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course Mth 277
9/9/12 11pm
Submit the following in the usual manner.`q001. Describe the path defined by the parametric equations
x(t) = - cos(t)
y(t) = sin(t)
0 <= t <= pi.
Your description should indicate where the path starts, where it ends, the direction of motion along the path if we assume that t represents clock time, and the shape of the path.
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t= 0. x= -cos(0) = -1. y= sin(0) = 0.
t= pi/6. x= -cos(pi/6) = -sqrt(3)/2. y= sin(pi/6)= 1/2.
t=pi/2. x= -cos(pi/2) = 0. y= sin(pi/2) = 1.
t= pi. x= -cos(pi) = 1. y = sin(pi) = 0.
This is a semicircle starting at (-1,0) and traveling clockwise to (1, 0).
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`q002. Describe the path defined by the parametric equations
x(t) = cos(t)
y(t) = t^2
0 <= t <= 4 pi.
Your description should indicate where the path starts, where it ends, the direction of motion along the path if we assume that t represents clock time, and the shape of the path.
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t= 0. x= cos(0) = 1. y= 0.
t= pi/2. x= cos(pi/2) = 0. y = pi^2/4.
t= pi. x=cos(pi) = -1. y = pi^2.
t= 3pi/2. x=cos(3pi/2)= 0. y = 9pi^2/4.
t= 2pi. x= cos(2pi)= 1. y = 4pi^2.
t= 3pi. x= cos(3pi)= -1. y = 9pi^2.
t= 4pi. x= cos(4pi) = 1. y = 16pi^2.
On a polar graph this is a spiral starting at (1,0) and curving outwards in a clockwise direction, ending at (1, 16pi^2).
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This graph wouldn't be a polar graph. x and y are rectangular coordinates.
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What is the expression for the distance of the point (x(t), y(t)) from the origin?
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sqrt( x(t)^2 + y(t)^2 )
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That would be the answer for any set of parametric equations for x(t) and y(t). It doesn't tell us anything about this set of equations.
x and y are defined in terms of t; your result should be expressed in terms of t.
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What is the derivative of the distance-from-the-origin function?
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( sqrt( x(t)^2 + y(t)^2 ) )'
If x and y are treated as constants.
1/(2*sqrt( x(t)^2 + y(t)^2 )
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x and y are both functions of t; the derivative would be with respect to t.
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If x and y were constants, the expression would be constant and the derivative would be zero.
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`q003. Describe the path defined by the parametric equations
x(t) = - 3 cos(2 t)
y(t) = 5 sin(2 t)
0 <= t <= pi.
Your description should indicate where the path starts, where it ends, the direction of motion along the path if we assume that t represents clock time, and the shape of the path.
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t=0. x= - 3 cos(2 *0) = -3. y(t) = 5 sin(2 *0)= 0. (-3,0).
t=pi/4. x= - 3 cos(2 *pi/4)= 0. y(t) = 5 sin(2 *pi/4)= 5. (0,5).
t=pi/2. x= - 3 cos(2 *pi/2)= 3. y(t) = 5 sin(2 *pi/2) = 0. (3,0).
t= 3pi/4. x= -3cos(2 * 3pi/4) = 0. y(t) = 5 sin(2 *3pi/4)= -5. (0,-5).
t=pi. x= - 3 cos(2 *pi)= -3. y(t) = 5 sin(2 *pi)= 0. (-3,0).
This is an ellipse with vertices at (+- 3,0) and (+- 5,0), center at the origin. Starts at (-3,0) and travels clockwise back to its original point.
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`q004. Describe the path defined by the parametric equations
x(t) = cos(2 t)
y(t) = sin(3 t)
0 <= t <= 2 pi.
Your description should indicate where the path starts, where it ends, the direction of motion along the path if we assume that t represents clock time, and the shape of the path.
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t= 0. x= cos(0) = 1. y = sin(0)= 0.
t=pi/6. x= cos(2* pi/6) = 1/2. y=sin(3* pi/6) = 1.
t=pi/3. x= cos(2* pi/3) = -1/2. y= sin(3* pi/3) = 0.
t= pi/2. x= cos(2* pi/2)= -1. y= sin(3* pi/2) = -1.
t=2pi/3. x= cos(2* 2pi/3)= -1/2. y = sin(3* 2pi/3) = 0.
t= 5pi/6. x=cos (2*5pi/6) = 1/2. y= sin(3* 5pi/6) = 1.
t= pi. x= cos(2*pi) = 1. y = sin(3* pi) = 0.
t= 3pi/2. x=cos(2* 3pi/2) = -1. sin(3* 3pi/2)= -1.
t= 2pi. x= cos(2* 2pi) = 1. sin(3*2pi) = 0.
This sortve goes in a counterclockwise motion and the curve is somewhat fish-shaped... It's difficult to describe.
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`q005. An amusement park ride consists of long arms of length 5 meters radiating from a central axis, with another axis at the far end of the arm and shorter arms of length 2 meters radiating from that axis. The central axis rotates the system at the rate of .6 radian per second, so that the end of the first arm follows the path defined by x(t) = 5 meters * cos( .6 rad/sec * t), y(t) = 5 meters * sin(.6 rad/sec * t).
The axis at the end of the long arm rotates independently at .9 radians per second.
What are the parametric equations of the position of a rider at the end of the short arm?
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following the pattern of the long arm equations.
short arm x(t) = 2 * cos (.9 t). y(t) = 2* sin(.9 t).
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What function describes the distance of the rider from the center as a function of t?
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This would be the equations of the long arm plus the equations of the short arm.
Total position function.
x(t) = 5 meters * cos( .6 rad/sec * t) + 2 * cos (.9 t).
y(t) = 5 meters * sin(.6 rad/sec * t) + 2* sin(.9 t).
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Good.
Now what is the distance function?
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How would you go about finding the maximum rate at which the rider is moving away from the center? You don't have to actually do this, just describe as best you can how you would go about it.
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Rate implies a derivative.
You would find the derivative of the total position function and maximize it.
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Right.
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