Zooming in on a light ray bouncing off a curved mirror.
The most important image in calculus is the movie below. This movie shows a section of a curved mirror with a light ray coming in and bouncing off. As the movie plays, the scene is magnified and under high magnification we see that the curved mirror looks flat. This is the key feature of a differentiable function
Sometimes we use the word smooth rather than differentiable.
Zooming in on a light ray bouncing off a curved mirror.
Because such curves look flat under high magnification, when we are looking at a phenomenon -- like a light ray bouncing off a mirror -- that is a "local" phenomenon -- that is, that happens in a small neighborhood of a point on a curve -- we are able to think of the curve as being flat. Because flat curves are easy to understand, this is often exactly the lever that enables us to solve or at least begin solving a problem that at first seems very difficult. This is the basic principle behind Newton's Method. It was also the basic principle behind our work with curved mirrors.
We can do exactly the same thing with the graph of a function z = f(x, y) of two variables that we did in the movie above for the graph of a function y = f(x) of one variable -- we can pick a spot on the graph and zoom in on a small neighborhood of the point using higher and higher magnification. Look at the movie below. Notice that as the magnification increases the surface looks more and more like a flat plane.
Zooming in on a surface.
Your CAS window looks at the example above and then at another point on the same surface. Follow the instructions in your CAS window to look at these two examples. Then look at some other examples of your own choice.
Not all functions y = f(x) are smooth. For example, the function f(x) = |x| has a sharp point at the origin.
and under high magnification this point stays sharp.
Notice that despite progressively higher magnification the graph never looks flat.
near the point (0, 0, 0) using your CAS window. Notice that under high magnification this surface does not appear flat.
near the point (0, 0) under high magnification in your CAS window. Notice that even though this function does not have any sharp points or corners it still does not look flat near (0, 0) under high magnification.
Definition:
A function
is said to be smooth or differentiable near the point (x0, y0) if under high magnification the graph of the function near this point looks like a flat plane -- that is, it looks like the graph of a function of the form
where a, b, and c are constants. This is exactly analogous to the situation for a function
of one variable, which is smooth or differentiable at a point x0 if under high magnification its graph looks like a function of the form
where m and b are constants. In this situation the constant m is called the slope of the function T(x) and is f'(x0). The constant b is f(x0) - m x0.
Use your CAS window to compare the functions f(x) and T(x) near the indicated point x0 in each of the examples below.
Partial Derivatives
One way to understand how a function
behaves near a point (x0, y0) is to fix one of the two variables -- say, x at its value x0 and then change just the other variable. For example, suppose that we are interested in the function
near the point (0, 1). Consider the three graphs below.
The graph above shows the function f(x, y) with -2 <= x <= 2 and -2 <= y <= 2
.
The graph above is part of the first graph. The first graph has been sliced so that the plane x = 0 is the right face of the graph. Notice that along this face the graph looks like z = -y2.
You can look at these same slices using two-dimensional graphs rather than three-dimensional graphs as shown in the two figures below.
By changing just the variable x and leaving y fixed, we can focus our attention on the way in which z depends on x. In particular, we can compute the partial derivative of z with respect to x.
Similarly, by changing just the variable y and leaving x fixed, we can focus our attention on the way in which z depends on y. In particular, we can compute the partial derivative of z with respect to y.
These partial derivatives are computed using exactly the same methods and formulas as ordinary derivatives. During the computation of the partial derivative with respect to x we must remember that y is a constant. Similarly, during the computation of the partial derivative with respect to y we must remember that x is a constant. For example, if
Then
and
Your computer algebra system will compute partial derivatives as easily as ordinary derivatives.
Compute both partial derivatives of each of the following functions "by hand" and then check your answer using your CAS window.
The two partial derivatives give us a lot of geometric information about the function z = f(x, y). For example, the Mathematica log below shows how we can use the partial derivatives to draw the tangent plane to the surface
at the point (-0.5, -0.5).
Since the partial derivative of z with respect to x tells us the slope of the function z = f(x, y) in a plane parallel to the zx-plane the vector
is tangent to the surface z = f(x, y). Similarly, since the partial derivative of z with respect to y tells us the slope of the function z = f(x, y) in a plane parallel to the yz-plane the vector
is tangent to the surface z = f(x, y).
We can use these two tangent vectors to give us a vector, Q, perpendicular to the surface.
and a normal vector, N, to the surface
Show that the vector Q and, hence, the vector N point generally upward.
By putting this together with the work we did on shading the sphere you can use your CAS window to produce nicely shaded and colored pictures of a surface like
