University of Texas M 408D Rusin HW 11

University of Texas M 408D Rusin HW 11 - byrne (cmb3744) HW...

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Unformatted text preview: byrne (cmb3744) HW 11 rusin (55565) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Sorry for the delay I got wrapped up grading the tests... This HW will be due Monday evening. 001 10.0 points Use the Chain Rule to find the partial derivative w s for w = x 2 + y 2 + z 2 , x = st, y = s cos t, z = s sin t when s = 9, t = 0. 1. w s = 16 2. w s = 18 correct 3. w s = 21 4. w s = 19 5. w s = 14 Explanation: By the Chain Rule for Partial Differentia- tion w s = w x x s + w y y s + w z z s . Here, we have w x = 2 x, x s = t while w y = 2 y, y s = cos t and w z = 2 z, z s = sin t. Thus w s = 2 xt + 2 y cos t + 2 z sin t. Note that when s = 9 and t = 0, it follows that x = 0, y = 9, z = 0. Consequently, for these values, w s = 18 . keywords: 002 10.0 points Use partial differentiation and the Chain Rule applied to F ( x, y ) = 0 to determine dy/dx when F ( x, y ) = cos( x 5 y ) xe 6 y = 0 . 1. dy dx = sin( x 5 y ) 6 e 6 y 6 sin( x 5 y ) 5 xe 6 y 2. dy dx = sin( x 5 y ) 6 xe 6 y 6 sin( x 5 y ) 5 e 6 y 3. dy dx = sin( x 5 y ) + 6 xe 6 y 5 sin( x 5 y ) e 6 y 4. dy dx = sin( x 5 y ) + e 6 y 5 sin( x 5 y ) 6 xe 6 y correct 5. dy dx = sin( x 5 y ) + e 6 y 6 xe 6 y 5 sin( x 5 y ) 6. dy dx = sin( x 5 y ) + e 6 y 5 xe 6 y 6 sin( x 5 y ) Explanation: Applying the Chain Rule to both sides of the equation F ( x, y ) = 0, we see that F x dx dx + F y dy dx = F x + F y dy dx = 0 . Thus dy dx = F x F y = F x F y . byrne (cmb3744) HW 11 rusin (55565) 2 When F ( x, y ) = cos( x 5 y ) xe 6 y = 0 , therefore, dy dx = sin( x 5 y ) e 6 y 5 sin( x 5 y ) 6 xe 6 y . Consequently, dy dx = sin( x 5 y ) + e 6 y 5 sin( x 5 y ) 6 xe 6 y . 003 10.0 points Use the equation z y = F y F z to find z y for xe 7 y + 10 yz + ze 3 x = 0 . 1. z y = 7 e 7 y + 10 y 10 z + e 3 x 2. z y = e 3 x + 10 y 10 z + 7 xe 7 y 3. z y = 7 xe 7 y + 10 z 10 y + e 3 x correct 4. z y = 7 xe 7 y + 10 z 10 y + 3 e 3 x 5. z y = 7 e 7 y + 10 y 10 z + 3 e 3 x Explanation: 004 10.0 points The temperature at a point ( x, y ) in the plane is T ( x, y ) C. If a bug crawls on the plane so that its position in the plane after t minutes is given by ( x ( t ) , y ( t )) where x = 2 + t, y = 5 + 1 2 t, determine how fast the temperature is rising on the bugs path at t = 2 when T x (2 , 6) = 20 , T y (2 , 6) = 2 ....
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This note was uploaded on 05/08/2011 for the course MATH 408D taught by Professor Chu during the Spring '09 term at University of Texas at Austin.

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University of Texas M 408D Rusin HW 11 - byrne (cmb3744) HW...

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