Example 4: Estimating Intersection Points and Shading Regions for Inequalities

* Objectives*: To graphically estimate
the intersection point of two curves, and to shade solutions
regions for inequalities

* Example 4A*: Using DPGraph,
graphically estimate the points of intersection of the curves

Click here to download the .dpg file. The range for

x, namely, from 0 to 7 is chosen to show thatln(x)clears the maximum of cosine after the first intersection point. Hence there can be only one intersection point between the two curves. This is illustrated in Figure 1 below:In the last line of the .dpg file, notice the equation

x = a. This plots a vertical line through the plot at the value specified in the file for the valuea. Using the scrollbar, change the value ofa, thus moving the vertical line until it passes through the intersection point. The value ofaat this point is given at the bottom of the DPGraph screen and noted in Figure 1. Based on this estimate, we may conclude that the equalityln(1.304) = cos (1.304)holds approximately (to two decimal places according to the calculator with cosine computed in radian mode).

* Example 4B*: Graph the domain of the
function

*z = f(x,y) = ln(4(x**^2** + y**^2**) - 2(x + y) - x**^4** - y**^4**)).*

Since the function

ln(..)is defined only when inside the parentheses is positive, we need to graph the following maininequality:

4(x^2+ y^2) - 2(x + y) - x^4- y^4 > 0

This is easily done with the DPGraph. Click here for the .dpg file, which produces the following graph:

In the last line of the .dpg file, note the inequality z < 0 that is separated by "&" from the main inequality. This

zinequality is responsible for the "shading" while the boundary of the shaded region is determined by the main inequality. All the white areas, including the circular hole, represent regions where the main inequality doesnothold.