HW_11_Key

HW_11_Key - Version 083 – Homework 11 – Gilbert...

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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([email protected]) 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 bug’s 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 bug’s 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...

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