HW_11_Key - Version 083 Homework 11 Gilbert (59825) 1 This...

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Unformatted text preview: Version 083 Homework 11 Gilbert (59825) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. You are the first students to try out this new system. Please be patient - bugs will occur, but they will be fixed very quickly! Report any bugs to Shane Lewis(gsl@mail.utexas.edu) and send a copy to me. Thanks. 001 10.0 points Use the Chain Rule to find dw dt for w = xe y/z when x = t 2 , y = 7 t, z = 7 + 2 t. 1. dw dt = parenleftBig t x z 7 xy z parenrightBig e y/z 2. dw dt = parenleftBig 2 t x z 2 xy z 2 parenrightBig e y/z correct 3. dw dt = parenleftBig 2 t x z 2 xy z parenrightBig e y/z 4. dw dt = parenleftBig t + x z + 7 xy z 2 parenrightBig e y/z 5. dw dt = parenleftBig 2 t + x z + 2 xy z 2 parenrightBig e y/z 6. dw dt = parenleftBig t + x z + 7 xy z parenrightBig 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 = 7 t, z = 7 + 2 t, therefore, dw dt = 2 te y/z x z e y/z 2 xy z 2 e y/z . Consequently, dw dt = parenleftBig 2 t x z 2 xy z 2 parenrightBig e y/z . 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 6 y ) xe 2 y = 0 . 1. dy dx = sin( x 6 y ) + e 2 y 6 xe 2 y 2 sin( x 6 y ) 2. dy dx = sin( x 6 y ) 2 xe 2 y 2 sin( x 6 y ) 6 e 2 y 3. dy dx = sin( x 6 y ) 2 e 2 y 2 sin( x 6 y ) 6 xe 2 y 4. dy dx = sin( x 6 y ) + 2 xe 2 y 6 sin( x 6 y ) e 2 y 5. dy dx = sin( x 6 y ) + e 2 y 6 sin( x 6 y ) 2 xe 2 y correct 6. dy dx = sin( x 6 y ) + e 2 y 2 xe 2 y 6 sin( x 6 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 . Version 083 Homework 11 Gilbert (59825) 2 When F ( x, y ) = cos( x 6 y ) xe 2 y = 0 , therefore, dy dx = sin( x 6 y ) e 2 y 6 sin( x 6 y ) 2 xe 2 y . Consequently, dy dx = sin( x 6 y ) + e 2 y 6 sin( x 6 y ) 2 xe 2 y . 003 10.0 points The temperature at a point ( x, y ) in the plane is T ( x, y ) measured in degrees Celsius. If a bug crawls so that its position in the plane after t minutes is given by x = 2 + t, y = 4 + 1 2 t, determine how fast is the temperature rising on the bugs path after 2 minutes when T x (2 , 5) = 16 , T y (2 , 5) = 6 . 1. rate = 7 C / min correct 2. rate = 9 C / min 3. rate = 11 C / min 4. rate = 8 C / min 5. rate = 10 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...
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HW_11_Key - Version 083 Homework 11 Gilbert (59825) 1 This...

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