Part 2: Interpreting and Using r'(t)
Complete this worksheet to help you answer the questions for Part 2 of the project. The following two graphs are examples of the pictures that you should be able to create and include in your project after completing this worksheet.
The relationship between r(t) and r'(t)
Instructions for plotting and analyzing your rider function
Below these instructions you'll find a blank GeoGebra applet ready for you to use. You may need to adjust the y-axes in order to get good pictures of your rider function and its derivative, so here are a few tips for how to do so.
Adjusting the scale on the y-axesTo adjust the y-axes of the applet, right click on the graph and select "Graphics." To the right, you should see two pictures that look like this:
If you click the top graphic, you'll adjust the settings for the top graph. Likewise, highlighting the bottom graphic will allow you to adjust the settings for the bottom graph. I recommend only adjusting the entry for "y Max" as the x-axis and the "y Min" entries are already pre-set. When you've adjusted the settings appropriately, close the settings window by clicking the "X" which appears just above the two graphics shown above.
Using GeoGebra to graph and analyze r(t) and r'(t)
- First, click on the top graph to let GeoGebra know you'd like to add a function to it. Then, click near the top of the left side of the screen in the input bar. This will allow you to input functions and points into the graphs. Type in your rider function "r(t) = ..." You can type the word "pi" to denote pi in an equation, and you should use the asterisk * to denote multiplication.
- When you've graphed r(t), you may need to adjust the y-axis of the top graph using the instructions given above. Remember to select the upper graphic to make sure that you are adjusting the y-axis of the top graph.
- Next, we want to graph the derivative r'(t) on the lower graph. First, use your knowledge of derivative rules to calculate r'(t) by hand. Then, click the lower graph to let GeoGebra know you'll enter a function that should be graphed there. Using the input bar, enter your derivative "r'(t) = ..." Again, you may want to adjust the y-axis of the lower graph at this point.
- Next, we want to examine the relationship between r(t) and r'(t) at two different points, so we want to create two sliding points on r(t) that are connected to corresponding points on r'(t). To do this, first use the point tool, , and click at two random places along the graph of r(t). This should create two points along your graph called "A" and "B." If you click back on the move tool, , you should be able to slide points A and B along the graph of r(t).
- To link the points A and B with the corresponding points along the derivative, we'll need to use the input bar. GeoGebra recognizes the input "x(A)" to be the x-coordinate of the point A. Let's create a point called A' in the bottom graph that has x-coordinate x(A), and y-coordinate given by the value of the derivative at that x-value, which is r'(x(A)). Notice, we've just plugged x(A) into the derivative r'(t). In summary, first click on the bottom graph, and then type "A'=(x(A),r'(x(A)))" into the input bar. This should create a point A' in the bottom graph that lives on the graph of r'(t). Furthermore, as you move the point A on the upper graph of r(t), you should see A' move along with it.
- Create another point B' in the same way that is linked to the motion of the point B. As you slide A and B along r(t), notice the values of the slopes at different points. Do the y-values of A' and B' behave as you would expect?
- Drag A to a place on r(t) where the slope is the largest. At what point on the graph of r'(t) does A' move? Drag B to a place on r(t) where the slope is the smallest, and notice where B' moves on the graph of r'(t).
Finding the local linearization at a point T_0
- Choose a day and time that you frequently visit this subway stop, and let T_0 represent this day and time. For instance, if you visit the stop on Tuesdays at 8:30am, then T_0=32.5. For this example, we'll use 32.5, but you should choose your own T_0 value. Click on the top graph, and graph the point "T_0=(32.5,r(32.5))" (make sure to use a capital T). Notice that this point cannot slide, because you have specified that the x-coordinate is equal to a particular value.
- Click on the bottom graph, and input a similar point called "T_0'" which uses the derivative r'(t) for the y-value instead of the original function r(t).
- Next, we want to graph the local linearization of r(t) at the point T_0. To do this, first click on the top graph. Remember that the point-slope form of a line is y=m(t-t_0)+y_0, where m is the slope and (t_0,y_0) is a point on the line. Using the expressions x(T_0), y(T_0), and y(T_0'), enter the local linearization into the input bar. Your entry should begin with "L(t) = ...". When done correctly, you should see that L(t) is the graph of the tangent line to r(t) at the point T_0.
Using L(t) to approximate r(t)
Next, we want to see how good of an approximation we can get using L(t) to estimate r(T_0+1). That is, we'll want to compute L(T_0+1) and r(T_0+1), and then find the percent error between the two. See if you can use the upper graph in the applet you've created to make a picture of the function zoomed in on the point T_0 like the second applet above (the one called "Linearization"). To do so, you'll have to create points on L(T_0+1) and r(T_0+1). Since you've had some experience creating points, I won't give you step-by-step instructions here, but I'll give you a few extra GeoGebra tips that might be helpful.
- Zooming - If you have a scroll wheel on your mouse, you can use it to zoom in and out. You can also right click on the graph to use the zoom function. Finally, you can adjust the window as discussed before.
- Point labels - If you'd like a point to display its coordinates, you can right click on the point and go to "Object Properties." A menu will appear with a checked box that says "Show Label." Next to that, you can require that the point show its value as well.
- Text boxes - You can use the text box tool, , to add text to your applet.
- Line segments - You can create a line segment between two points by using the line segment tool, . You can use this tool to draw a line segment between the two points L(T_0+1) and r(T_0+1) to visualize the size of the error.