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Math Help - Changing least rapidly?

  1. #1
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    Red face Changing least rapidly?

    At (2,1), I have to find the direction where f(x,y)=x^2+3xy-y^2 is increasing fastest.

    So I did:

     \triangledown f(2,1)= \langle 2(2)+3, 3(2)-2 \rangle = \langle 7,4 \rangle

    Next I have to find the rate of growth in this direction, so I did:

     ||\triangledown f(2,1)|| = \sqrt{7^2+4^2} = \sqrt{65}

    Finally I have to find the direction a this point where f is changing "least rapidly." Not sure what to do here...
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  2. #2
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    Quote Originally Posted by MathSucker View Post
    At (2,1), I have to find the direction where f(x,y)=x^2+3xy-y^2 is increasing fastest.

    So I did:

     \triangledown f(2,1)= \langle 2(2)+3, 3(2)-2 \rangle = \langle 7,4 \rangle

    Next I have to find the rate of growth in this direction, so I did:

     ||\triangledown f(2,1)|| = \sqrt{7^2+4^2} = \sqrt{65}

    Finally I have to find the direction a this point where f is changing "least rapidly." Not sure what to do here...
    What do you mean by "changing least rapidly"? A function increases most rapidly in the direction of the gradient, which you calculated, decreases most rapidly in the opposite direction, and has 0 rate of change at right angles to the gradient. Does "has 0 rate of change" qualify as "changes least rapidly"?
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  3. #3
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    I'm not quite sure what it means, hence the name of the thread. I assume the magnitude of when it is decreasing fastest is the same as when it is increasing fastest. I was thinking "zero because it isn't moving sideways" but that seems to easy. Or maybe that's the point. I dunno.
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