calculus

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a)
If x3 + y3 - xy2 = 5, find dy/dx.

(b)
Using your answer to part (a), make a table of approximate y-values of points on the curve near x = 1, y = 2. Include x = 0.96, 0.98, 1, 1.02, 1.04.

(c)
Find the y-value for x = 0.96 by substituting x = 0.96 in the original equation and solving for y using a computer or calculator. Compare with your answer in part (b).

(d)
Find all points where the tangent line is horizontal or vertical.

  • calculus -

    a) Assuming you meant

    x^3 + y^3 - x y^2 = 5
    3x^2 + 3y^2 dy/dx - x(2y)dy/dx - y^2 = 0
    dy/dx(3y^2 - 2xy) = y^2 - 3x^2
    dy/dx = (y^2 - 3x^2)/(3y^2 - 2xy)

    b) just arithmetic, you do it
    c) yours

    d) for a horizontal tangent, dy/dx = 0
    then y^2 - 3x^2 = 0
    y^2 = 3x^2
    y = ± x√3

    sub into the first:
    x^3 + 3√3x^3 - x(x√3) = 5
    x^3 + 3√3 x^3 - √3 x^2 - 5 = 0
    Nastiness!!!!
    wolfram says x = 1.03412 correct to 5 decimals
    now sub that into the original and find y, equally nasty.

    for vertical asymptote , denominator of dy/dx = 0
    3y^2 - 2xy) = 0
    y(3y - 2x) = 0
    y = 0 , that is not bad
    or
    y = 2x/3

    more mess ahead

    I suggest you use the Wolfram page or your built-in equation solver of your fancy calculator to solve these equations.

    (it might be a good idea to check my work, lately I have been making silly errors by not writing out the solution first)

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