Geogebra for Exercise #60 on Page 112 of Stewart

First, enter the equation for the circle (x - 1)² + y² = 1 in the input box, as shown in Figure 1.

Press the Enter key and the graph of the circle (x - 1)² + y² = 1 will appear, as shown in Figure 2.

Press the Enter key to record the value of r, as shown in Figure 4. Note how r = 1.5 is added to the list of Free Objects.

Next, we create a circle of radius r centered at the origin. First, enter the equation x² + y² = r² in the input box, as shown in Figure 5.

Press the Enter key to create the graph of the circle x² + y² = r², as shown in Figure 6. Because this circle "depends" upon the "Free object" r, note how the equation of this circle is added to the list of "Dependent objects" at the left.

Now we will use the intersection tool to find some points of intersection. The second tool from the left has a little down-arrow in its lower right-hand corner. Click the little arrow and a drop-down list of tools appears. Select the tool that says "Intersect two objects." Proceed as follows:

• Use your mouse to click the vertical y-axis and the circle centered at the origin with radius r. This will create two points of intersection, A and B, as shown in Figure 7.
• Next, click each of the circles with the intersect tool. This will create two points at C and D where the circles intersect, also shown in Figure 7.

Next, we want to draw a ray, starting at the point A and passing through the point C. To do this, click the little down-arrow in the third tool from the left, then select "Ray through two points" from the drop-down list that appears. Use the tool as follows:

• Use your mouse to first click the point A, then the point C, in that order. This will create the ray starting at point A and passing through point C, as shown in Figure 8.

Now we want to find the point where the ray AC intersects the x-axis. Select the "Intersect two points" tool from the drop down list of the second tool from the left. Click the ray, then click the x-axis to produce a point of intersection at E, as shown in Figure 9.

The goal of this exercise is to determine what happens to the point E as the radius of the circle centered at the origin shrinks to zero. There are several ways to proceed at this point.

• A brute force approach would be to double-click the value of r in the list of "Free objects" at the left and manually change its value, first to 1, then 0.5, then 0.2, then 0.1, etc., keeping track of what happens to the point at E as you move r ever nearer to zero.
• A second approach is to right-click the r = 1.5 in the list of "Free objects," then select "Show object" from the drop-down list that appears. This produces the slider for r shown in Figure 10.

• A simple method is to just move the point on the slider with your mouse, keeping track of what happens to the point E as the value of r moves towards zero.
• You can also select the "Move tool" form the drop down list of the first tool on the left, then click r = 1.5 in the list of "Free objects." Once selected, you can change the value of r with the arrow keys on your keyboard.
• To change the increment on r, right-click r = 1.5 (the value of r has probably changed at this point) in the "Free objects" list, then select "Properties" from the drop-down list. Select the "Slider" tab in the Properties dialog. Change the value of "min" to 0, "max" to 1, and "Increment" to 0.001, then Close the dialog.

  You can now return to moving your slider or using your arrow keys to change the value of r. Note that the change in increment now allows more fine-grain control as you force r to approach zero.

Just one more question to ask: "What happens to the point E as r approaches zero?"