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Unformatted text preview: lawrence (cdl678) – Homework 8 – ODELL – (56280) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find lim t →∞ r ( t ) when r ( t ) = (Big 8 tan − 1 t, e − 4 t , ln t t )Big . 1. limit = ( 8 π, , 1 ) 2. limit = ( , , 1 ) 3. limit = ( 4 π, , ) correct 4. limit = ( 8 π, − 4 , ) 5. limit = ( 4 π, − 4 , ) 6. limit = ( , − 4 , 1 ) Explanation: For a vector function r ( t ) = ( f ( t ) , g ( t ) , h ( t ) ) , the limit lim t →∞ r ( t ) = ( lim t →∞ f ( t ) , lim t →∞ g ( t ) , lim t →∞ h ( t ) ) . But for the given vector function, lim t →∞ f ( t ) = lim t →∞ 8 tan − 1 ( t ) = 4 π , while lim t →∞ g ( t ) = lim t →∞ e − 4 t = 0 , and lim t →∞ h ( t ) = lim t →∞ ln t t = 0 , using L’Hospital’s Rule. Consequently, lim t →∞ r ( t ) = ( 4 π, , ) . 002 10.0 points A space curve is shown in black on the surface x y z Which one of the following vector functions has this space curve as its graph? 1. r ( t ) = ( cos t, sin t, sin 4 t ) correct 2. r ( t ) = ( cos t, sin t, cos 4 t ) 3. r ( t ) = ( cos t, sin t, t ) 4. r ( t ) = ( cos t, sin t, cos 2 t ) 5. r ( t ) = ( sin t, cos t, − t ) 6. r ( t ) = ( sin t, cos t, cos 2 t ) Explanation: If we write r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) , then x ( t ) 2 + y ( t ) 2 = 1 for all the given vector functions, showing that their graph will always lie on the cylindrical cylinder x 2 + y 2 = 1 lawrence (cdl678) – Homework 8 – ODELL – (56280) 2 To determine which particular vector function has the given graph, we have to look more closely at the graph itself. Notice that the graph oscillates with period 4, so r ( t ) is one of ( cos t, sin t, sin 4 t ) , ( cos t, sin t, cos 4 t ) . On the other hand, it passes it through (1 , , 0) and (0 , 1 , 0). Consequently, the space curve is the graph of r ( t ) = ( cos t, sin t, sin 4 t ) . keywords: 003 (part 1 of 2) 10.0 points The vector function r ( t ) = (1 + 4 cos t ) i + 5 j + (2 − 4 sin t ) k traces out a circle in 3space as t varies. In which plane does this circle lie? 1. plane x = − 5 2. plane y = 5 correct 3. plane y = − 5 4. plane z = − 5 5. plane x = 5 6. plane z = 5 Explanation: Writing r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , we see that y ( t ) = 5 for all t . Consequently, r ( t ) traces out a curve in the plane y = 5 . 004 (part 2 of 2) 10.0 points Determine the radius and center of the cir cle traced out by r ( t ). 1. correct 2. 3. 4. 5. 6. Explanation: Writing r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , we see also that Consequently, r ( t ) traces out a circle with ....
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 Spring '10
 odell
 Derivative, Lawrence

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