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Math Help - Curve Analysis w/Variables-what values of k, etc

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    Curve Analysis w/Variables-what values of k, etc



    The first one I'm having problems with because k is a constant and I'm not used to finding that in derivatives. The second one I don't even know where to start.

    How do you approach these problems which have a variable such as k, a b c, etc in terms of derivatives and curve analysis anyways? I couldn't find anything about them on the tutorial.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    okay, k, a, b and c are just constants. you treat them as you would any number when finding the derivative or whatever.

    (1) We find the maximum of a function by finding it's derivative and setting it equal to zero, so let's do that

    f(x) = x + k/x
    => f ' (x) = 1 - k/(x^2)
    for max, set f ' (x) = 0
    => 1 - k/(x^2) = 0
    => k = x^2
    we want max at x = -2
    so k = (-2)^2 = 4
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post

    Here's the second one.

    the possibility of inflection points occur when the second derivative is 0, so let's find the points where the second derivative is 0

    y = x^3 + bx^2 + c
    => y' = 3x^2 + 2bx
    => y'' = 6x + 2b

    set y'' = 0
    => 6x + 2b = 0
    => 3x + b = 0
    => b = -3x
    but x = -1 (since we are dealing with (-1,5)
    so b = 3

    so we have y = x^3 + 3x^2 + c
    we know that (-1,5) is a point on the curve, so

    5 = (-1)^3 + 3(-1)^2 + c
    => 5 = -1 + 3 + c
    => c = 5 + 1 - 3
    => c = 3

    so our curve is y = x^3 + 3x^2 + 3

    now, the slope of the tangent line is given by y'

    y' = 3x^2 + 2bx = 3x^2 + 6x
    at (-1,5)
    y' = 3(-1)^2 + 6(-1) = 3 - 6 = -3

    so the slope of our tangent line at (-1,5) is -3

    now using m = -3, (x1,y1) = (-1,5), we have by the point-slope form:

    y - y1 = m(x - x1)
    => y - 5 = -3(x + 1)
    => y = -3x - 3 + 5
    => y = -3x + 2 ..............the equation of the tangent line at P
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