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What is the derivative of 4x(16-x^2)^0.5?

  1. Jul 12, 2004 #1
    I did this three times, and always come up with utter nonsense.

    What is the derivative of 4x(16-x^2)^0.5?
    (Root of (16 minus x-squared) by 4x.) What are its zilches? How did you calculate that?



    This derivative is part of the solution to this problem: "What are the dimentions of the largest rectangle (by area) inscribable in a circle with radius 4?"

    I drew the circle and the square (see attachment), divided the square into four parts, drew a diagonal through one, and called it a hypotenuse, which is equal to 4. X and y are the sides of the mini squares. A = 4xy (since there are 4 mini-squares). Then I used Pythagoras to come up with that equation above. But I can't differentiate it now...

    The final answer is supposed to be 32 square units.
     

    Attached Files:

  2. jcsd
  3. Jul 12, 2004 #2

    matt grime

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    you can differentiate it, just keep trying, or you could make your life easier: the maximizing the area is the same as maximizing the square of the area.
     
  4. Jul 12, 2004 #3
    Use the product rule and chain rule. I'll get you started:

    (d/dx)(4x(16-x^2)^0.5)=4((16-x^2)^0.5)+(4x)(...

    As for the zeros, the only important one is 2(2)^.5

    matt grime is right, it ends up being a square, but you should start with a rectangle to prove it to yourself.

    Sure you can, finish the above differentiation, simplify and then find a root that makes sense in this problem. Use the root to find y and then you'll know the area (which is, as you stated, 32 square units).

    Good Luck,

    Kevin
     
  5. Jul 12, 2004 #4
    (d/dx)(4x(16-x^2)^0.5)=4((16-x^2)^0.5) - (4x^2)(16-x^2)^.5 =

    -4x^2
    -------------- + 4sqrt(16-x^2) = 0;
    sqrt(16-x^2)

    4x^2
    -------------- = 4sqrt(16-x^2) = 0;
    sqrt(16-x^2)

    4x^2 = 4(16-x^2)

    x^2 = 16 - x^2

    0 = 16 :yuck:

    BTW, this means x = sqrt(16-x^2), which is y.
    So x=y, and if this is a square, that's perfectly true.

    So there you have it: a full mathematical proof that 0=16.
     
    Last edited: Jul 12, 2004
  6. Jul 12, 2004 #5
    The above is good

    This is bad. "add" x^2 to both sides to get 2(x^2)=16 which yields x=2sqrt(2) and not 0=16
    So there you have it: a full mathematical proof that you can't add
    :biggrin:

    Cheers,

    Kevin
     
  7. Jul 12, 2004 #6
    You needed calculus to prove THAT?

    God, I'm sure glad I'm not studying arithmetic any more.
     
    Last edited: Jul 12, 2004
  8. Jul 12, 2004 #7
    10 = 2 really
    I mean:

    102=210

    1*2^1+0*2^0=2*10^0
     
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