Finding the Tangent Line

This program draws lines between pairs of points and computes the slope of that line. The points used are forced, by the program, to lie on the green curve that is plotted on the graph. When you click on the screen, the programs records the x (left and right) coordinate of the location of your click. It then substitutes that x value into the formula for the green line and determines the y-coordinate. Therefore, the coordinates used by the program always satisfy the equation of the green curve. As a result, ALL of the points plotted by the program lie on the green curve, because they satisfy the equation. Hence when you click on the screen a point will be plotted on the green curve, directly above or below the location of your click, since the x-coordinate is not changed. The equation for the first example below is y = 16 - xx ( "sixteen minus x squared" ).

We are interested in finding the tangent line to the green curve at a specific point, called the "anchor" point. That generally does not move although you can change it by clicking on the button labeled, surprisingly, "Set Anchor". The next click you make in the window will be plotted as before, except in red instead of blue and that point will subsequently be used as the anchor. When you click at any other time, a blue point will be plotted. Then the line that connects the blue point and the anchor (red) point will be drawn. Such a line, connecting to two points of the green curve, is called a "secant" line to the curve. Finally the slope of that line will be computed and displayed on the panel to the right. Other slopes may be displayed there as well, in fact, the twenty most recent slopes. If there are fewer than twenty, then the most recent will be at the bottom of the list. If the list or screen become to cluttered, clicking reset will erase everything except the green curve.

The "tangent" line to a curve at a point (here, the anchor point) is a line that, near the anchor point, only touches the curve once. Imagine that the curve is a green highway and you are driving down it in a car. When you reach the anchor point, you suddenly straighten the wheel. The car will then not stay on the road, but will continue in the direction in which it was going at the anchor point. This in fact describes the "direction" of the curve at the anchor point and is the reason why the tangent line is of such great importance.

Again, in a neighborhood of the anchor point, the tangent line only touches the curve once. On the other hand, every line drawn by this program connects the curve at two points. Hence, NONE of the curves drawn by this program will be tangent lines! So how can it help you to find the tangent lines? The answer lies in what you learn in your attempt to get close. You will find that the secret to getting a line that is "nearly" tangent, is simply to click on a point that is near to the anchor point. A little practice with this program will hopefully make that concept feel obvious but please don't make the mistake of believing that there isn't much to it. The subtlety lies in the analogy, in what it says about the real world. It took a man of the stature of Sir Isaac Newton to first see that analogy, so it behooves you to give it some serious consideration. A discussion of this analogy can be found by clicking here. For now I will continue with the discussion of the program.

Because "close" is important you may desire to input x-coordinates in a way that is more precise than clicking on the screen. You can do this with the text areas on the right. They are not just to show you values. If you type into the window and hit "return", that value will be input into the program and plotted, just as if you had clicked. The left-hand text area can be used to read or change the x-coordinate of the anchor. The right-hand text area can be used to read or change the x-coordinate of a regular blue point. These areas can deal with more decimal places than they display, but typing in letters or an out of bounds number results in an error.

Since close is of such importance, a fun tool has been included to help in this regard. Auto-generate box, marked simply "generate" produces a long sequence of points each time you click. After the point where you click, it then finds the point that is half way to the anchor. Then it finds the point that is half way to the anchor from the last one, and so on. It continues this until it is closer than the screen can display. The last line drawn is almost certain to be very near to the tangent line. It is important to look at the sequence of slopes generated in this process. Notice that after the first few values, the slopes in the sequence don't change very much. This is presumably because the slope is close to the value of that for the tangent line. Since that number is a goal, this is an important point. In the first example below, if the anchor is at a whole number, ( use the text area to input one ) the slope of the tangent line will also be a whole number. See if you can figure out what it is. In fact, by figuring out what it is for the numbers, .5, 1, 1.5, 2, 2.5, 3, 3.5, and 4, you should even be able to determine the formula for the slope of the tangent line. This is your first assignment.

The two examples of the program can be seen by clicking below: