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Parametric Equations and Polar Coordinates

Problems

Parametric Equations

Polar Coordinates

Problem : Is the following plane curve a function: y = 3t 2 , x = , 0≤t≤5 ?

Yes. By examining the graph, you can see that for every x , there is only one f (x) .

Problem : The following plane curve is a circle: x = 2 cos(t) , y = 2 sin(t) , 0≤t < 2Π . Is its orientation clockwise or couterclockwise? What happens when you reverse the parametric equations, so that x = 2 sin(t) , y = 2 cos(t) ?

The orientation of the first curve is counterclockwise. When the functions for x and y are exchanged, the curve's orientation becomes clockwise.

Problem : Convert the parametric equation x = 2t , y = , t > 0 , to a rectangular equation.

y = .

Problem : Convert the parametric equation x = 3t + 1 , y = , t , to a rectangular equation.

y = .

Problem : How many times does the graph of x = t 2 - t - 6 , y = 2t , -5 < t < 5 cross the y -axis?

Twice, when t = - 2 at (0, - 4) and when t = 3 at (0, 6) .

Problem : Jim and Bob are racing from the origin to the point (5, 10) . Let t be the number of seconds after the start of the race. Jim's position at any time t is given by the parametric equations x = t , y = 2t . Bob's position at any time t is given by the parametric equations x = 5t , y = 10t . Who will win the race? How long does it take each competitor to finish the race?

Jim reaches the point (5, 10) after t = 5 seconds. Bob will reach the point (5, 10) after t = 1 second. Bob will win the race.

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