hw 12 - rhodes (ajr2283) HW12 Gilbert (56380) This...

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rhodes (ajr2283) – HW12 – Gilbert – (56380) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Determine the vector Function r ( y ) whose graph is the cross-section oF the graph oF z = f ( x, y ) = x 2 3 y 2 x + 2 y by the plane x = 2. 1. r ( y ) = a y, 2 , 2 3 y 2 + 2 y A 2. r ( y ) = a y, 2 , 6 3 y 2 + 2 y A 3. r ( y ) = a− 2 , y, 3 y 2 + 2 y A 4. r ( y ) = a− 2 , y, 6 3 y 2 + 2 y A correct 5. r ( y ) = a− 2 , y, 2 3 y 2 + 2 y A 6. r ( y ) = a y, 2 , 3 y 2 + 2 y A Explanation: The graph oF z = f ( x, y ) = x 2 3 y 2 x + 2 y is the set oF all points ( x, y, f ( x, y )) as x, y vary in 3-space. So the intersection oF this graph with the plane x = 2 is the set oF all points ( 2 , y, f ( 2 , y )) , −∞ < y < . But f ( 2 , y ) = 6 3 y 2 + 2 y . Consequently, the cross-section is the graph oF r ( y ) = a− 2 , y, 6 3 y 2 + 2 y A . keywords: GraphsContoursMV, GraphsCon- toursMVExam, 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 ) = a cos t, sin t, t A 2. r ( t ) = a cos t, sin t, cos 2 t A 3. r ( t ) = a sin t, cos t, cos 2 t A 4. r ( t ) = a cos t, sin t, cos 4 t A 5. r ( t ) = a sin t, cos t, t A 6. r ( t ) = a cos t, sin t, sin 4 t A correct Explanation: IF we write r ( t ) = a x ( t ) , y ( t ) , z ( t ) A , 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
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rhodes (ajr2283) – HW12 – Gilbert – (56380) 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 a cos t, sin t, sin 4 t A , a cos t, sin t, cos 4 t A . On the other hand, it passes it through (1 , 0 , 0) and (0 , 1 , 0). Consequently, the space curve is the graph of r ( t ) = a cos t, sin t, sin 4 t A . keywords: 003 (part 1 of 2) 10.0 points The vector function r ( t ) = (1 + 2 cos t ) i + 2 j + (5 2 sin t ) k traces out a circle in 3-space as t varies. In which plane does this circle lie? 1. plane z = 2 2. plane y = 2 3. plane y = 2 correct 4. plane x = 2 5. plane x = 2 6. plane z = 2 Explanation: Writing r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k , we see that y ( t ) = 2 for all t . Consequently, r ( t ) traces out a curve in the plane y = 2 . 004 (part 2 of 2) 10.0 points Determine the radius and center of the cir- cle traced out by r ( t ). 1.
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hw 12 - rhodes (ajr2283) HW12 Gilbert (56380) This...

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