{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

University of Texas M 408D Rusin HW 11

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

This preview shows pages 1–3. Sign up to view the full content.

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 10.0points 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 10.0points 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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 10.0points 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 10.0points 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 bug’s path at t = 2 when T x (2 , 6) = 20 , T y (2 , 6) = 2 . 1. rate = 4 C / min 2. rate = 5 C / min 3. rate = 2 C / min 4. rate = 3 C / min 5. rate = 6 C / min correct Explanation: By the Chain Rule for partial differentia- tion, the rate of change of temperature on the bug’s path at ( x ( t ) , y ( t )) is given by dT dt = dT ( x ( t ) , y ( t )) dt = ∂T ∂x dx dt + ∂T ∂y dy dt . Now dx dt = 1 2 2 + t , dy dt = 1 2 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 15

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online