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Thread: Please help me on a problem about particular antiderivatives.

  1. #1
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    Please help me on a problem about particular antiderivatives.

    Hello everyone. Our professor discussed this particular problem the other day:

    At any point (x,y) on a curve, d2y/dx2 = 1 - x^2. The equation of the tangent line to the curve at (1,1) is y = -x + 2. Find the equation of the curve.

    I have notes on the solution but Iím stuck. I donít understand why dy/dx equals to -1. Can someone please lead me in the right direction? Attached is the picture of my lecture notes.

    Thank you very much!
    Attached Thumbnails Attached Thumbnails Please help me on a problem about particular antiderivatives.-410b93b6-fbf6-4fdd-9357-c71e1777595b.jpeg  
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  2. #2
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    Re: Please help me on a problem about particular antiderivatives.

    It looks like you correctly derived $\dfrac{dy}{dx}$, now just integrate it.

    $\dfrac{dy}{dx} = -1$ at $x=1$ because you selected the constant of integration, $-\dfrac 5 3$ to make it so.
    Thanks from topsquark and kaonashi
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  3. #3
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    Re: Please help me on a problem about particular antiderivatives.

    Quote Originally Posted by kaonashi View Post
    Hello everyone. Our professor discussed this particular problem the other day:
    At any point (x,y) on a curve, d2y/dx2 = 1 - x^2. The equation of the tangent line to the curve at (1,1) is y = -x + 2. Find the equation of the curve.
    I have notes on the solution but Iím stuck. I donít understand why dy/dx equals to -1. Can someone please lead me in the right direction? Attached is the picture of my lecture notes.
    Sorry but I cannot read your images.
    But based upon the post, if $\dfrac{d^2y}{dx^2}=1-x^2$ then $\dfrac{dx}{dy}=x-\tfrac{1}{3}x^3+C$ Now surely you know that $\dfrac{dx}{dy}$ is slope.
    And we are given that $y=-1x+2$ is the tangent at $(1,1)$ which has slope $-1$.
    Thus $\dfrac{dx}{dy}(1,1)=1-\tfrac{1}{3}1^3+C=-1$ so $C=~?$

    The find the primitive(anti-derivative) again. Use the point $(1,1)$ to find the constant in the primitive.
    Thanks from topsquark and kaonashi
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  4. #4
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    Re: Please help me on a problem about particular antiderivatives.

    Thanks everyone! So it was the slope all along!
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