020. `query 20 Applied Calculus I 11-27-2007
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10:59:46 2.7.16 dy/dx at (2,1) if x^2-y^3=3
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RESPONSE --> the deriv of x^2 is 2x the deriv of y^3 is 3y^2dy/dx (i think) 2x-3y^2dy/dx=0 -3y^2dy/dx= -2x dy/dx=2x/3y^2 dy/dx=4/3 confidence assessment: 2
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10:59:57 The derivative of x^2 with respect to x is 2 x. The derivative of y^3 with respect to x is 3 y^2 dy/dx. You can see this by realizing that since y is implicitly a function of x, y^3 is a composite function: inner function is y(x), outer function f(z) = z^3. So the derivative is y'(x) * 3 * f(y(x)) = dy/dx * 3 * y^3. So the derivative of the equation is 2 x - 3 y^3 dy/dx = 0, giving 3 y^2 dy/dx = 2 x so dy/dx = 2 x / ( 3 y^2). At (2,1), we have x = 2 and y = 1 so dy/dx = 2 * 2 / (3 * 1^2) = 4/3. **
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RESPONSE --> yup self critique assessment: 3
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11:03:35 2.7.30 slope of x^2-y^3=0 at (1,1) What is the desired slope and how did you get it?
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RESPONSE --> 2x-3y^2dy/dx=0 dy/dx=2x/3y^2 at(-1,1) you get -2/3 confidence assessment: 3
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11:03:43 The derivative of the equation is 2 x - 3 y^2 dy/dx = 0. Solving for dy/dx we get dy/dx = 2x / (3 y^2). At (-1,1) we have x = 1 and y = 1 so at this point dy/dx = 2 * -1 / (3 * 1^2) = -2/3. **
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RESPONSE --> yup self critique assessment: 3
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11:07:07 2.7.36 p=`sqrt( (500-x)/(2x))
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RESPONSE --> this actually #44, which wasnt assigned, but... p^2=(500-x)/2x 2xp^2=500-x 2xp^2+x-500=0 prod rule (2x)(2p)+(p^2)(2dx/dp)-dx/dp=0 4xp+2p^2dx/dp-dx/dp=0 2p^2dx/dp-dx/dp=-4xp dx/dp(2p^2-1)=-4xp dx/dp=-4xp/2p^2-1 confidence assessment: 2
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11:07:19 You could apply implicit differentiation to the present form, and that would work but it would be fairly messy. You have lots of choices for valid ways to rewrite the equation but I would recommend squaring both sides and getting rid of denominators. You get p^2 = (500-x) / (2x) so 2x p^2 = 500-x and 2x p^2 + x - 500 = 0. You want dx/dp so take the derivative with respect to p: 2x * 2p + 2 dx/dp * p^2 - dx / dp = 0 (2 p^2 - 1) dx/dp = - 4 x p dx / dp = -4 x p / (2p^2 - 1) **
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RESPONSE --> yup self critique assessment: 3
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