the path of a particle in the xy-plane is vector r = (cos2t, sint) for t for all [-pi/2, pi] where t represents time. Sketch the path. Is it a smooth curve?
How do i sketch the path? do i just plug in random points between [-pi/2, pi]? how would i connect it together? and how do i know if it's a smooth curve? Thank you.
If that pair of coordinates represents (X,Y), then the easiest way is to do pretty much what you already suggested, but do it systematically in small increments. That is, work out a set of points as follows. -pi/2 is approximately -1.57, so
t=-1.57, X=cos(2t)=-1, Y=sin(t)=-1
t=-1.4, X=cos(2t)=-0.942, Y=sin(t)=-0.985
t=-1.3, X=cos(2t)=-0.856, Y=sin(t)=-0.963
etc etc. Then just plot X vs Y. If you've got access to a spreadsheet you'll find it very quick to do that.
Just one question though: are you sure that interval isn't [-pi/2, +pi/2]? If so, then yes, it's a smooth curve between those limits. But if you're correct about the upper limit for t being pi, then the curve is rather more complicated: it runs smoothly from (-1,1) through (0,0), up to (-1,1) and then bounces back on itself again to (0,0), overwriting itself in the process. If I had to bet, I'd guess that the question has been copied down wrong.
To sketch the path of the particle defined by the vector r = (cos2t, sint), you can follow these steps:
1. Start by selecting a range of values for t that covers the given interval [-π/2, π]. For example, you can choose t = -π/2, -π/3, -π/4, ...π/3, π/2, π, etc.
2. Plug each value of t into the vector r = (cos2t, sint) to obtain corresponding x and y coordinates for each point on the path.
3. Plot these (x, y) coordinates on the xy-plane, creating points on your sketch.
4. Connect the plotted points in the order they appear on the graph, maintaining the sequence defined by the values of t. This will allow you to connect the points smoothly, representing the path of the particle.
5. Once you have completed connecting the points, assess the resulting curve. A smooth curve typically means that there are no abrupt changes, discontinuities, or sudden breaks in the path. It should flow continuously, without any sharp turns or corners. Observe the sketch to determine if it meets these criteria.
In this particular case, the vector r = (cos2t, sint) represents an elliptical path since the x-coordinate is defined by the function cos(2t) while the y-coordinate is simply sin(t). Therefore, the resulting path should be smooth and represent an elliptical shape on the xy-plane when you connect the points.