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Math Help - Find the equation of the tangent

  1. #1
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    Find the equation of the tangent

    Find the equation of the tangent?
    I got really stuck in some equations and I've been trying to figure out how to solve them for a long time now and would really appreciate help. The equations are:

    Find the equation of the tangent to each of the following curves at the point indicated by the given value of x

    y = (x+5)/√x where x = 25

    and the other equation is

    y= 1/∛x, where x = 1/8

    It would help a lot if you could explain the solution in case it's very complicated. These equations really confuse me because the answer is really different from the others in the book.

    For those who want to check if they're right:
    1st equation answer: 25y-2x=100
    2st equation answer: 3y+16x=8

    Also the other thing that confuses me is that with the other equations the y never changes such as (x+4)/x x = -2 and the answer is y = -x-3 but on those two equations there is a number before y and etc. Sorry if what I'm writing is confusing, I find it hard to explain what is confusing me.

    I'd appreciate any help I could get!
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  2. #2
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    Re: Find the equation of the tangent

    1. Find the line tangent to the graph of f(x) = \frac{x+5}{\sqrt{x}} where x = 25.

    Solution:

    Recall that a line can be written in the form (y - y_0) = m(x - x_0) where m is the slope of the line and (x_0, y_0) lie on the line. In this case, we're looking for a line that passes through the point (25, f(25)) and has the slope of f at the point x = 25.

    So, (x_0, y_0) = (25, f(25)) = (25, \frac{30}{5}) = (25, 6).

    Now let's find m. The slope of f at the point (25, 6) is f'(25).

    So,

    f'(x) = \frac{d}{dx}( \frac{x + 5}{\sqrt{x}})
     = \frac{ \sqrt{x} \frac{d}{dx}(x + 5) - (x + 5)\frac{d}{dx} (\sqrt{x})}{ (\sqrt{x})^2}

     = \frac{ \sqrt{x} - (x+5) \frac{1}{2} x^{-\frac{1}{2}}}{x}

    So f'(25) = \frac{5 - \frac{30}{2} \frac{1}{5}}{25}

    = \frac{5 - 3}{25}

    = \frac{2}{25}

    So the equation of the line will be:

    (y - 6) = \frac{2}{25}(x - 25),

    25(y - 6) = 2x - 50, or finally

    25y - 2x = 150 - 50 = 100


    Can you do the second one?
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