Chapter 11: The Dynamic Geometry of Calculus
In Chapter 6 we introduced you to Sketchpad’s function plotting capabilities. In this chapter we shall use those capabilities to explore the geometry of calculus, starting with the relation between secant lines to a curve, tangents to that same curve and the derivative of the function that generates the curve. We also introduced in Chapter 6 the idea of using sliders (variable horizontal line segments) to vary the parameters of a function. The following section leads you though the steps to create your own slider tool that you will use for creating many functions in this chapter.
It is fairly easy to create sliders (dynamic line segments) to provide values that can be manipulated simply by moving the end-point of the slider.
You are now ready to create a slider tool from this construction. Follow the steps below:
A window similar to Figure 11.1 below should appear:
Figure 11.1: Script View window for the Slider_Tool
Your slider_tool will now be available for use whenever you open GSP 4.
Use your slider tool to create four sliders in a new sketch. The coordinate axes will be created automatically. Your Sliders (segments) will have only one end-point labeled. Four measures will appear. Re-label these measures according to the labels on your sliders. So as not to confuse which endpoint is the moveable one you can hide the unlabelled endpoint of each slider. Use the measures of each slider to create and plot a new function: f(x) = ax3+bx2+cx+d. You should have something like figure 11.2 below.
Figure 11.2: Plot of a cubic function using 4 sliders as parameters.
Explore the roles of the four parameters on the cubic function by adjusting each of the sliders. Create a new cubic function using three of the sliders as roots of the function: e.g. g(x) = a(x-b)(x-c)(x-d). Explore how these two cubic functions differ and how they are similar.
In this activity we shall use our slider tool to create a control point that will be used to measure small changes in x. Call this control point dx. Place a free point on the x-axis of your function plot for the cubic function f(x) = ax3+bx2+cx+d. Call this free point x. Measure the abscissa of this point and label this measurement x. In the calculator create the measure of (x - dx/2) and (x + dx/2). Using the GSP calculator and each of these measures and the function f(x), create the values f(x - dx/2) and f(x + dx/2) respectively. The following steps will plot these values of f(x) as points on the graph and the secant between them.
1. Select the measures of (x - dx/2) and f(x - dx/2) in that order and Plot as (x, y) under the Graph menu. A point should appear on your function graph.
2. Do the same for (x + dx/2) and f(x + dx/2) A second point should appear on your function graph.
3. Select these two new points on the graph and construct a line through them.
4. Vary the length of dx to see what happens to this secant line.
As dx becomes very small this secant line appears to become tangent to the curve representing the cubic function. With dx very small (<0.1) move your free x point along the x-axis to see how the secant line travels along the curve (see Figure 11.3).
Lengthen dx so that you can see the two points distinctly on the graph of the function. Plot the point x, f(x) on your graph. This point should appear between the two points that define your secant line. Select the secant line and the point between the two points defining the line. Construct a line parallel to the secant line through the point x, f(x). Measure the slope of this “tangent” line. As you make dx very small the secant line becomes the tangent line. If you make dx large (dx > 2.0) and move your free point x, it will be obvious that the line parallel to the secant line through f(x) is not always tangent to the cubic function. Make dx small again. Now plot the point defined by the measure of x and the slope of the line through the point f(x) [Select the measure of x and the slope of the “tangent” line in that order and Plot as (x, y) from the Graph menu.] With this new point still selected, construct its locus with respect to the free point x on the x-axis. You should now have a sketch that looks something like figure 11.4. What shape is this locus curve? How does this curve relate to the derivative of the cubic function?
Figure 11.4: Constructing the tangent to a cubic function and plotting its slope
Create the expression for the derivative of your cubic function by selecting the expression for f(x) and choosing Derivative from the Graph menu. You should get the following expression: f’(x) = 3ax2+2bx+c. Use the calculator to obtain the value of this expression for point x. Make dx very small. What do you notice about the slope of the tangent line and the value of this expression at point x? Move your free point x. Do these values remain the same? Do they remain equal to each other?
The value of the derivative varies with the position of x, so we can graph this relationship. Select the expression for the derivative (f’(x) = 3ax2+2bx+c) and choose Plot Function from the Graph menu. What shape is the graph of the derivative function? Does this curve co-inside with the locus of the slope of the tangent line through f(x)? Enlarge dx until you see some separation between these two curves. Do the shapes of the curves remain similar even when they are separated? Reduce dx until the two curves are again coincident.
Vary your coefficients and then write three things about the relationship between the graphs of the cubic function and its derivative (see Figure 11.5).
Figure 11.5: Graphs of a cubic function and its derivative
Start a new sketch, create the axes and place a free point on the x-axis. Re-label this point x. Use your slider tool to create three control points, a, b and c. Follow the method you used for the cubic function to create the function f(x) = ax2+bx+c and construct its graph. Create a slider dx and use this to construct a secant to your quadratic function. Plot the point x, f(x) on your graph. Construct the line through f(x) parallel to your secant line. Move point x to see the “tangent” and secant lines move around your parabola. Vary dx. Make it as large as you can. Move x again. Does the “tangent” line ever appear not to be tangent to the curve? Go back to your cubic sketch and try the same thing. What seems to be special about the quadratic function?
Assignment 11.1: Create the expression for the derivative of your quadratic f'(x)=2ax+b (or ask GSP for the derivative of your function) and plot its graph. Does the value of f'(x) equal the slope of the secant line for any x?
Prove (algebraically) that the secant line through the points ((x-dx/2), f(x-dx/2)) and ((x+dx/2), f(x+dx/2)) will always be parallel to the tangent at the point (x, f(x)) for any quadratic function!
Using the construction methods above create a graph of a general sine function: f(x)= a•sin(b•x+c) and plot the points (x+dx/2, f(x+dx/2)) and (x-dx,/2 f(x-dx/2)) to construct the secant to the sine curve. [Note: You will need to change the Preferences for the angle measure under the Edit menu to Radians.] Make dx very small and measure the slope of the secant-tangent line. Plot x and slope as (x, y). Construct the locus of this new plotted point as x varies. In what ways is this new curve similar to your sine curve? Vary a and c. How do these coefficients affect the two curves? Vary b. How does this coefficient affect the two curves? Make b = 1.00. What do you notice about the two curves? How are they different and how are they alike? Vary a and c again. What do you notice? Set both a and b equal to 1; set c equal to zero. Make a conjecture about the function that would produce the curve that graphs the slope of a tangent to sine x.
Assignment 11.2: Create the derivative function, g(x) of a•sin(b•x+c) using the parameter measures and the New Function option from the Graph menu. Plot g(x). The graph of your derivative function should coincide with the graph of the slope of the tangent to f(x) for very small dx. If it does not, vary the coefficient b. What role does b play in the slope function?
The word Integration carries with it the notion of joining together. If we consider the situation of driving a car on a journey for two hours, for instance, the total distance traveled during the two hours would be our average speed multiplied by the time traveled. If we were able to travel the whole journey at this one speed (say 50 mph) then the speed-time graph would be a horizontal straight line and the distance traveled would be the area of the rectangle indicated in Figure 11.7. – that is 100 miles (50x2).
Return to a quadratic function controlled by your sliders or parameters for a, b and c. Construct the reflected image of point LL about the vertical axis. Move point UL to the reflection of LL using a movement button. Now vary the parameter (b) for the coefficient of the x-term in your function. What do you notice about the accumulation rectangle or the value for the area under the curve? Figures 11.16 and 11.17 illustrate the situation for two different values of b.
Figure 11.16: Area Under a Quadratic with Lower Bound = -Upper Bound
 The idea of negative area makes sense in the case of the graph representing the speed of a car versus time traveled, where the area under the curve represents the total distance traveled during the journey. When the speed becomes negative, the car is traveling backwards (or back towards its starting point) thus the distance it travels during this time must be subtracted from the distance traveled towards its end point.