geometric_limit.html

An Interesting Geometric Limit

In this lab you will determine the limit of a point P on the positive x -axis where the location of P is determined geometrically. We consider a circle with a fixed radius k centered at the point ( k ,0). Note that this circle is tangent to the y -axis at the origin. Then we take an arbitrary circle of radius r centered at the origin. Let Q denote the positive y -intercept of the circle centered at the origin and let S denote the point in the first quadrant at which the two circles intersect. Draw a line L through Q and S and define P as the x- intercept of L. A picture of the two circles in the case k = 1 is shown below. The question is to determine what happens to the point P as the radius of the circle centered at the origin shrinks to 0.

The picture above is in fact an animation. If you click in the graphic area, you will get a different toolbar at the top of your screen for playing animations. If you click on the button, you can play the animation. If you click on the button, you can step through the animation one frame at a time. There are other buttons on the animation toolbar that control the direction in which the animation is played (forwards or backwards) and the speed at which the animation is played. Does the animation seem to suggest that the point P has a limit as r approaches 0? What does the limit appear to be?

Projcet

Let k denote the number of letters in your first name. Let C denote the circle of radius k centered at ( k ,0). Let D = D ( r ) denote the circle of radius r centered at the origin.

1. Define k and write down the equation of the circle C . Write down the equation of the circle D ( r ). What are the coordinates of the positive y -intercept Q of the circle D ( r )?

2. Determine the coordinates of the point of intersection S in the first quadrant of the circles C and D ( r ). Your answer should depend only on the variable r . Please copy and execute the following Maple command to force Maple to solve equations as completely as possible..

> _EnvExplicit:=true;

3. Determine the slope of the line L through the points Q and S in terms of R. Use the slope-intercept form of the equation of a line to write down an equation for the line L . Use the equation of the line to solve for P , the x -intercept of the line L , in terms of r .

4. Define a function f ( r ) as the x -coordinate of P . Make a table of values of f ( r ) for small values of r . Does f appear to have a limit as r approaches 0 from the right? Is it obvious from your formula for f ( r ) that f ( r ) has a limit as r goes to 0? Use the limit command to calculate the limit of f ( r ) as r approaches 0 from the right.

5. Check the result from Maple by simplifying your expression for f ( r ) to an equivalent expression that makes sense when r = 0.

Extra Credit

Repeat the above exercise with the circle centered at ( k ,0) replaced by the parabloa . Determine a formula for the x -intercept as a function f of r and make a table of values of f ( r ) and then use Maple to evaluate the limit from the right. As a challenge, see if you can simplify your expression for f ( r ) to an equivalent expression that makes sense when r = 0.

The Most Common Maple Commands

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.

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I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

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