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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Thursday, April 20, 2006

Sarah's Blog Assignment

Let C represent the piece of the curve y = (64-16x^2) ^ (1/3) that lies in the first quadrant.
Let S be the region bounded by C and the coordinate axes.
a) Find the slop of the tangent line to C at y=1.
b) Find the area of S.
c) Find the volume when S is rotated arount the x-axis.
d) Find the volume when S is rotated arouind the line x=-2.

I have tried to find an equation editor that will show the following math signs properfly but the files seem to be given out incomplete. As for point, it really is hard to use and I have no time to fiddle with it, so bare with how my answers will be, and sorry in advance.

a) slope @ y=1
1 = cube root ( 64 - 16x^2 )
cube both sides
1 = 64 - 16x^2
-63 = - 16x^2
63 / 16 = x^2
square root both sides
square root ( 63 / 16 ) = x
x = 1.9843
( 1.9843 , 1 )
y = ( 64 - 16x^2 )^(1/3)
y' = (1/3)( 64 - 16x^2 )^(-2/3) * ( -32x)
y' = (-32x) / [( 3 ) ( cube root [ ( 64 - 16 x^2)^2])]
therefore, y' (1.9843) = - 21.1538
(**NOTE: It is always good to store exact values into your calculator. If you are using programs like the Riesum programs, be careful where you store it, the alpha letter you may be using might be used by the program as well. **)
b) Area of S.
x - intercept ( 2 , 0 )

c) Volume of S when rotated around the x-axis.

Taking a slice, it would look like a disc. A circle. Area of a circle is the pi times the radius squared. Pi becomes a constant. The radius is measured by the function itself so therefore,

d) volume when S is rotated around the line x= -2.

When revolving it around the vertical line, we get a cylinder formed. Method that we use is cylindrical shells. Thinking of a piece of paper and just wrapping it around some center, How do we find the volume of that piece of paper? L * W * H. Height in this case is still the function value. Width is the itty bitty x difference, dx. Lenght though is a the circumference of the circle, 2 pi radius or 2 pi x. Therefore we have the integrand,

P.S. To that post, what else makes me focus? Other than candies, if I'm not at home its the music, not just any particular ones right now are korean r&b and pop. As long as it sounds good to me and something that I could never sing or know what its about. =D

See you all Monday.

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  • At 8:32 PM CDT, Blogger Lani said…

    Hi Sarah,

    I was hooked on M&Ms for many years!! You are right, they can help with focus, especially on long commutes. Music is probably more healthy although I must admit when I really want to focus, neither music, nor quotes always help me there. Right now it's peanut butter for me! And sometimes visualizing; have you tried that?



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