GeoGebra Tutorial 17 – Functions, Tangent Lines and Derivatives

This is the 17th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, I strongly suggest that you read the GeoGebra Essentials Series.    In this tutorial, we are going to use slider control a, b, c, d and e and graph the function f(x) = ax4 + bx3 + cx2 + dx + e.

Figure 1

We then  construct a line tangent to the function and passing through  point A and trace the graph of the point whose x-coordinate is the x-coordinate of A, and whose y-coordinate as the  slope of the tangent line. We compute for the derivative of f(x), and see if  there is a relationship between the trace and the derivative. If you want to follow the this tutorial step by step, you can open the GeoGebra window. Before following the tutorial, you may want to see the final output.

1. Open GeoGebra. We will need the Algebra view and the Axes so be sure that they are displayed. If not, use the View menu from the menu bar to show them.
2. To label points only, click the Options menu, click Labeling, and then click New Points Only.
3. To create slider a, type a = 1 in the input box and press the ENTER key. Right click the equation a = 1 in the algebra window (leftmost window pane) and click Show object from the context menu. Slider a should appear on your drawing pad.
4. Using step 3, create 4 more sliders namely b, c, d and e.
5. To graph the function f(x) = ax4 + bx3 + cx2 + dx + e, type f(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e in the input box, then press the ENTER key.
6. Move the sliders and observe what happens.

Figure 2

7. To construct point A on function f, select the New Point tool and click graph of the function.
8. To construct a line tangent to f and passing through A, select the Tangents tool, click point A and click the graph of f. A tangent line should appear passing through point A.
9. Move point A on the function, and move the sliders. What do you observe?
10. To get the slope of the tangent line, select the Slope tool and then click the tangent line. This will produce m (see Algebra window). Given similar values of the numerical coefficients and the right place for point A, your graph should look like Figure 3.

Figure 3

11. We now create point D, which will trace the ordered pair (x(A),m) where m is the slope of the tangent line. Note that x(A) means the x-coordinate of A and m was automatically assigned by GeoGebra to the value of the slope.  To create the point type D = (x(A), m).
12. We now change the color of point D. To do this, right click point D, and click Object Properties from the context menu. In the dialog box, select the Color tab and select a color you want from the color palette. Next, select the Basic tab, be sure to check the Show trace check box, then click the Close button.

Figure 3

13. Move point A along the function. What do you observe about the traces of point D?
14. To graph the derivative of f(x), type f’(x) = derivative[f], then press the ENTER key. What do you observe about the derivative of the function f?
15. Right click the derivative function and click Properties. In the dialog box, go to the Color tab and select the color you want, preferably the same color as point D. Drag point A. After this step, your drawing should look like the one shown in Figure 1.
16. What can you say about the relationship of the derivative and the path traced by point D and the derivative function?
17. How can you relate the tangent function, its slope, the derivative function and the line traced by point D?
18. Based on the activity above, how will you describe the derivative of a function at a particular point and derivative in general?

GeoGebra Basic Construction 5– Rectangle

In this GeoGebra tutorial, we use the Perpendicular Line and Parallel Line tools to construct a rectangle. The idea is to construct segment AB, construct two lines perpendicular to it, one passing through segment A and the other through segment B. Next, we will construct point on the line passing through B, then construct a line parallel to AB passing through C. The fourth intersection will be our point D. If you want to follow this construction step-by-step, click here to open GeoGebra on your browser.

Step-by-Step Construction

1. Open GeoGebra and select Geometry from the Perspective menu on the side bar.
2. To automatically show the labels of points and not the other objects, click the Options menu, click Labeling, then click New Points Only.
3. To construct a rectangle, select the Segment between Two Points tool and click two distinct places on the Graphics view to construct segment AB.
4. We construct two lines that are both perpendicular to AB, one passing through A and the other through B. To do this, click the Perpendicular Line tool, click point A and click the segment. To construct another line, click point B then click the segment. After these steps, your drawing should look like the one shown in Figure 1.

Figure 1

5. Next, we construct a point on the line passing through point B. To do this, click the New Point tool, and then click on the line passing through B.
6. Now, to create a line parallel to AB, passing through point C, click the Parallel line tool, click on point C and then click on the segment.

Figure 2

Your drawing should look like the one shown in Figure 2 after this step.

7. Using the intersect tool, we construct the intersection of the line passing C (parallel to AB) and the line passing through A. To do this, click the Intersect Two Objects tool and click the two lines mentioned.
8. Now, we hide the three lines. To do this, click the Show/Hide Object tool and click the three lines. Notice that the lines are highlighted. Click Move tool to hide the lines.
9. Use the Segment between Two Points tool to connect the points to construct the rectangle.
10.  Drag the points of the vertices of the rectangle. What do you observe?

Invitation to the Mathematics and Multimedia Blog Carnival

Hi guys. I am happy to announce that I will be hosting my own blog carnival — the Mathematics and Multimedia Blog Carnival. M&M Carnival will include K-12  topics in mathematics. Although, any math  topic is allowed, the following topics will be prioritized:

  1. connections between and among different topics in mathematics as well connection of mathematics to other fields
  2. use of mathematics in solving real life problems
  3. clear, non-bookish conceptual explanations of mathematical concepts, particularly those which are hard to teach and difficult to learn
  4. integration of technology in teaching mathematics
  5. introduction of new software and Web 2.0 technologies
  6. software reviews and tutorials

Please spread the word. The first issue will be posted on July 12, 2010.  To submit your article, click here.

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