Appendix A Using a Graphing Calculator. Section 4: The CALCULATE Menu

Transcription

1 Appendix A Using a Graphing Calculator Section 4: The CALCULATE Menu The CALC menu provides access to many features that will be regularly used in the class. value returns a single y value when the user enters an x value. zero provides a means for finding the zeros or x intercepts of a function. minimum provides a means for finding the local or relative minima of a function. maximum provides a means for finding the local or relative maxima of a function. intersect provides a means for finding any points of intersection when two or more functions are graphed. value: Enter a function at Y=. Then press GRAPH. Once the graph is on the screen, press 2 nd TRACE and select value. Enter the desired value next to X=. The calculator will return the associated y value. Note, the point in question must be displayed on the screen. Otherwise, the calculator will return an error message. Example 1: Use the value feature to find f (3) if f ( x) = x 5x x + 9. Solution: Start by graphing the function. Then press 2 nd TRACE and select value. Now press 3, ENTER. Math 1314 Page 1 of 17 Appendix A, Section 4

2 Note the outcome when X= 12 is entered. The highest value of x displayed is 10, so value will not return a y value for X = 12. zero: Enter a function at Y= and graph it. Press 2 nd TRACE and select zero. The calculator asks, Left Bound? Use the left arrow key to move the cursor so that it is to the left of the zero. Press ENTER. The calculator asks, Right Bound? Use the right arrow key to move the cursor so that it is to the right of the zero. Press ENTER. The calculator asks, Guess? Press ENTER. The location of the zero will appear at the bottom of the screen. Repeat the process to find the other zeros (if any) of the function. *** Example 2: Find the leftmost zero of the function f ( x) = x 5x x + 9. Solution: Start by graphing the function and adjusting the viewing window, if needed. Math 1314 Page 2 of 17 Appendix A, Section 4

3 Press 2 nd TRACE and select zero. Move the cursor so that it is to the left of the zero. Press ENTER. Note the arrowhead pointing right near the top of the screen. Now use the right arrow key to move the cursor to the right of the zero. Press ENTER. Math 1314 Page 3 of 17 Appendix A, Section 4

4 Note the arrowhead pointing left. The two arrowheads are setting the interval within which the calculator will search for the zero. Notice that the first point had a negative y value and the second point has a positive y value. This guarantees that there is a zero in between. Now press ENTER to find the zero. The zero we found is ( ,0). Example 3: Use the same function, zero that is near x = f ( x) = x 5x x + 9. Now find the Solution: Graph the function, then press 2 nd TRACE and select zero. Move the cursor so that it is to the left of the zero. The location of the zero means the left bound needs to have a positive y value. Math 1314 Page 4 of 17 Appendix A, Section 4

5 Press ENTER. Now move the cursor to that it is to the right of the zero. The point will need to have a negative y value. Now press ENTER. Now press ENTER and read off the zero. Math 1314 Page 5 of 17 Appendix A, Section 4

6 This zero is ( ,0 ). *** minimum: This feature will find the x and y coordinates of a relative (or local) minimum of the function. A relative or local minimum is a point on the graph that is lower than the points around it. Enter a function at Y= and press GRAPH. Move the cursor so that it is to the left of the low point. Press ENTER. Move the cursor so that it is to the right of the low point. Press ENTER. Then press ENTER and read off the coordinates of the relative (local) minimum. Example 4: Find the coordinates of the relative (local) minimum of the graph of f ( x) = x 5x x Solution: Enter the function in the Y= menu and press GRAPH. Adjust the viewing window if needed. The relative (local) minimum is located in the 4 th Quadrant. Press 2 nd TRACE and select minimum. Press ENTER. Math 1314 Page 6 of 17 Appendix A, Section 4

7 Move the cursor so that it is to the left of the low point. Press ENTER. Move the cursor so that it is to the right of the low point. Press ENTER. Note the two arrowheads at the top that are marking the interval within which the calculator will look for a low point. Press ENTER. Math 1314 Page 7 of 17 Appendix A, Section 4

8 The x and y coordinates of the relative (local) minimum are displayed at the bottom , of the screen. The relative minimum is ( ) maximum: The process of finding a relative (local) maximum is identical to the process for finding a relative (local) minimum. Identify an interval that contains the local maxim, then ask the calculator to return the answer. Example 5: Find the coordinates of the relative (local) minimum of the graph of f ( x) = x 5x x Solution: Enter the function in the Y= menu and press GRAPH. Adjust the viewing window, if needed. *** Press 2 nd TRACE and select maximum. Math 1314 Page 8 of 17 Appendix A, Section 4

9 Press ENTER. Move the cursor so that it is to the left of the high point. Press ENTER. Move the cursor so that it is to the right of the high point. Math 1314 Page 9 of 17 Appendix A, Section 4

10 Note the two arrowheads at the top of the screen define an interval within which the calculator will look for a maximum value. Press ENTER. Press ENTER and read off the coordinates of the relative (local) maximum. The local maximum is ( ,9.0278). intersect: This feature finds the point(s) at which two functions intersect. Enter the two functions on the Y= menu. Press GRAPH. Adjust the viewing window if needed to see the point(s) of intersection. Press 2 nd TRACE and select intersect. The calculator asks First curve? and wants confirmation that the cursor is on one of the two functions whose intersection is sought. Look at the top of the screen to see which of the functions is selected. If this is correct, press ENTER. It will be necessary to move the cursor close to the point of intersection before beginning the process. (If there are several equations entered in the calculator, use the up and down arrow keys to toggle between the functions and press ENTER when the *** Math 1314 Page 10 of 17 Appendix A, Section 4

11 correct one is displayed at the top of the screen). Now the calculator asks Second curve? and wants confirmation that the cursor is on the other desired function. If this is correct, press ENTER. (If not, toggle through the functions until the correct one is displayed at the top of the screen and press ENTER). Press ENTER one more time. The point of intersection will be displayed at the bottom of the screen. If two functions have more than one intersection point, repeat the process to find the others. Example 6: Find the point of intersection if 1 f ( x) = 3x + 5 and g( x) = x 1. 2 Solution: Enter the two functions on the Y= menu and press GRAPH. Press 2 nd TRACE and select intersect. The cursor is on one of the functions (see the display at the top), so press ENTER. The cursor has moved to the other function (see the display at the top), so press ENTER. Math 1314 Page 11 of 17 Appendix A, Section 4

12 The calculator is waiting for verification. Press ENTER. The point of intersection ( 2.4, 2.2) is displayed at the bottom of the screen. *** Example 7: Find the points of intersection if 1 g( x) = x f ( x) = x 5x x + 9 and Solution: Enter the two functions on the Y= menu and press GRAPH. Use a standard viewing window. Note that we are interested in the points of intersection, and we can see all three of them. We can t see the minimum that s located in Quadrant 4, but we don t need to in order to find the points of intersection. Start by finding the leftmost point of intersection. Press 2 nd TRACE and select intersect. Math 1314 Page 12 of 17 Appendix A, Section 4

13 Press ENTER. Move the cursor so that it is close to the Quadrant 3 point of intersection. Press ENTER. The cursor will move to the other function. Note the function displayed at the top of the window. Press ENTER. Then press ENTER again. Math 1314 Page 13 of 17 Appendix A, Section 4

14 The third quadrant point of intersection( , ) is shown at the bottom of the screen. Next, find the intersection point that appears to be near x = 1. Press 2 nd TRACE and select intersect. Move the cursor close to x = 1. Press ENTER. The cursor will move to the other function. (Note the function displayed at the top of the screen.) Math 1314 Page 14 of 17 Appendix A, Section 4

15 Press ENTER. Press ENTER. The coordinates of the second point of intersection, ( , ), are shown at the bottom of the screen. Now repeat the process to find the third and final point of intersection. Press 2 nd TRACE and select intersect. Move the cursor so that it is near the third intersection point. Math 1314 Page 15 of 17 Appendix A, Section 4

16 Press ENTER. Press ENTER. Press ENTER. The third point of intersection, ( , ), is given at the bottom of the screen. *** Math 1314 Page 16 of 17 Appendix A, Section 4

17 Note: There are two other features on the CALC menu, dy/dx and intf(x)dx. Other features do the same operations and are easier to use. Math 1314 Page 17 of 17 Appendix A, Section 4