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Math Help - Implicit differentiation.

  1. #1
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    Implicit differentiation.

    Here's the problem:

    "Consider the curve defined by the implicit equation y+5cosy=x^2+2.

    a) Find dy/dx in terms of x and y.

    b) Find the equation of the tangent line at the point shown where x=2.

    c) Find two positive values of x in the portion of the curve shown at which there is a vertical tangent line." (There's a picture here of the graph, given so that one could visually look to see where they think a vertical tangent line might be.) "Use calculus to find the values to 3 decimal places. You can use your calculator to solve the equation that must be true for a vertical tangent line."

    Any help on how to solve these, please?
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  2. #2
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    Quote Originally Posted by Rumor View Post
    Here's the problem:

    "Consider the curve defined by the implicit equation y+5cosy=x^2+2.

    a) Find dy/dx in terms of x and y.

    b) Find the equation of the tangent line at the point shown where x=2.

    c) Find two positive values of x in the portion of the curve shown at which there is a vertical tangent line." (There's a picture here of the graph, given so that one could visually look to see where they think a vertical tangent line might be.) "Use calculus to find the values to 3 decimal places. You can use your calculator to solve the equation that must be true for a vertical tangent line."

    Any help on how to solve these, please?
    For part 'a'

    y+5cosy=x^2+2

    Differentiating:

    y'+5\frac{d}{dx}cos(y)=2x

    The derivative \frac{d}{dx}cos(y) must be evaluated using the chain rule.

    \frac{d}{dx}cos(y)=-y'sin(y)

    So the equation becomes:

    y'-5y'sin(y)=2x

    y'=\frac{2x}{1-5sin(y)}
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  3. #3
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    b) Find the equation of the tangent line at the point shown where x=2.

    It seems like there is a problem here. There is no way to isolate y in any of these equations. So it's impossible to find the tangent line. When x=2, we have y+5cos(y)=6. Do you know what the y coordinate of the point is?
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  4. #4
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    Quote Originally Posted by adkinsjr View Post
    b) Find the equation of the tangent line at the point shown where x=2.

    It seems like there is a problem here. There is no way to isolate y in any of these equations. So it's impossible to find the tangent line. When x=2, we have y+5cos(y)=6. Do you know what the y coordinate of the point is?
    Thank you for the help with part a.

    There was no y coordinate given so no, I don't know what it is.
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