HW11S - Gauthier, Joseph Homework 11 Due: Nov 10 2007, 3:00...

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Unformatted text preview: Gauthier, Joseph Homework 11 Due: Nov 10 2007, 3:00 am Inst: JEGilbert 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Use the Chain Rule to find dw dt when w = xe y/z and x = t 2 , y = 5- t, z = 5 + 4 t. 1. dw dt = t + x z + 5 xy z 2 e y/z 2. dw dt = t + x z + 5 xy z e y/z 3. dw dt = 2 t + x z + 4 xy z 2 e y/z 4. dw dt = 2 t- x z- 4 xy z e y/z 5. dw dt = 2 t- x z- 4 xy z 2 e y/z correct 6. dw dt = t- x z- 5 xy z e y/z Explanation: By the Chain Rule for Partial Differentia- tion, dw dt = w x dx dt + w y dy dt + w z dz dt . When w = xe y/z when x = t 2 , y = 5- t, z = 5 + 4 t, therefore, dw dt = 2 te y/z- x z e y/z- 4 xy z 2 e y/z . Consequently, dw dt = 2 t- x z- 4 xy z 2 e y/z . keywords: 002 (part 1 of 1) 10 points Use partial differentiation and the Chain Rule applied to F ( x, y ) = 0 to determine dy/dx when F ( x, y ) = cos( x- 2 y )- xe 6 y = 0 . 1. dy dx = sin( x- 2 y ) + e 6 y 2 sin( x- 2 y )- 6 xe 6 y correct 2. dy dx = sin( x- 2 y )- 6 e 6 y 6 sin( x- 2 y )- 2 xe 6 y 3. dy dx = sin( x- 2 y ) + 6 xe 6 y 2 sin( x- 2 y )- e 6 y 4. dy dx = sin( x- 2 y )- 6 xe 6 y 6 sin( x- 2 y )- 2 e 6 y 5. dy dx = sin( x- 2 y ) + e 6 y 2 xe 6 y- 6 sin( x- 2 y ) 6. dy dx = sin( x- 2 y ) + e 6 y 6 xe 6 y- 2 sin( x- 2 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 . When F ( x, y ) = cos( x- 2 y )- xe 6 y = 0 , Gauthier, Joseph Homework 11 Due: Nov 10 2007, 3:00 am Inst: JEGilbert 2 therefore, dy dx =-- sin( x- 2 y )- e 6 y 2 sin( x- 2 y )- 6 xe 6 y . Consequently, dy dx = sin( x- 2 y ) + e 6 y 2 sin( x- 2 y )- 6 xe 6 y . keywords: partial differentiation, Chain Rule, implicit differentiation, Implicit Function Theorem 003 (part 1 of 1) 10 points The temperature at a point ( x, y ) is T ( x, y ) measured in degrees Celsius. If a bug crawls so that its position after t minutes is given by x = 11 + t, y = 3 + 3 5 t, with x, y measured in centimeters, determine how fast the temperature rising on the bugs path after 5 minutes when T x (4 , 6) = 24 , T y (4 , 6) = 10 . 1. rate = 9 C / min correct 2. rate = 10 C / min 3. rate = 7 C / min 4. rate = 6 C / min 5. rate = 8 C / min Explanation: By the Chain Rule for partial differentia- tion, the rate of change of temperatuure T on the bugs path is given by dT dt = dT ( x ( t ) , y ( t )) dt = T x dx dt + T y dy dt = 1 11 + t T x + 3 5 T y ....
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This homework help was uploaded on 03/19/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas at Austin.

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HW11S - Gauthier, Joseph Homework 11 Due: Nov 10 2007, 3:00...

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