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Math Help - Calculus/tangent line

  1. #1
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    Calculus/tangent line

    Find all values of x at which the tangent line to the given curve passes through the origin for y = 1/(x + 4)
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  2. #2
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    Quote Originally Posted by pantera View Post
    Find all values of x at which the tangent line to the given curve passes through the origin for y = 1/(x + 4)
    The tangent is a line of slope "m".

    The derivative of the function gives this slope.

    As the tangent passes through the origin, it's equation is y=mx, as c=0.

    Hence, the point of intersection of this tangent and the curve gives us x.

    f'(x)=\frac{(x+4)(0)-1(1)}{(x+4)^2}

    The equation of the tangent(s) through (0,0) is y=mx=\frac{-x}{(x+4)^2}

    Solving f(x) for the curve = mx for the line finds x.

    \frac{1}{x+4}=\frac{-x}{(x+4)^2}

    Solving this, bearing in mind that x=-4 is not part of the domain of f(x) or f'(x) discovers x.

    Hence multiply both fractions by (x+4) before solving for x.
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  3. #3
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    Lines look like this: (y-y_{0}) = m(x-x_{0})

    We know we need the Origin: y_{0} = m \cdot x_{0}

    The derivative gives the slope: m = -\frac{1}{(x_{0}+4)^{2}}

    We know: y_{0} = \frac{1}{x_{0}+4}

    I'm a little surprised to find only one solution. I expected two solutions when I started the problem. I was wrong.

    Let's see what you get.
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  4. #4
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    Here's why there is only one solution
    Attached Thumbnails Attached Thumbnails Calculus/tangent line-tangent-through-origin.jpg  
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