Chapter 2. Getting Started with the TI-Nspire CX CAS

Transcription

1

2 Your TI-Nspire CX CAS will serve as your partner while you journey through the course. This handheld device operates much like a small computer. The more familiar you become with this handheld device, the better you will understand the concepts and ideas presented. Just like anything new, you will need to take some time to become familiar with the TI and some of its basic operations. This chapter should help you gain the familiarity you need to start using the TI today. Using the TI as a calculator We can use our handheld device for basic calculations, for entering and evaluation expressions and for defining variables and functions. The first time you turn on your TI you will see the home screen. See Figure 1. Figure 1 Opening and Closing the Scratchpad The TI-NSpire CX CAS has a Scratchpad calculator which will allow you to make quick calculations without affecting a TI-Nspire document (more on Documents later). To open this 1

3 scratchpad from the Home screen, press the key. The first time you open the Scratchpad, a blank page opens. See Figure 2. Figure 2 Press to alternate between the Calculate and Graph pages (more on graphs later). Press menu to see the Scratchpad Calculate or Scratchpad Graph menu. These menus are sub-menus of the TI-Nspire menus for the Calculator and Graphs & Geometry applications. Finally, to close the Scratchpad, press esc. Open your Scratchpad in calculate mode. We are now ready to enter some basic expressions and have our TI help us to evaluate them. To get the feel for the device let us start by entering an expression. Enter 3 5 and press the enter key. Was the result 15 returned? Now it is time to start looking at some more complicated expressions. Example 2.1: Enter the expression 6 + 2! + 5 and press the enter key. What value was returned? You should see the answer: 27. 2

4 If you were unable to successfully find the value of the expression 6 + 2! + 5 you may have found the value of 6 + 2!!! = 518. To avoid this mistake you must make sure that you inform the calculator that the exponent is 4 not To do this you must hit the u after you finish entering the exponent, in this case after 4. The keystroke sequence is ^ 4 u enter Note that the symbol appears on the center keypad. Example 2.2: Enter the following: 6 + 2! +! and press the enter key. What value is!" returned? You should see the answer:!"!. Example 2.3: If you had intended to enter the expression: what would you have to do differently? Try it and see what answer you get. Did you get? (See Figure 3.) Figure 3 3

5 If you were unable to successfully enter the expression you may have forgotten to use parentheses appropriately. Remember that the importance of parentheses was mentioned in Chapter 1. The correct use of parentheses is essential when you use this, or any other, calculator. To correctly enter the expression you must use the following sequence of key presses: 6 + ( ) ctrl 16 enter Note that the square root symbol is above the x 2 key. Take some time now to explore some different expressions and evaluate each. Additional Practice Problems Answers 1. 5 (5 8) = (36 6) 3 = !!"!"!!" + 6 = 13/2 ( ) = 5 8 You may have noticed that some of the answers above were given in fractional rater than decimal form. If you ask for the value of! the calculator will return!. When the decimal form!! of this fraction is desired you must use ctrl enter rather than simply enter. This returns the decimal approximation ( ) of the fraction. 4

6 1 4 ctrl enter ctrl enter It should be noted that in some cases the decimal equivalent is an exact equivalence, as in the case of! = However, in many cases the decimal equivalent serves as an approximation of! the exact value of the fraction. For example,! = (6 s forever), but!!! ctrl enter

7 Homework = ( ) (3+7) 3 = 3. -(5+1) 2 = ( ) 2 = = = = = = = ( ) = 5 ( ) 3 ) = =

8 Defining and Evaluating Functions Your TI-Nspire device also has the ability to define and evaluate functions. In addition, functions can be represented both numerically and graphically. To demonstrate some of these features we will begin by considering the situation described in Example 1.2. Example 1.2 Revisited: The Gold-Plated Square Charms You may remember the function concerning gold-plated charms discussed in chapter 1. In that discussion we saw that the function f(x) = 2x 2 could be used to calculate the surface area of a square charm with side of length x. This function can be defined on the TI-nSpire and this definition can be used to find the surface area for squares of any size in the domain. Go to the home screen Press Press menu. Select 1:Actions Select 1:Define Type the function: g x = 2x! Press enter. You can now calculate the surface area of a square gold-plated charm with any length side. To find the surface area of a 2mm by 2mm charm simply: Type g(2), then press ENTER. 7

9 Figure 4 Did you see the screen above indicating that g(2) = 8. If not try again. For practice find the following: g(3) = g(5.25) = g(b) = g(b + 4) = To clear a particular function from this handheld device you must Hit menu Choose 1: Actions Pick 3: Delete Variables Type the name of the function you wish to delete. (In this case g.) Hit enter. The function definition is now deleted. If you now attempt to use it the calculator does not return the results of a calculation, it simply returns the expression you typed. 8

10 Figure 5 Example 2.4: The Vegetable Garden This spring you are planning to plant a vegetable garden in the back yard. To keep the animals from devouring your vegetables, you will erect a fence around the garden. The shape of the garden will be rectangular. You will be constructing the fenced-in garden up against the side of the garage so you will only need fencing on three sides. Let s start by trying to understand this situation. Are we interested in the perimeter or the area of the space? Perimeter or Area? Since we are looking foe the amount of fence needed to enclose to the space, and not the amount of space, this is a perimeter problem. We will decide on the amount of fencing we will purchase later in the problem. Our next step is to picture the situation. Be sure to include the garage in your sketch. Label all attributes in the sketch. One such design might look like the one in Figure 6: 9

11 Figure 6 Now that we have developed a rough sketch of the situation, we want to translate our sketch into an appropriate algebraic representation. What are the variables in this situation? Use these variables to create an algebraic expression that can be used to represent the situation described above. Perimeter = One such expressions can be: Perimeter = width + width + length or P = 2W + L We can generate a few different scenarios for this situation. Let s say you don t really have a predetermined amount of fencing that you plan on purchasing. Given different values for the length and width, how much fencing would be needed? We will use the TI-Nspire to assist us in answering this. 10

12 Clear Scratchpad History, as described earlier. Press menu Select 1:Actions Select 1:Define Type the function:. Press enter. Now it s time to select some values for our length and width. One possible configuration would be a garden with a length of 20 ft. and a width of 15 ft. To find the amount of fence needed for this garden you can type p(20,15)and press enter. (See Figure 7) Figure 7 This informs us that 50 feet of fence is needed for an enclosure 20 feet long and 15 feet wide. Please notice that if you had entered p(15,20) you would not get the same answer (what answer do you get? ) since this function would calculate the amount of fencing needed for an enclosure 15 feet long and 20 feet wide. 11

13 Find three additional sets of values for the length and width of the vegetable garden that use 50 feet of fence for the garden. Width 1 Length 1 uses 50 ft of fence. Width 2 Length 2 uses 50 ft of fence. Width 3 Length 3 uses 50 ft of fence. Is there a particular choice of values for the length and width that seems optimal? What might we mean by optimal? One possible interpretation for optimal, in this case, might be finding maximum area that can be enclosed by 50 feet of fence. 12

14 Homework Create the following function on the TI-Nspire CX CAS: f x = 6x! 10. Then evaluate the following: 14. f(-3) = 15. f(2) = 16. f(! ) =! 17. f(b) = 18. f(b + h) = Create the following function on the TI-Nspire CX CAS: f x =!!!. Then evaluate the 2 following: x 19. f(-3) = 20. f(2) = 21. f(! ) =! 22. f(b) = 23. f(b + h) = Create the following function on the TI-Nspire CX CAS: f x = 21. Then evaluate the following: 24. f(-3) = 25. f(2) = 26. f(b) = 13

15 Tables As mentioned in Chapter 1, functions can also be modeled using tables. The TI-Nspire has the capability of creating and displaying such tables. You will have to access this capability from the home screen. The table capabilities of the calculator will be demonstrated using the perimeter function you created above. From the home screen scroll down to the icons found on the bottom of this screen and find the icon labeled: Add Lists & Spreedsheet. Press Enter. Cursor to the top of column A and type width for the width values. Press Enter. Cursor to the top of column B and type length for the length values. Press Enter. Cursor to the top of column C and type perimeter for the perimeter values. Press Enter. Move to the cell below perimeter column and type the formula for the perimeter in our garden problem: =2 width + length. Be sure to begin with an =. Press Enter. Move the cursor to cell A1, and type 15. Move the cursor to cell B1, and type 20. Press Enter. The perimeter, given this input, will appear in cell C1. See Figure 8. Note: Column names can not be single letter names such as w, l, or p. However names such as ww or ll will work. 14

16 Figure 8 Once again, try different values for width and length and discuss the different options for perimeter. Do you get a perimeter of 50 for any of your other choices of width and length? If not, experiment to find a pair that yields a perimeter of 50. Now, let s practice entering a formula into a column. What is the formula for the area of the garden we have been discussing? Area = Enter that formula into column D. Now use your table to find the perimeter and area of a garden with: 15

17 Width = 20 feet Length = 20 feet Perimeter = Area = Width = 10 feet Length = 20 feet Perimeter = Area = Width = 10 feet Length = 30 feet Perimeter = Area = Do you now have the table in Figure 9? Figure 9 16

18 Homework 27. Create a table for the function in Example 1.1. Fill-in the blanks below: 28. Create a table for the function in Example 1.2. Fill-in the blanks below: 17

19 29. Create a table for each of the functions below. x f(x)= 2x x g(x) = x 2 x h(x)=1/x x s(x)=3x 3-2x 2-5x Describe, in words, the behavior of function f. 31. Describe, in words, the behavior of function g. 18

20 32. Describe, in words, the behavior of function h. 33. Describe, in words, the behavior of function s. 19

21 Viewing / Clearing the Scratchpad History Each expression and result becomes part of the Scratchpad history and it is displayed above the entry line. Press or to scroll through the Scratchpad history. You can clear a line in the history by highlighting the line (moving the cursor to the specified line) and hitting the del key. You can clear the entire history using the following sequence of keystrokes: Hit menu Choose 1: Actions Select 5: Clear History. When you clear the history, all variables and functions defined in the history retain their current values. If you clear the history by mistake, use the undo feature: ctrl esc. Graphing Functions on the Scratchpad As you would imagine, your TI-Nspire also has the ability to graph various types of functions. Graphing will be one of the most useful and important functions you will utilize on your handheld device. Let s look at how your TI-Nspire can help you better understand and represent functions in both graphic and/or table formats. To get started, let s consider the function: f (x) = x To use our TI-Nspire to graph this function. 20

22 Go to the Home Page Highlight B: Graph. Hit ctrl then +page Choose 2:Add Graphs In the entry line, you will see f1(x)=. Enter the function, x 2 +1 and press enter. Another path to the same place is: Go the Home page Highlight B: Graph Hit: enter Hit: menu Choose 3:Graph Entry/Edit Choose 1:Function In the entry line, you will see f1(x)=. Enter the function, x 2 +1 and press enter. The function is displayed with its equation as a label. (see Figure 10) Figure 10 21

23 To display a table of values for the function: Press menu Choose 7:Table Select 1:Split-Screen Table. You can scroll up and down to inspect different values. (See Figure 11) Figure 11 You can adjust the table start value and the table step value. Press menu. Select 2:Table Choose 5:Edit Table Settings. Change the table settings to a start value of 3 and a step of 0.5. Does your screen now look like the one In Figure 12? 22

24 Figure 12 To remove the table from your display Click menu Choose 2:Table Select 1:Remove Table You have the ability to change and enhance the graph window in various ways. To see or change the window settings: Press menu. Select 4:Window/Zoom, Select 1:Window Settings. As an example of the capabilities of this option we will modify the graph of f x = x! + 1. In Figure 13 will picture the graph of f(x) on the TI-Nspire standard axis. Figure 14 will picture 23

25 the graph of f(x) on a graph with a x-axis ranging from -5 to 15 with tick marks counted off by units of one, and a y-axis ranging from -15 to 25 with tick marks counted off by units of five. Figure 13 Figure 14 To change the appearance of the window: Press menu. Select 2:View. 24

26 As an example of the capabilities of this option we will modify the graph in Figure 14 by adding grid lines. (See Figure 15) Figure 15 To move a function label around the screen: Move the cursor over the function label. When you are in the right place, the word label will appear along with a hand symbol. Press CTRL then the hand symbol. The hand will close to grab the label. Use the cursor to move the label. When you have placed it in the position you desire hit enter. 25

27 Homework 34. Graph the function f(x) = 0.5x 3-5x 2 +x + 3 on the standard TI-Nspire axis. (-10 < x < 10 and < y < 6.67) 26

28 35. Graph the function f(x) = 0.5x 3-5x 2 +x + 3 on an axis system with -10 < x < 25 and -75 < y <

29 36. What XScale was used in problem 35? 37. What YScale was used in problem 35? 38. Comment on what was learned by expanding the view of the Domain and Range on the axes of the graph. 28

30 Plotting More than One Graph on a Single axis Let s investigate another very useful graphing option adding a second graph and finding the intersection of the two graphs. We graphed the function, f (x) = x This function graphed as a parabola opening up and crossed the y axis at the point (0, 1), (The point at which a graph crosses the y axis is called the y intercept and, for a function called f, always occur at the point (0, f (0)). ) A second function can be entered on the same graph. To enter a second graph you should: Click on menu Select 3:Graph Entry/Edit. Select 1:Function. You now have the option of adding a second function to this graphing window. Notice that the entry line no longer is asking you for f1(x). It is asking for the definition of f2(x). Type in the function, 2 3 x + 3 and press enter. Notice that your TI understood that 2 3 is the coefficient on the variable x. You should see the graph in Figure 16 below. You can use the options already explored to move the function labels and to change the window settings. 29

31 Figure 16 Notice that the graph of f 1(x) = x 2 +1, and the graph of f 2(x) = 2, appear to have 3 x + 3 two points of intersection. We can use our TI-Nspire to find those points of intersection. This feature will be both useful and important for many applications we will encounter in this course. To find the intersection point(s) for two functions, do the following: From the graph window displaying the two functions click menu. Choose 6: Analyze Graph Select option 4: Intersection. You will be prompted to identify a lower bound. Use the mouse pad to select an x value slightly to the left of the first point of intersection. In this example, select a value near x = -2 and click enter. (See Figure 17) 30

32 Figure 17 Use the mouse pad again to select an upper bound. Choose any x value slightly to the right of the point of intersection and press enter. In this example, select a value near x = 0. (See Figure 18) Figure 18 31

33 After the label of the intersection point is moved to be more clearly visible we see that the first intersection point found is approximately (-1.12, 2.25). Repeat the above process to find the second point of intersection. You should get an intersection point of (1.79, 4.19). See figure 19 below. Figure 19 This is a good time to introduce you to another feature on the TI-nspire CX CAS, its ability to do algebra. In the example above you saw that the points of intersection were approximately (-1.12, 2.25) and (1.79, 4.19). Algebra is a tool that can be used to find the exact points of intersection. You will notice that when we ask for the points of intersection we are looking for the points at which the functions have the same y values. Thus, we are looking for the x s for which f1(x) = f2(x). In this example we are looking for the solutions of the equation x 2 +1 = 2. To find these values let use begin with a clean scratchpad. 3 x

34 Hit menu Choose 3: Algebra Select 1: Solve Type in the equation you wish to solve. In this case type in x^2u +1=2 3x+3,x. (Note that you could also have typed in f1(x)=f2(x),x.) Hit enter. Did you get the results seen in Figure 20? Figure 20 Task: Show that ( 19 1) Task: Show that ( 19 +1)

35 Task: Explain how the x s found above can be used to find the associated y s. Looking at the graph in Figure 19 one sees a point of intersection that has not been discussed, the point at which the line crosses the x-axis. At this point f(x) = 0. This observation suggests both the method of finding such values and the name that we might assign them. We will call all the values x i at which f(x i ) = 0 the Zeros of the function (they are sometimes called the roots). They can be found by using the Solve subroutine of the Algebra package in the TI Nspire and setting the function equal to zero. Solve! x + 3 = 0, x x! = What happens when you try to find the zeros of f x = x! + 1? 34

36 Homework 1. Graph the functions f(x) = 2x and g(x) = x 2 on a single graph. Be sure to label and scale your axes. 2. Mark the points of intersection on the graph. 3. Find the approximate points of intersection using the graphical intersection tool. 4. Find the exact points of intersection using the algebraic method. 35

37 5. Graph the functions h(x) = 1/x and s(x) =3x 3-2x 2-5x +2 on a single graph. 6. Mark the points of intersection on the graph. 7. Find the approximate points of intersection using the graphical intersection tool. 8. Find the exact points of intersection using the algebraic method. 36

38 Clearing Graphs To delete all the graphs created on your scratchpad: Click menu Choose 1:Actions Choose 4:Delete All Click on Yes To delete a single graph created on your scratchpad: Place the function s rule you wish to delete in the entry line and Hit ctrl clear Hit enter Clearing Scratchpad Contents To delete the calculations and graphing work from the Scratchpad application, perform the following steps: Press DOC, then select B: Clear Scratchpad. Press ENTER to delete the Scratchpad contents. Your TI-Nspire has many other great graphing features. The previous example is provided to get you started. Later in the course, we will use some of these features in calculus when we are finding the area between two curves and looking for the lower and upper limits when integrating a function. In addition, you can explore many of the other functions and options as you move through the course, including scatter plots and regression, two useful techniques used in statistics. 37

39 Creating Sub-Folders and Documents One of the advanced features of the Nspire is its ability to create folders and documents. Creating a New Folder Go to Home Highlight 2: My Documents Hit enter Highlight the folder in which you wish you create a sub-subfolder and hit menu. For this example we will create a sub-folder in the My Documents folder. Choose 1: New Folder Enter the name you wish to give the new Folder. (We will call this folder Chap 2.) Hit enter Saving a Document to a Folder Go to Home Highlight 1: New Documents Hit enter Choose the action you wish to take. In this example we will choose 1: Add Calculator Enter the contents you wish to save. Let us enter enter. Hit doc Choose 1: File Choose 5: Save As (make sure the correct folder appears in the Save in box 38

40 Type in the file name, we will call this sample 1. Highlight save Hit enter Retrieving (Opening) Documents Go to Home Highlight 2: My Documents Hit enter Highlight the document you wish to retrieve (open) Hit enter Open sample 1 Adding a Page to a Document With sample 1 open: Hit ctrl doc Choose the action you wish to take. In this example we will choose 2: Add Graphs Enter the contents you wish to save. Let us enter the function f1(x)=2x+1. Hit doc Choose 1: File Choose 4: Save 39

41 Deleting a Document Go to Home Highlight 2: My Documents Hit enter Highlight the document you wish to delete Hit ctrl menu Choose 6: Delete Answer Yes if questioned about proceeding. Homework 1. Create a folder named HW Chap 2 2. Insert into that folder a document called tens containing the results of the following calculations: 3 + 7, 60 6, 25 15, Add a page to that document containing the graph of f (x) = x