Winter 2008 - Cioaba's Class - Exam 1 (Version 2)

# Winter 2008 - Cioaba's Class - Exam 1 (Version 2) - Name...

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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(2 t ) , sin(2 t ) , t ) in the direction of increasing t . At the same time, Tom starts running on the curve c ( t ) = ( cos(4 t − π 2 ) , sin(4 t − π 2 ) , 2 t − π 4 ) 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 ) = (− 2 sin(2 t ) , 2 cos(2 t ) , 1 ) which means that its speed at time t equals | m ′ ( t ) | = radicalbig ( − 2 sin(2 t )) 2 + (2 cos(2 t )) 2 + 1 2 = √ 4 + 1 = √ 5. The velocity of Tom is c ′ ( t ) = (− 4 sin(4 t − π 2 ) , 4 cos(4 t − π 2 ) , 2 ) which means that its speed at time t equals | c ′ ( t ) | = radicalbig ( − 4 sin(4 t − π 2 )) 2 + (4 cos(4 t − π 2 )) 2 + 2 2 = √ 16 + 4 = √ 20. The time when Tom catches Jerry is the time t > 0 when their coordinates are equal. This means that ( cos(2 t ) , sin(2 t ) , t ) = ( cos(4 t − π 2 ) , sin(4 t − π 2 ) , 2 t − π 4 ) which happens when t = 2 t − π 4 . This means t = π 4 . 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 4 . This equals integraltext π 4 | c ′ ( t ) | dt = integraltext π 4 √ 20 dt = π √ 20 4 . 2.a.(3 points) Consider the points P such that the distance from P to the point (4 ,...
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Winter 2008 - Cioaba's Class - Exam 1 (Version 2) - Name...

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