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Unformatted text preview: Name: PID: Discussion Section No: Time: TA’s name: Midterm 1, Math 20C  Winter 2008 Duration: 50 minutes Please close your books, turn off your calculators and phones and put them away. To get full credit you should explain your answers. 1. At t = 0, Jerry starts walking on the curve m ( t ) = ( cos( t + π ) , sin( t + π ) , 2 t ) in the direction of increasing t . At the same time, Tom starts running on the curve c ( t ) = ( cos(3 t ) , sin(3 t ) , 6 t − 2 π ) in the direction of increasing t . a.(3 points) Find the speed of Jerry, the speed of Tom and the time when Tom catches Jerry. Solution The velocity of Jerry is m ′ ( t ) = (− sin( t + π ) , cos( t + π ) , 2 ) which means that its speed at time t equals  m ′ ( t )  = radicalbig ( − sin( t + π )) 2 + (cos( t + π )) 2 + 2 2 = √ 1 + 4 = √ 5. The velocity of Tom is c ′ ( t ) = (− 3 sin(3 t ) , 3 cos(3 t ) , 6 ) which means that its speed at time t equals  c ′ ( t )  = radicalbig ( − 3 sin(3 t )) 2 + (3 cos(3 t )) 2 + 6 2 = √ 6 + 36 = √ 45. The time when Tom catches Jerry is the time t > 0 when their coordinates are equal. This means that ( cos( t + π ) , sin( t + π ) , 2 t ) = ( cos(3 t ) , sin(3 t ) , 6 t − 2 π ) which happens when 2 t = 6 t − 2 π . This means t = π 2 . b.(2 points) What is the distance covered by Tom from t = 0 until he catches Jerry ? Solution The distance equals the distance covered by Tom from t = 0 until t = π 2 . This equals integraltext π 2  c ′ ( t )  dt = integraltext π 2 √ 45 dt = π √ 45 2 . 2.a.(3 points) Consider the points P such that the distance from P to the point ( − 1 , 5 , 3) is twice the distance from P to the point (6...
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This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Winter '08 term at UCSD.
 Winter '08
 Helton
 Math

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