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Math Help - Quadratic Approximation

  1. #1
    calc3
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    Quadratic Approximation

    The question asks to find a general formula to approximate f(x+h,y+h) as h is almost 0, (h (squigly equals sign) 0) using the quadratic approximation.

    If you could please help me understand how to approach this problem, I would appreciate it. I'm not quite sure how I'm supposed to use the quadratic approximation. The h throws me off. Thanks in advance!
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  2. #2
    Flow Master
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    Quote Originally Posted by calc3 View Post
    The question asks to find a general formula to approximate f(x+h,y+h) as h is almost 0, (h (squigly equals sign) 0) using the quadratic approximation.

    If you could please help me understand how to approach this problem, I would appreciate it. I'm not quite sure how I'm supposed to use the quadratic approximation. The h throws me off. Thanks in advance!
    You want the Taylor series of a real function in two variables f(x, y) up to the quadratic term in h:


    f(x+h, y+h) \approx f(x, y) + \left( \frac{\partial f}{\partial x} \, h + \frac{\partial f}{\partial y} \, h \right) + \frac{1}{2} \, \left( \frac{\partial^2 f}{\partial x^2} \, h^2 + 2h^2 \, \frac{\partial^2 f}{\partial x \partial y}  + \frac{\partial^2 f}{\partial y^2} \, h^2 \right)



    = f(x, y) + h \, \left( \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \right) + \frac{h^2}{2} \, \left( \frac{\partial^2 f}{\partial x^2} + 2 \, \frac{\partial^2 f}{\partial x \partial y}  + \frac{\partial^2 f}{\partial y^2} \right) .
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