BERG HW01-solutions - sagbini (ces3287) HW01 berg (55811) 1...

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Unformatted text preview: sagbini (ces3287) HW01 berg (55811) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points If the constant C is chosen so that the curve given parametrically by parenleftBig C 2 12 t 2 , Ct parenrightBig , t 4 , is the arc of the parabola y 2 = 12 x from (0 , 0) to (3 , 6), find the coordinates of the point P on this arc corresponding to t = 2. 1. P = parenleftBig 3 4 , 3 parenrightBig correct 2. P = parenleftBig 3 16 , 3 parenrightBig 3. P = parenleftBig 3 2 , 3 4 parenrightBig 4. P = parenleftBig 3 , 3 4 parenrightBig 5. P = parenleftBig 3 2 , 3 16 parenrightBig 6. P = parenleftBig 3 16 , 3 2 parenrightBig Explanation: The point P has coordinates parenleftBig C 2 12 t 2 vextendsingle vextendsingle vextendsingle t =2 , Ct vextendsingle vextendsingle vextendsingle t =2 parenrightBig = parenleftBig C 2 3 , 2 C parenrightBig , so we need to find C . But we are told that the graph of parenleftBig C 2 12 t 2 , Ct parenrightBig passes through (3 , 6) when t = 4. Thus 4 C = 6 , i . e ., C = 3 2 . Consequently, P = parenleftBig C 2 3 , 2 C parenrightBig = parenleftBig 3 4 , 3 parenrightBig . keywords: parametric curve, parabola 002 10.0 points Find a Cartesian equation for the curve given in parametric form by x ( t ) = 1 8 t 2 , y ( t ) = 1 8 t 3 . 1. x = 1 2 y 3 / 2 2. x = 1 4 y 3 / 2 3. x = 1 2 y 2 / 3 correct 4. x = 1 4 y 2 / 3 5. x = 1 2 y 4 / 3 6. x = 1 4 y 4 / 3 Explanation: We have to eliminate the parameter t from the equations for x and y . But from the equation for y , it follows that t = 2 y 1 / 3 , in which case x = 1 8 parenleftBig 2 y 1 / 3 parenrightBig 2 = 1 2 y 2 / 3 . 003 10.0 points Determine a Cartesian equation for the curve given in parametric form by x ( t ) = 3 e t , y ( t ) = 2 e 2 t . 1. x 2 y = 12 sagbini (ces3287) HW01 berg (55811) 2 2. x 2 y = 18 correct 3. x 2 y = 12 4. x 2 y = 18 5. xy 2 = 6 6. x y 2 = 6 Explanation: We have to eliminate the parameter t from the equations for x and y . Now from the equation for x it follows that e t = x 3 , from which in turn it follows that y = 2 parenleftBig 3 x parenrightBig 2 . Consequently, x 2 y = 18 . 004 10.0 points Which one of the following could be the graph of the curve given parametrically by x ( t ) = t 3- 2 t, y ( t ) = 3- t 2 , where the arrows indicate the direction of increasing t ? 1. x y 2. x y correct 3. x y 4. x y 5. x y sagbini (ces3287) HW01 berg (55811) 3 6. x y Explanation: All the graphs are symmetric either about the y-axis or the x-axis. Lets check which it is for the graph of ( x ( t ) , y ( t )) = ( t 3- 2 t, 3- t 2 ) ....
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BERG HW01-solutions - sagbini (ces3287) HW01 berg (55811) 1...

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