Minimization Problem: Distance from a Point to a Curve

This post is a further exploration of the optimization problem of finding a point on f(x) = sinh(x) – xe^-3x closest to the point (1,7).  The problem is part of our Teaching Concepts with Maple web site, a collection of video examples and Maple worksheets designed to illustrate how Maple can be used to generate greater insight and understanding with its ease-of-use features.

The problem is stated as:

After looking over the solution to this problem, I was moved to write the animation in figure 3.8.1 below.

 

Click on the graph, then move the slider in the animation toolbar that opens at the top of the worksheet. The number written above the red dot (the point (1, 7)) is the distance from that point to the curve. The slider controls the x-coordinate along the x-axis. The code for the animation is hidden behind the cell containing the animation.

Frankly, this animation appeared at the end of a worksheet in which a number of preliminary calculations first were realized. After defining the function and an expression for d(x), the distance from (1, 7) to the graph of f(x), a graph of d(x) was drawn. Surprisingly, it showed two minima. These two minima had to be found and compared. All the while, the thought of the animation in Figure 3.8.1 was lurking in the background. So then, here are the additional calculations related to the posted solution to the problem.

Apply the Minimize command from the Optimization package:

Obtain the leftmost minimum:

 

Clearly, X₁ is the global minimum, and the corresponding y-coordinate on the graph of f is

 

Of course, the line from (1,7) to the distance-minimizing point on the graph of f is orthogonal to that graph, but let’s see if the orthogonality can be demonstrated. The slope of the minimizing line is:

The slope of the tangent line at the minimizing point:

Show that m M ≐ (-1) to establish orthogonality.

Likewise, the reader can check that a line from (1,7) to the “other” minimum is also orthogonal to the graph of f.

 

Download this worksheet: Teaching_Concepts-Minimization Problem.mw