course Mth 271 020. `query 20*********************************************
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Given Solution: `a 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. ** ------------------------------------------------ Self-critique Rating: I took the derivative and plugged in the values of x to obtain dy/dx ********************************************* Question: `q2.7.28 (was 2.7.22) slope of x^2-y^3=0 at (1,1) What is the desired slope and how did you get it? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^2 – y^3 = 0 2x – 3y^2 (dy/dx) = 0 dy/dx = 2x/3y^2 at (1,1) dy/dx = 2(1)/3(1)^2 = 2/3 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ I feel very confident about this problem.
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Given Solution: `a 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. ** ------------------------------------------------ Self-critique Rating: I took the derivative of the problem, solved for dy/dx, and plugged in the point to get the desired slope. ********************************************* Question: `q2.7.42 (was 2.7.36) p=`sqrt( (500-x)/(2x)) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: p = ‘sqrt((500-x)/(2x)) p^2 = (((500-x)/(2x))^(1/2))^2 p^2 = 500-x/2x 2xp^2 = 500 – x 2xp^2 + x – 500 = 0 2x * 2p + xdx/dp * p^2 – dx/dp = 0 dx/dp = -4xp/(2p^x -1) confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ I feel fairly confident about this problem.
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Given Solution: `a 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) ** ------------------------------------------------ Self-critique Rating: I had to look ahead at the answer to better understand what you were looking for in an answer.