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anonymous

  • 5 years ago

f(x,y) = x/y, P=(2,1) and v= -1i -1j. Find the maximum rate of change of f at P. Find the (unit) direction vector in which the maximum rate of change occurs at P.

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  1. anonymous
    • 5 years ago
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    Find the gradient of f(x,y), evaluate it at P, and then find the dot product with v

  2. anonymous
    • 5 years ago
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    Grad(f)=<1/y, -x/y^2> at P: <1, -2>

  3. anonymous
    • 5 years ago
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    so the dot product is: <1, -2>*<-1, -1> = -1 + 2 = 1

  4. anonymous
    • 5 years ago
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    oops. I skipped a step

  5. anonymous
    • 5 years ago
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    <\[<-1/\sqrt{2}, -1/\sqrt{2}>\]

  6. anonymous
    • 5 years ago
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    \[<1, -2>*<-1/\sqrt{2}, -1/\sqrt{2}>\], where * is a dot product

  7. anonymous
    • 5 years ago
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    so the final is \[1/\sqrt{2}\]

  8. anonymous
    • 5 years ago
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    I'm not asking for a directional derivative of f at P in the direction of V. I'm asking maximum rate of changing f at P and unit vector in which the maximum rate of changing occurs at P.

  9. anonymous
    • 5 years ago
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    ah, sorry. The magnitude of the gradient is the greatest rate of change

  10. anonymous
    • 5 years ago
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    \[<1, -2>=\]=\[\sqrt{5}\]

  11. anonymous
    • 5 years ago
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    where that is the magnitude (greatest rate)

  12. anonymous
    • 5 years ago
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    then the unit vector would be \[<1/\sqrt{5}, -2/\sqrt{5}>\]

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