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«Math 1314 Page 1 of 54 Appendix B, Section 3 The display has an algebra window on the left and a graphing area on the right. At the bottom is the ...»

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Appendix B – Using Other Technologies

Section 3: GeoGebra

So far, we have looked at the features of a graphing calculator and at Excel as aids

in working problems in this course. There are many other software packages

available online to assist students who are taking calculus. In this section, the

focus will be on a free software called GeoGebra. GeoGebra is a graphing utility

and more. The URL for the software is www.geogebra.org. There you will find a

link to download the software and links to some help materials. This section shows version of GeoGebra as displayed on a PC. GeoGebra can be downloaded on a Mac and working with it on a Mac is the same as on a PC.

GeoGebra is not yet available as a smartphone or iPad application.

When you open GeoGebra, this is what you will see.

Math 1314 Page 1 of 54 Appendix B, Section 3 The display has an algebra window on the left and a graphing area on the right. At the bottom is the input line. On the bottom at the right, you’ll find an arrow in a box. Click on it to display Input Help.

You work with GeoGebra using commands. Input Help organizes commands by topic and that can help you find the command you need if you don’t remember its name. If you aren’t sure where your command might be located, select All Commands and scroll down until you find it.

One of the advantages of GeoGebra is that graphing is easier. Instead of hunting for an appropriate window, you can just grab the axes and pull until you get a good view.

Math 1314 Page 2 of 54 Appendix B, Section 3 Example 1: Enter the function f ( x) = 2 x 3 + 5 x 2 − 4 x + 3 using GeoGebra, and format the graph so that you have can see all zeros and extrema of the function.

Solution: Type the equation into the input line.

Press Enter on your keyboard to view the graph.

Math 1314 Page 3 of 54 Appendix B, Section 3 This does not give a good view of the function. There is a relative maximum in Quadrant 2, and we can’t see it. To get a good view of the graph, we need more positive y values. Click on the last icon in the list of icons near the top of the screen.

Then put the cursor near the y axis somewhere between Quadrants 1 and 2 and pull down towards the origin. Keep pulling down until you have a good view of the graph of the function. The y axis labels you see may vary depending on how far down you pull the graph.

Math 1314 Page 4 of 54 Appendix B, Section 3 Notice that the function appears in the algebra list to the left of the graph as a “Free Object.” *** Now that you have a graph of a function to work with, you can find relevant points on the graph of the function, such as the zero(s), relative extrema and inflection points. When using GeoGebra, you will need to know the names that the program has assigned to the value you wish to find. Often these will be different that the names that are given in class or in the text. For example, if you want to find the zero of a function, you won’t find “Zero” given in the list of commands. Geogebra calls zeros of a function “Roots.”

–  –  –

Solution: We can find these values using the list of commands in the Input Help box. We’ll find all of these in the list under Functions & Calculus. Click on the plus sign to the left of the title to display the list.

Here is the result. You can scroll up and down the list using the scroll bar on the right side of the box. You can click on the little minus sign that now appears to the left of Functions & Calculus if you want to collapse the list.

–  –  –

Click on Paste to display the Root command in the Input line.

The red brackets indicate that you need to enter more information, called the arguments. You can find the roots of either a polynomial function or any general function. In this example, you want the roots of a polynomial function, so you only need to type in the letter name for the function, in this case, f, for f ( x).

You’ll see what to do for other types of functions in later examples.

–  –  –

Note that the zero is listed as a dependent object in the algebra window and is plotted on the graph of the function. If you want or need to display more decimal places, you can change from the default of two decimal places to 3 or 4 or more.

Click on Options in the menu at the top.

–  –  –

Move the cursor over the word Rounding until another menu pops up. You can then choose the number of decimal places or significant digits that you wish to display.

–  –  –

*** Math 1314 Page 10 of 54 Appendix B, Section 3 Once you are more familiar with GeoGebra and know the commands that you want to use, you can just start typing the word “Root” until Root[] appears in the Input line. The same options that you saw in the box on the left will pop up as you type.

Select the command that you want to use and press Enter.

Put the cursor inside the brackets, and type f and then press Enter on your keyboard. GeoGebra will return the same display that you saw in the last example.

In this example, the function had only one zero. If your polynomial function has more than one zero, GeoGebra will return all of them at once. For other types of functions, you will need to find one zero at a time and include a start value of x and an end value of x inside the brackets in the input line to tell GeoGebra which zero you wish to find.

Next you can find the relative maximum and the relative minimum of the graph of the polynomial function. These are called the extrema of a function. You’ll find the word “Extremum” in the Input Help list under Functions and Calculus.

–  –  –

To find the extrema of a polynomial function, as we have in this example, put the cursor inside the brackets and type the name of the function you are working with, in this case, f. Then press enter. GeoGebra will return values for all relative extrema of the polynomial function at once. Note that points appear on the graph at the relative maximum and the relative minimum, and the coordinates for these points are listed under Dependent Objects in the algebra window.

Math 1314 Page 12 of 54 Appendix B, Section 3 Note that the items in the list of Dependent Objects are not labeled. Just by looking at the list, you don’t know what A, B and C represent (although you can probably figure them out by looking at the graph). Put your cursor on point B on the graph and right click. You’ll see a menu that gives the details of the point.

–  –  –

Math 1314 Page 14 of 54 Appendix B, Section 3 You can change many things, such as the color of the point or the size of the point, using the Color and Style options at the top. You can also change the label. Click on the drop down menu next to Show Label, and you’ll see four options, Name (the default, which just gives the letter from the list), Name & Value (which gives both the name and the coordinates), Value (which just gives the coordinates) and Caption (which allows you to manually enter a caption in the Caption line above the menu, and that is what will appear on the graph). In the graph on the next page, you’ll see a customized caption which gives all information about the point.

Captions are given in text boxes, so you can move them by left-clicking on the caption and dragging to the desired location.

Math 1314 Page 15 of 54 Appendix B, Section 3 Once you are more familiar with GeoGebra, you can by-pass the Input Help box and just enter your command in the input line. For the relative extrema, put the cursor in the input line and start typing “Extremum.” GeoGebra wants you to specify the function name, so put the cursor inside the brackets and type f(x). Then press Enter on your keyboard. You will get the same result that you saw when you used the Input Help box.

Math 1314 Page 16 of 54 Appendix B, Section 3 Finally, you want to find the location and coordinates of any inflection points for the graph of this function. An inflection point is a point on the graph of a function where the graph changes concavity, that is, where it changes from looking like an upward-opening parabola to looking like a downward-opening parabola, or vice versa. The change is sometimes very subtle, and you can use calculus methods to locate these. You can also use GeoGebra.

Using the Input Help box, expand Functions & Calculus and scroll down until you see Inflection Point.

–  –  –

Math 1314 Page 18 of 54 Appendix B, Section 3 Point D appears on the graph and in the list of Dependent Objects. This is the inflection point. Note that you can only use this method to find inflection points of polynomial functions using GeoGebra (no other options appeared in the Input Help box). You will see how to use GeoGebra to find inflections points of other types of functions in the next example.

You can find the same values with a general function, but the selections that you make and the required inputs will be different.

–  –  –

Solution: This is a rational function, and the numerator contains an exponential term. This is not a polynomial function, so using GeoGebra has a few more

–  –  –

Then graph the function and use the Move Graphics View icon to grab the axes and resize the graph, so that you have a good view of the graph of the function.

Next, we need to find the zeros of the function. Open the Input Help menu and select Functions and Calculus. Then scroll down until you see “Root.” Alternatively, you can just type “Root” in the input line below the graph. You’ll see that you have several options for this command, and since you no longer have a polynomial function to work with, you’ll need a different selection.

Math 1314 Page 20 of 54 Appendix B, Section 3 To find the zeros of a general function, you’ll need to choose the third option, that is, the option where you will enter the function, the starting x value and the ending x value. This is very similar to what you do when using a graphing calculator – you need to mark the interval within which you want the software to look for a zero. Note that you have to do these one at a time, which is different from the method used for polynomials. You will use the same method for finding extrema.

Click on Paste, and then insert the inputs inside the brackets. In this case, the function name is f, and the left-most zero occurs between −1 and 0.

Now press Enter, and the zero will appear as a point in the algebra window and on the graph of the function. You can select more decimal places using the Rounding feature that is on the Options menu. You can also change the labeling, as show in Example 2, if you’d like to.

–  –  –

Next, you want to find the relative maximum and the relative minimum. The relative maximum occurs between the two zeros, and the relative minimum occurs in Quadrant 2. You’ll use the Extremum command, and you will need to specify the function and the interval on which you want GeoGebra to look for an extremum. Geogebra will only give you one answer at a time.

Find the relative maximum first. Using the Input Help menu, you can see the options. This is not a polynomial function, so you’ll use the second option.

Paste Extremum into the input line and fill in the information. The zero is in the interval [ −1, 1].

Now press enter to display to relative maximum.

–  –  –

Math 1314 Page 24 of 54 Appendix B, Section 3 The relative minimum occurs at ( −2.7482, 14.6816 ). If you use the Input Help menu to find the Inflection Point command, you will see that it works only for polynomial functions. You will need to use a different method for finding inflection points of general functions. We will defer discussing inflection points until after seeing how to using GeoGebra to find derivatives.

*** GeoGebra can also find the symbolic derivative (i.e., the derivative in terms of the variables).

–  –  –

Solution: Derivative is one of the commands in the Functions and Calculus list.

Start by entering the function. Then either find the Derivative command using the Input Help menu or just put the cursor in the Input line and start typing “Derivative”.

You want to find the derivative of f ( x) so type that in the brackets. F gNow press Enter onyour keyboard.

Math 1314 Page 25 of 54 Appendix B, Section 3 GeoGebra graphs f ′( x) and displays the symbolic derivative as f ′( x) in the algebra window. Now to find the value of the derivative at the two points that are given, you will need to evaluate f ′( x). Type f ′(2.08) in the Input line and press Enter on your keyboard. Then type f ′(10.47) (shown below) in the Input line and press Enter on your keyboard.


–  –  –

*** If you need to find the second derivative of a function, you will include this in the brackets in the input line. The command prompts you what to enter.

So in this example, you would type “f Tab 2” inside the brackets to find the second derivative.

–  –  –

*** If you want to limit the items that are displayed on the graph, you can easily do so.

Note that the bullets to the left of many of the items in the Dependent Objects list are shaded. That means that the items will appear on the graph. In the graph

–  –  –

If you want to change the way GeoGebra labels points, right-click on the point in the algebra window, select Object Properties, click on the pull down menu titled Show Label and select the type of labeling you want.

–  –  –

In the examples you’ve seen so far, points have been labeled with just the name, which is the letter given in the algebra window. In the graph shown below,

–  –  –

You can also give the point a customized caption. From the Show Label menu in Object Properties, select Caption and type in what you want to display. In the graph shown below, the relative minimum at Point C has been captioned “Relative minimum.” The labels are text boxes and can be dragged to another location on the graph, if needed.

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