course mth174
I have a pretty well understanding of mostly everything that is on my upcoming math test. But the only things I'm having slight trouble with is usage of the Fourier Series and using the coefficients to sketch graph. I understand like the constants and the integrals. Also, using the differential equation to model situations is pretty easy, but maybe I just haven't read the text thoroughly enough, can you give me some information for using the phase plane and using trajectories to answer questions that will be answered.
The phase plane for y vs. x has a direction field constructed by evaluating dy/dx = (dy/dt) / (dx/dt) at a grid of points.
If you start at a point and follow the direction field you obtain a trajectory.
A closed trajectory is called an orbit.
A trajectory can be closed, or unbounded, or can lead to a finite equilibrium point.
Equilibrium points occur when dy/dt = dx/dt =0. The equilibrium is stable if the trajectories of points near the equilibrium point go toward the equilibrium point, and unstable if the trajectories lead away.
The set of points where dy/dt = 0 form nullclines, along which the direction field is horizontal.
The set of points where dx/dt = 0 form nullclines, along which the direction field is vertical.
Inside a region bounded by nullclines the trajectories have similar behavior, which can be analyzed by considering the signs of dy/dt and dx/dt in each region.
Do you have questions on any of the assigned problems related to phase plane and trajectories?
That's pretty much it - I think that I will make a high A on the next test but I just generally want to pin down these confusions first and foremost.
Since I want to make sure my 'theoretical' understanding is decent, what will be the best way to show that an integral of, like, cosxsinx (something of that nature) will be equal to zero, using a graph and symmetry arguments.
To graph Fourier series, first you need to understand the graphs of the sine and cosine functions. Be sure you can easily graph any function of the form A sin(m x) or B cos(n x).
Then you need to be able to construct the graph of the sum of two or more graphed functions (e.g., construct the graph of .5 sin(3x) + .2 cos(5x) from the graphs of .5 sin(3x) and .2 cos(5x)).
Graph cos(x) * sin(x) on the interval [0, 2 pi). You should see why the graph of cos(x) sin(x) is positive on the interval [0, pi/2), negative on the interval [pi/2, pi} and again on the interval [pi, 3 pi/2), then positive on the interval [3 pi/2, 2 pi), and it should also be clear from the graphs of the sine and cosine why the areas of the four regions are all equal.
Then graph cos(2x) and sin(3x) on the same interval and think about how you would construct the graph of cos(2x) sin(3x).
Ask me on break tomorrow and I'll look at your work and show you the rest.