Functions and graphs that are critically important.

For the rest of this course, understanding the sine and cosine functions, their graphs, and their graphical relation to each other, will be critically important.  That will help you understand our exploration of the nature and behavior of oscillating LC  circuits.

Let's begin with a broader discussion of the essential graphs and functions which are intertwined with the physics of this course. For this course there are two types of graphs that are important. It will be essential for you to become familiar and comfortable with these graphs, and with the functions that create them. If you can achieve that familiarity, then you will probably do well in this course.

The first type, which you've already seen, are the exponential functions. These comes in two varieties:
1) a growing exponential that usually starts at zero, and grows, first at a high rate and then more slowly, to an asymptotic limiting value.
2) a decreasing exponential that begins at a finite value––not zero––and decreases exponentially to zero (or, in some cases, to a finite limiting value which is not zero).
These graphs are associated with the functions exp(-t/tau) and (1-exp(-t/tau)), respectively. If you do not feel intimately familiar with these graphs, their appearance and their functions, then you might want to review that in the near future. For the coming week, however, it is the sine and cosine functions that will be critically important, as discussed next.

The second kind of graphs that we will use extensively are the oscillating functions: sine and cosine. These will be essential for our study of oscillating circuits which begins on Monday and will be a major part of the rest of this course.

Consider the function: sin(wt), where w has units of 1/seconds, also known as frequency and t is time. What is its value at t=0? Can you graph this function?  What is its highest value? What is its lowest value? What is its period? Do you know what period means? These questions are best answered by looking at a graph and seeing and understanding what the function looks like -- how it goes up and down and repeats as a function of time.

The function, cos(wt), is equally important. Unlike the sine function it starts out at 1, at t=0, but just like the sine function, it oscillates up and down and repeats as a function of time.
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Appendix.  Advanced topics related to the graphs and discussion above:

Derivatives:
First, note that the derivative of the increasing exponential from the previous section is a decreasing exponential. One can relate that to what one sees in the graph. The increasing exponential has a positive slope which slope starts out large and decreases in magnitude as t increases. (The slope and the derivative are intimately related.)

The cosine and sine functions are similarly related. Specifically, the derivative of sin(wt) is w*cos(wt). It is also true that the derivative of cos(wt) is -w*sin(wt).

Curvature:
We say that the increasing exponential function has concave downward curvature. On the other hand, the decreasing exponential can be described as having concave upward curvature. The cosine and sine functions have more complex curvature which oscillates betweeen upward to downward.