This is about my math class - Essay Example

This paper will focus on a specific function in three variables. In order to make our discussion systematic, we shall divide the discussion into three cases, one for each value or range values of the given parameter. The function in focus is given by, (1) where b is the parameter of the given function. Let us first consider the trivial case in which b is equal to zero, giving us (2) . Figure 1 shows a sketch of the graph of this function. As seen from the graph, it is simply a quadric surface, generated by a parabola on the xz plane, moved along the y-axis.

When taking the implicit characterization of the graph on the xyz space, we see that we come up with a line on the y-axis. Fig. 1. Graph of . Next, we take the case in which b is greater than zero. The graph of this function is an elliptic paraboloid, with the standard form , illustrated in Figure 2. Figure 2. Graph of . The elliptic paraboloid is a quadric surface, with a distinctive “nose-cone” appearance. Horizontal sections of this graph turn out to be ellipses while vertical sections are parabolas, hence its name.

When b is equal to one, that is, the standard form given here, horizontal sections are actually circles. Narrower ellipses are generated when b is less than 1 and wider ones are formed when b is greater than 1. Suppose we want to find the tangent plane of this function at the point (1,2). Recall that to find the equation of the tangent plane to the surface at the point , we first need to get partial derivative of f with respect to x and the partial derivative of f with respect to y and plug in the values to the formula of the tangent plane which is (3) .

And so we get, Thus, the equation of the tangent plane is, Figure 3 illustrates a sketch of the graph of the function with the corresponding tangent plane at (1,2). The saddle and extremum points of the function are at the origin. Fig. 3. Graph of with tangent plane at (1,2), having the equation The last case that we will consider is when b is less than zero. Again, for the sake of simplicity, let us take the case when b is equal to -1 and make our generalizations from that case. Figure 4 shows the graph of , which is a hyperbolic paraboloid.

As with the elliptic paraboloid, such a graph got its name because its horizontal sections are hyperbolas while its vertical sections are parabolas. Fig. 4. Graph of . Taking the same steps in the previous section to find the tangent plane to the graph at (1,2), we have the following calculations. Thus, the equation of the tangent plane is, Figure 5 illustrate the tangent plane of the function at (1,2). Fig. 5. Graph of with tangent plane at (1,2), having the equation The saddle point of the given function is located at the origin, and the extremum is also at the origin.

In conclusion, the function generates three different kinds of graphs on the xyz space depending on the value of the parameter b. The surface generated may either be a parabola moved along the y axis when b is zero, an elliptic paraboloid when b is positive, or a hyperbolic paraboloid when b is negative. Traces on the two dimensional plane may be inferred from the names of their graphs, with the trace on the horizontal plane denoted by the