You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
The movie below shows a series of light rays bouncing off a flat mirror. Each of the light rays is vertical. The first light ray hits the mirror at its left edge and as the movie runs (or as you move the slider bar at the bottom of the movie) the light rays move from left to right and hit the mirror in different places. All the incoming light rays are parallel to each other and because the mirror is flat the reflected or outgoing light rays are also parallel. The last frame of the movie shows all the light rays in one frame. Notice that all the reflected light rays are parallel, as advertised.
Light reflections from a flat mirror
The next movie shows a series of light rays bouncing off a circular mirror. Each light ray is vertical and, as in the first movie, as the movie plays the light rays move from left to right. As in the first movie, all the incoming light rays are parallel but now because the mirror is not flat the reflected light rays are no longer parallel to each other. When we look at all the light rays together in the last frame we see an interesting pattern, called a light caustic formed by the reflected light rays. The shape of this particular caustic is called a cusp. You have probably seen similar caustics in many different places -- for example, caustics are often seen on the surface of coffee in a cup.
Light reflections from a circular mirror
Calculus can help us understand many interesting and often very surprising optical phenomena like caustics and like the distortions we see when we look at objects underwater. Optics is one of the recurring themes in Open Calculus and in Open Physics.
When a light ray hits a flat mirror as shown in the figure below, it bounces off so that the angle of incidence (the angle the incoming light ray makes with the mirror) is equal to the angle of coincidence (the angle that the outgoing, or reflected, light ray makes with the mirror.)
You can verify this fact experimentally with a flashlight and a mirror. Experiments like this or more precise experiments using lasers provide lots of evidence supporting the "law" --
The angle of incidence is equal to the angle of coincidence.
But no matter how many experiments we perform we can't be absolutely certain that a "law" like this is always true. Experimental physicists and theoretical physicists (with the help of a lot of mathematucs) work together to discover the basic ideas that enable us to explain and predict physical phenomena like light rays bouncing off mirrors. This particular "law" can be explained in terms of a more basic "law" -- Fermat's Principle.
Using the "law" that the angle of incidence is equal to the angle of coincidence and some trigonometry we can compute the slope of the reflected light ray when an incoming vertical light ray hits a mirror whose slope is m as shown in the figure below.
The formula is
Click here for more about this formula.
Answer each of the following questions geometrically by drawing pictures and then algebraically using the forumla above. Check that both answers agree.
Zooming in on a light ray bouncing off a curved mirror.
Notice that under high magnification this small section of the curved mirror looks like a straight line. This gives us an idea about how we might determine how a light ray is reflected from a curved mirror. We need to determine the slope of the line we see under extremely high magnification.
We can look at this problem in three ways -- geometrically, numerically, and algebraically. To look at the problem geometrically we add one more line to our movie -- the tangent line to the curve at the point where the light ray strikes the mirror.
Zooming in on a light ray bouncing off a curved mirror with tangent.
Notice that under high magnification the mirror looks almost exactly like its tangent.
The light caustic movie looks at a portion of a circular mirror. The radius of the circle is 8 units and its center is at the point (0, 8). The (lower half) of this circle can be described by the equation y = f(x) where f(x) is the function
Notice this function appears in your CAS window.
The figure below shows a typical point on this circle, the line from the center of the circle to this point (in blue and marked L), and (in red) the tangent to the circle at the same point.
The slope of the line L is easily calculated using the two points shown on the graph --
The line L and the tangent meet at a right angle and thus the slope of the tangent is the negative reciprocal of the slope of L giving us --
Notice that this formula appears in your CAS window.
We can approach this same problem -- finding the slope of the tangent -- numerically by looking at a second point s that is very close to the point x and computing the slope of the secant line through the two points (x, f(x)) and (s, f(s)). See the figure below.
The slope of this secant line is given by
f(s) - f(x)
slope of secant = -----------
s - x
Look at your CAS window in the cell that produces the light caustic for the circular mirror. In that cell there is a line like --
Mathematica --
slope[x_] := x/Sqrt[64 - x^2] (* Slope of tangent *)
Maple --
slope := x -> x/sqrt(64 - x^2): # slope of tangent
This line computes the slope of the tangent to the circular mirror at the point (x, f(x)) by using the formula we obtained geometrically above. For the purposes of this program we do not need the exact slope of the tangent -- just a good estimate. You could find a good enough estimate by computing
f(x + 0.001) - f(x)
-------------------
0.001
Change the appropriate line in your CAS window to estimate the slope of the tangent this way and then execute the cell to see another way of producing the same caustic.
We now have a way of producing estimates of the slope of the tangent to a curve y = f(x) that enables us to produce an estimate that is as good as needed for any particular purpose. Thus, we can find the exact slope of the tangent by finding the limit
f(s) - f(x)
lim -----------
s --> x s - x
or, equivalently, the limit
f(x + h) - f(x)
lim ---------------
h --> 0 h
You may want to look at the module Estimation and Limits -- Tangents for another discussion of this same problem.
This limit has a name -- it is called the derivative of the function f(x) and is written f'(x). It can be computed either as
f(s) - f(x)
f'(x) = lim -----------
s --> x s - x
or, equivalently, the limit
f(x + h) - f(x)
f'(x) = lim ---------------
h --> 0 h
For our circular mirror, the computations are fairly complicated --
Notice that the final answer agrees with the answer we obtained earlier by geometric means.
f(x + h) - f(x)
f'(x) = lim ---------------
h --> 0 h
For each of the following problems compute f'(x) by computing one of the two limits.
Because this function is a parabola this mirror is called parabolic.