:

    . . $\displaystyle y - 2 \;=\;\frac{4}{3}[x - (\text{-}1)]\quad\Rightarrow\quad y - 2 \:=\:\frac{4}{3}x + \frac{4}{3}$


    Multiply through by $\displaystyle 3\!:\;\;3y - 6 \:=\:4x + 4\quad\Rightarrow\quad \boxed{ 4x - 3y + 10 \:=\:0}$

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  3. #3
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    Quote Originally Posted by shepherdm1270 View Post
    b) $\displaystyle y^3 + xy - y = 8x^4$ ; x = 1
    First of all if you sub $\displaystyle x = 1$ you'll get
    $\displaystyle y^3=8$, which means $\displaystyle y=2$

    and if we differentiate:
    $\displaystyle y^3 + xy - y = 8x^4 $

    $\displaystyle 3y^2\frac{dy}{dx} + x\frac{dy}{dx} + y - \frac{dy}{dx} = 32x^3 $

    $\displaystyle \frac{dy}{dx}(3y^2 + x - 1) = 32x^3 - y$

    $\displaystyle \frac{dy}{dx} = \frac{32x^3 - y}{3y^2 + x -1}$

    sub $\displaystyle x=1$ and $\displaystyle y=2$ you'll get $\displaystyle \frac{dy}{dx} = \frac{5}{2}$

    $\displaystyle y-2 =\frac{5}{2}(x-1)$

    $\displaystyle 2y-4=5x-5$

    $\displaystyle 5x-2y-1=0$
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