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Math Help - problem in integral calculus(need help)

  1. #1
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    Arrow problem in integral calculus(need help)

    The points (-1,3) and (0,2) are on a curve, and at any point (x,y) on the curve d2y = 2- 4x. Find an equation of the curve.
    dx2

    (HINT: let d2y = dy' , and obtain an equation involving y', x and an arbitrary
    dx2 dx
    constant C1. From the equation obtain another equation involving y,x, C1 and C2. Compute C1 and C2 from the conditions)


    thnx.. i'll wait for your reply..
    =APriL=
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  2. #2
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    We start with the given second derivative of y, and work backwards.

    y'' = 2 - 4x

    Find an antiderivative of y'' to obtain the first derivative of y.

    y' = -2x^2 + 2x + C_1

    Find an antiderivative of y' to obtain y.

    y = -\frac{2}{3}x^3 + x^2 + C_1 \cdot x + C_2

    This equation has two unknowns for which we need to solve. We've been given the coordinates of two points which lie on the graph of y.

    Substitute these values for x and y to obtain a system of two equations.

    3 = -\frac{2}{3}(-1)^3 + (-1)^2 + C_1(-1) + C_2

    2 = -\frac{2}{3}(0)^3 + (0)^2 + C_1(0) + C_2

    Solving this system for the unknowns is simple algebra. Can you do it?

    Cheers,

    ~ Mark
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  3. #3
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    Quote Originally Posted by april29 View Post
    The points (-1,3) and (0,2) are on a curve, and at any point (x,y) on the curve d2y = 2- 4x. Find an equation of the curve.
    dx2

    (HINT: let d2y = dy' , and obtain an equation involving y', x and an arbitrary
    dx2 dx
    constant C1. From the equation obtain another equation involving y,x, C1 and C2. Compute C1 and C2 from the conditions)


    thnx.. i'll wait for your reply..
    =APriL=
    \frac{d^2 y}{dx^2} = 2 - 4x \Rightarrow \frac{dy}{dx} = 2x - 2x^2 + A \Rightarrow y = x^2 - \frac{2}{3} x^3 + Ax + B.

    Substitute (-1,3) and (0,2) to get two equations in A and B. Solve these equations simultaneously for A and B.
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