The following example shows how you can graphically represent the derivative of a curve, and have the value of the derivative automatically update as you move a point of tangency along the curve. The curve to be explored is y = 3sin(x).
Since the accuracy of our calculation in this example is not too important, we will first change the number format to fixed at 3 decimal places. This will also help keep our geometry workspace uncluttered.
Preparation 1. Press SH.
2. On the Home Setting screen set the number format to Fixed and the number of decimal places to 3.
Open the app and plot the graph
3. Press I and select Geometry.
If there are objects showing that you don’t need, press SJ and confirm your intention by tapping . 4. Select the type of graph you want to plot. In this example
we are plotting a simple sinusoidal function, so choose:
> Plot > Function
5. With plotfunc( on the entry line, enter 3*sin(x):
3seASsE
Note that x must be entered in lowercase in the Geometry app.
If your graph doesn’t resemble the illustration at the right, adjust the X Rng and YRng values in Plot Setup view
(SP).
We’ll now add a point to the curve, a point that
will be constrained always to follow the contour of the curve.
Add a constrained point
6. Tap and select Point On.
Choosing Point On rather than Point means that the point will be constrained to whatever it is placed on.
7. Tap anywhere on the graph, press E and then press J. Notice that a point is added to the graph and given a name (B in this example). Tap a blank area of the screen to
deselect everything. (Objects colored cyan are selected.)
Add a tangent 8. We will now add a tangent to the curve, making point B the point of tangency:
> More > Tangent
9. Tap on point B, press E and then press J. A tangent is drawn
through point B.
(Depending on where you placed point B, your illustration might be different from the one at the right.)
We’ll now make the
tangent stand out by giving it a bright color.
10. If the curve is selected, tap a blank area of the screen to deselect, and then tap on the tangent to select it.
11. Press Z and select Change Color.
12. Pick a color from the color-picker, press J and then tap on a blank area of the screen. Your tangent should now be colored.
13. Press E to select point B.
If there is only one point on the screen, pressing E automatically selects it. If there is more than one point, a menu will appear asking you to choose a point.
14. With point B selected, use the cursor keys to move it about.
Note that whatever you do, point B remains constrained to the curve. Moreover, as you move point B, the tangent moves as well. (If the moves off the screen, you can always bring it back by dragging your finger across the screen in the appropriate direction.)
15. Press J to deselect point B.
Create a
derivative point The derivative of a graph at any point is the slope of its tangent at that point. We’ll now create a new point that will be constrained to point B and whose ordinate value is the derivative of the graph at point B. We’ll constrain it my forcing its x coordinate (that is, its abscissa) to always match that of
point B, and its y coordinate (that is, its ordinate) to always equal the slope of the derivative at that point.
16. To define a point in terms of the attributes of other geometric objects, you need to go to Symbolic view:
Y
Note that each object you have so far created
is listed in Symbolic view. Note too that the name for an object in Symbolic view is the name it was given in Plot view but prefixed with a “G”. Thus the graph—labeled A in Plot view—is labeled GA in Symbolic view.
17. Highlight GC and tap .
When creating objects that are dependent on other objects, the order in which they appear in Symbolic view is important. Objects are drawn in Plot view in the order in which they appear in Symbolic view. Since we are about to create a new point that is dependent on the attributes of GB and GC, it is important that we place its definition after that of both GB and GC. That is why we made sure we were at the bottom the list of definitions before tapping . If our new definition appeared higher up in Symbolic view, the point we are about to create wouldn’t be drawn in Plot view.
18. Tap and choose Point > point
You now need to specify the x and y coordinates of the new point. The former is to be constrained to abscissa of point B (referred to as GB in Symbolic view) and the later is to constrained to the slope of C (referred to as GC in Symbolic view).
19. You should have point() on the entry line. Between the parentheses, add:
abscissa(GB),slope(GC)
You can enter the commands by hand, or choose them from the Catlg menu (one of the five Toolbox menus).
20.Tap .
The definition of your new point is added to Symbolic view. When you return to Plot view, you will see a point named D and it will have the same x coordinate as point B.
21. Press P.
If you can’t see point D, pan until it comes into view. The y coordinate of D will be the derivative of the curve at point B.
Since its difficult to read
coordinates off the screen, we’ll add a calculation that will give the exact derivative (to three decimal places) and which we can display in Plot view.
Add some
calculations 22. Press M.
Numeric view is where you enter calculations.
23. Tap .
24.Tap and choose Measure > slope 25.Between parentheses, add the name of the tangent,
namely GC, and tap .
Notice that the current slope is calculated and displayed.
The value here is dynamic, that is, if the slope of the tangent changes in Plot view, the value of the slope is automatically updated in Numeric view.
26.With the new calculation highlighted in Numeric view, tap .
Selecting a calculation in Numeric view means that it will also be displayed in Plot view.
27. Press P to return to Plot view.
Notice the calculation that you have just created in Numeric view is displayed at the top left of the screen.
Let’s now add two more
calculations to Numeric view and have them displayed in Plot view.
28.Press M to return to Numeric view.
29. Tap , enter GB, and tap . Entering just the name of a point will show its coordinates.
30.Tap , enter GC, and tap .
Entering just the name of a line will show its equation.
31. Make sure both of these new equations are selected (by choosing each one and pressing ).
32. Press P to return to Plot view.
Notice that your new calculations are displayed.
33.Press E and choose point GB.
34.Use the cursor keys to move point B along the graph.
Note that with each move, the results of the calculations shown at the top left of the screen change.
Trace the
derivative Point D is the point whose ordinate value matches the derivative of the curve at point B. It is easier to see how the derivative changes by looking at a plot of it rather than comparing subsequent calculations. We can do that by tracing point D as it moves in response to movements of point B.
First we’ll hide the calculations so that we can better see the trace curve.
35.Press M to return to Numeric view.
36.Select each calculation in turn and tap . All calculations should now be deselected.
37. Press P to return to Plot view.
38.Press E and select point GD.
39. Tap and select More > Trace 40.Press E and select point GB.
41. Using the cursor keys, move B along the curve.
You will notice that a shadow curve is traced out as you move B. This is the curve of the derivative of 3sin(x).