#$&* course 272 12/11 5pm If your solution to a stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `a You have to use the quotient rule. The derivative is taken with respect to y, so the ' stands for the derivative with respect to y. You get [ (xy)' (x^2 + y^2) - xy ( x^2 + y^2)' ] / (x^2 + y^2) ^ 2. Remembering that ' represents derivative with respect to y we get [ x ( x^2 + y^2) - xy ( 2y ) ] / (x^2 + y^2 ) ^ 2 or [ x^3 + x y^2 - 2 x y^2 ] / (x^2 + y^2) ^ 2, which simplifies to [ x^3 - x y^2 ] / (x^2 + y^2) ^ 2 or x [ x^2 - y^2 ] / (x^2 + y^2) ^ 2 . &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qQuery problem 7.4.32 wx, wy, wz at origin for w = 1/sqrt(1-x^2-y^2-z^2) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1/sqrt(1-x^2-y^2-z^2) inner function: (1-x^2-y^2-z^2) partial diff= -2x -2y -2z outer function: (1/-z^2) partial diff= -1/2z^(-3/2) wx: -2x * -1/2(1- x^2-y^2-z^2)^(-3/2) = x/ (1- x^2-y^2-z^2)^(3/2) wy: -2y * -1/2(1- x^2-y^2-z^2)^(-3/2) = y/ (1- x^2-y^2-z^2)^(3/2) wz: -2z * -1/2(1- x^2-y^2-z^2)^(-3/2) = z/ (1- x^2-y^2-z^2)^(3/2) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a `a Every partial derivative involves the chain rule with inner function 1 = x^2 - y^2 - z^2 and outer function 1/ sqrt(z) = z^(-1/2). The partial derivatives with respect to x, y and z of the 'inner function' (1 - x^2 - y^2 - z^2) are respectively -2x, -2y and -2z. The derivative of the 'outer function' z^-(1/2) is -1/2 z^(-3/2). So the partial derivatives of the entire function are wx = -2x * -1/2 ( 1 - z^2 - y^2 - z^2)^(-3/2) = x / (1 - x^2 - y^2 - z^2)^3/2 wy = -2y * -1/2 ( 1 - z^2 - y^2 - z^2)^(-3/2) = y / (1 - x^2 - y^2 - z^2)^3/2 wz = -2z * -1/2 ( 1 - z^2 - y^2 - z^2)^(-3/2) = z / (1 - x^2 - y^2 - z^2)^3/2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qWhat are the values of the three requested partial derivatives at the specified point? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Specified point = origin where x=0 y=0 z=0 0/ (1- 0^2-0^2-0^2)^(3/2): 0/1 =0 0/ (1- 0^2-0^2-0^2)^(3/2): 0/1 =0 0/ (1- 0^2-0^2-0^2)^(3/2): 0/1 =0 confidence rating #$&*: ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a At the origin we have x = y = z = 0 so that the three partial derivatives are all of form 0 / 1 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `qQuery problem 7.4.40 (was 7.4.36) fx = fy = 0 for 3x^3-12xy+y^3 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3x^3-12xy+y^3 f(x) = 9x^2 -12y = 0 so y=3x^2 /4 f(y)= -12x+3y^2 = 0 Subbing in (3x^2/4) for y: -12x+3 (3x^2/4)^2 = 0 -3x (4-9x^3 / 16) = 0 -3x =0 and (4-9x^3 /16) = 0 x= 0 x=4*3^(-2/3) Subbing x values into y=3x^2/4 3(0)=y 0 =y y= 9x^2-12y = 9 (4*3^(1/3)/3)^2 - 12y y= 9 (4*3^(1/3)/3)^2 / 12 y= ¾ * 16 * 3^(2/3)/9 = 4 * 3^(2/3)/ 3 so partials are both zero at: (0,0) and (4*3^(1/3)/3) , 4 * 3^(2/3)/ 3) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a Geometric solution: fx = 0 when y = 3/4 x^2, which describes a parabola with vertex at the origin, axis of symmetry conciding with the y axis, opening upward and passing thru the points (1, 3/4) and (-1, 3/4). fy = 0 when x = 1/4 y^2, which describes a parabola with vertex at the origin, axis of symmetry conciding with the x axis, opening to the right and passing thru the points (0, 1/4) and (0, -1/4). If you sketch these parabolas it should be clear that they coincide at one point in the first quadrant. You should estimate the coordinates of these points. Then proceed to solve these equations simultaneously to find the accurate coordinates: ** The partial derivatives are fx = 9x^2 - 12y fy = -12x + 3y^2. If the two partial derivatives are zero we get the equations 9x^2 - 12 y = 0 -12x + 3 y^2 = 0. Solving the first equation for y we get y = 3x^2 / 4. Substituting this expression for y in the second we have -12 x + 3 ( 3x^2 / 4)^2 = 0 so -12 x + 27 x^4 / 16 = 0. Factoring out -3x: -3x ( 4 - 9 x^3 / 16) = 0. This is so if -3x = 0 or 4 - 9 x^3 / 16 = 0. -3x = 0 gives solution x = 0. 4 - 9 x^3 / 16 = 0 if x^3 = 4 * 16 / 9, which happens when x = 64^(1/3) / 9^(1/3) = 4 * 3^(-2/3), which is expressed in standard form as 4 * 3 ^(1/3) / 3. Our solutions for x are thus x = 0 and x = 4 * 3 ^(1/3) / 3. Each of our x solutions is substituted into our solution y = 3x^2 / 4, in order to get the corresponding y coordinate: If x = 0 then since 9 x^2 - 12 y = 0 we have y = 0. If x = 4 * 3^(1/3)/3 then 9 x^2 - 12 y = 0 gives us 9 [ 4 * 3^(1/3)/3 ] ^2 - 12 y = 0 so y = 9 [ 4 * 3^(1/3)/3 ] ^2 / 12 = 3/4 * 16 * 3^(2/3)/9 = 4 * 3^(2/3) / 3. So the two partials are both zero at (0,0) and at ( 4 * 3^(1/3)/3, 4 * 3^(2/3)/3). STUDENT QUESTION Ok, how exactly did you solve for your partial derivatives? I notice I still had a variable expressed in each fx and fy. If fx, we equate to zero and solve for y, right? The opposite for fy, equating to zero, and solving for x? Clear on the factoring from here, finding our values, but did something wrong when attempting to find my partial derivatives. - Thank you. INSTRUCTOR RESPONSE We solve the first equation for y, obtaining y = 3x^2 / 4. We then substitute this expression for y in the second equation, which is -12x + 3 y^2 = 0. Replacing y with 3 x^2 / 4 gives us the equation -12 x + 3 ( 3x^2 / 4)^2 = 0, which we expand to get-12 x + 27 x^4 / 16 = 0. We then solve this for x, as indicated above, and obtain solutions x = 0 and x = 4 * 3 ^ (1/3) / 3. We substitute these x values into our expression y = 3 x^2 / 4 to get our y values. This gives us our two critical points (0,0) and ( 4 * 3^(1/3)/3, 4 * 3^(2/3)/3). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating: 3 " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!