Math 178Worksheet 1: Parametric Curves
Directions: Show all work, being sure to support your answers. Plots and tables must be technology-generated and printed. Plots must have the same horizontal and vertical scales (axis equal in Matlab).
Example: Let and . Plot the graph of the parametric curve for the interval . Indicate (by hand) the starting point and place arrows on the graph to indicate the direction in which the curve is traced.
»t = 0:.01:4.*pi; x = 2.*cos(t./2); y = 3.*sin(t);
»plot(x,y); axis equal
Example: Create a table in Matlab using increments of 0.5 on .
»t = 0:0.5:2.*pi; x = 2.*cos(t./2); y = 3.*sin(t);
»[t' x' y']
ans =
0 2.0000 0
0.5000 1.9378 1.4383
1.0000 1.7552 2.5244
1.5000 1.4634 2.9925
2.0000 1.0806 2.7279
2.5000 0.6306 1.7954
3.0000 0.1415 0.4234
3.5000 -0.3565 -1.0523
4.0000 -0.8323 -2.2704
4.5000 -1.2563 -2.9326
5.0000 -1.6023 -2.8768
5.5000 -1.8486 -2.1166
6.0000 -1.9800 -0.8382
Problems:
In problems 1 - 5 use the following parametric equations:
1. Plot separately each curve given above for . Make the horizontal and vertical scales equal (axis equal in Matlab).
2. All of the parameterized curves given above lie on the graph of a single Cartesian equation. What is the Cartesian equation? Explain how you can derive this Cartesian equation from the third set of parametric equations.
3. Make a table of values in increments of 0.5 for each curve for . Include columns for t, x and y and label each. Be sure to include endpoints in your tables. (It is permissible to generate each table using technology without the endpoints, and then add the endpoint values to your table.)
4. On each of the plots obtained in your solution to problem 1, indicate (by hand) the following:
a. Label the starting and ending points of the curve.
b. Use arrows to show the direction in which the graph is being traced.
5. Shown below are the graphs of x(t) and y(t) for one of the three curves given above. Notice that the horizontal axis represents t, not x. Decide which set of curves (above) was used, which curve below is x, and which curve below is y. Describe how the graph can be used to determine the starting point and direction of the parametric curve.
6. A bug is traveling along a path described by the parametric equation .
a. Plot the path of the bug for the first 6 seconds (). Make the horizontal and vertical scales equal (axis equal in Matlab).
b. Where does the bug begin its travels? Where is it after 6 seconds?
c. Plot points that show the location of the bug at every .1 seconds over the six-second interval. Again, make the horizontal and vertical scales equal.
d. How does the spacing of the dots relate to the bug's speed? Indicate (roughly, by hand) where the bug appears to be moving slowest.
7. A second bug is traveling along the path at the same time as the bug in problem 6.
a. Graph the two paths on the same plane. How many times does the path of the second bug cross the path of the first bug? These are points of intersection.
b. Are any of these point(s) of intersection collision point(s)? In other words, are the bugs ever at the same place at the same time? If so, find the collision point(s).