Useful TI89 Skills
Solving a system of linear equations
We shall plot the graph of the function .
- To enter this function, access the Y= key. Do this by pressing the green diamond key followed by F1. Enter a new function at this point by entering the right side of the above expression. Edit an existing function definition by pressing F3. Use F4 to indicate which function(s) are to be graphed. Note that multiple functions can be entered at this stage. All functions with checks will be graphed.
- Draw the graph using the GRAPH key. Do this by pressing the green diamond followed by F3.
- If you cannot see the graph, it may be that the graph window is inappropriate. Begin by using F2 ZoomStd. This will show a portion of this particular graph. If the graph had still not appeared we would look at the formula to get some clues about what the ranges on the coordinate system should be. We note that, due to the logarithm, the graph has a vertical asymptote on the y-axis.
- Now look for interesting features of the graph. In this case, we will look for x-intercepts. Use F3 to activate the trace function. The coordinates of a highlighted point will be shown on the screen. The left and right cursor control keys can be used to move the point. Doing this we see that there are two x-intercepts between 0 and 3. To get a better look at where these intercepts are located press F2 ZoomBox Enter. Set the first corner of the zoom box at approximately (0,-0.5). The cursor control keys more the corner. When the location is correct, press Enter. Locate the second corner at approximately (4,1). Now you should see clearly (with the help of F3 Trace) that this graph has two x-intercepts at approximately 1 and 2.15. To get more accurate estimates of these intercepts see the next section.
We shall solve the equation .
- Press Home to return to the home screen. The solve command can be used to solve this equation. Press CATALOG and scroll to the solve command and notice the proper syntax for this command given at the bottom of the screen. Press Enter to place the solve command on the command line of the home screen. Then enter the equation and the variable. Your command should look like
Solve(x-1-3*ln(x)/2=0,x).
- Press Enter to get the approximate solutions 1 and 2.14403.
- In
some cases, the solver may have problems finding a solution. It will help the solver if you give it
two good estimates of the solution, and you can often get such estimates
graphically. To do this, plot the
graph of using the instructions of the previous section. While viewing the graph, use F3 Zero. Use the left and right cursor control keys to set the lower bound as the largest value you can find that is smaller than the solution you seek. Similarly, the upper bound should be set as the smallest value you can find that is larger than the solution. The solver uses these two values to find an accurate solution.
Solving a System of Linear Equations
As an example, we shall solve the system
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- To access the Data/Matrix editor, press the APPS key. Then scroll down to Data/Matrix Editor and press the right arrow key. Scroll down to New and press the Enter key.
- Once in the matrix editor, select the type Matrix , skip over the folder option with the down-arrow key, enter a name for your array (such as A). Next, enter 3 for the row dimension and 4 for the column dimension, and then press the Enter key.
- Now enter the coefficients of your variables and the numbers on the right-hand side into the matrix editor. When you are finished, the array of numbers should be
- Now,
press the Home
key, and enter rref(A) on the input line.
An array will be returned, and the first three columns should have
ones on the diagonal and zeros elsewhere.
If that is the case, the last column will contain the values of x,
y, and z arranged from top to bottom. The answers should be , respectively.
Finding Maximum or Minimum Values of Functions
We shall find the minimum of the
function .
- Enter this function using the Y= key. Do this by pressing the green diamond key followed by F1. Enter a new function at this point by entering the right side of the above expression. Edit an existing function definition by pressing F3. Use F4 to indicate which function(s) are to be graphed. Note that multiple functions can be entered at this stage. All functions with checks will be graphed.
- Draw the graph using the GRAPH key. Do this by pressing the green diamond followed by F3.
- Find
the minimum using F5
Minimum. Use the left and right
cursor control keys to enter x-values to the left and right of the x-value
that minimizes the function. Input
these values with Enter. The minimum of this function is
approximately . This function has no maximum, and it might be instructive for you to attempt finding a maximum using the calculator.
Series
We will find sums of the form
.
- Press F3 to bring up calculus applications.
- Select the fourth menu item with sigma.
- Now
enter the above expression so that the command line has the following
appearance: .
Curve Fitting
We shall enter a data matrix of two columns and plot points and the regression line for the data.
- To access the Data/Matrix editor, press the APPS key. Then scroll down to Data/Matrix Editor and press the right arrow key. Scroll down to New and press the Enter key.
- Once in the matrix editor, select the type Data , skip over the folder option with the down-arrow key, enter a name for your array (such as A), and then press the Enter key. Now enter the x-coordinates of your data in the first column; enter the y-coordinates in the second column.
- After you have entered the data, press F2 to set up your plot. Next, press F1. Select Scatter Plot as the plot type, and enter C1 and C2 in the rows identifying x and y. Press Enter to accept this plot structure.
- Now it’s time to calculate the regression line. Go back to the Data/Matrix editor and select the Current Matrix. You should see the data that you entered earlier.
- Press F5
for calculations. Select
the Calculation Type that is appropriate for your problem perhaps LinReg for linear regression. Enter C1 for x and C2 for y. Next, enter a graph variable in the Store ReSEQ to lineperhaps y2. This action stores the regression equation in one of your graph variables so that you can easily construct the graph without entering the equation yourself.
- Press Enter to see the regression equation.
- To graph the data points and the regression line (or curve), press the green diamond key followed by F1. Make sure that your regression equation and your point set are checked.
- Draw the graph using the GRAPH key. Do this by pressing the green diamond followed by F3.
- If you cannot see the graph, it may be that the graph window is inappropriate. Zoom out by using F2 ZoomOut. You should be able to see the data points and the regression line or curve.
Matrix Inverses
We’ll outline the process of finding the inverse of a square matrix.
- Access the Data/Matrix editor by pressing the APPS key. Then scroll down to Data/Matrix Editor and press the right arrow key. Scroll down to New and press the Enter key.
- Select matrix rather than data type. Enter a name, say A, for your matrix in the field labeled “Variable:” Next, enter the row and column dimensions.
- Once in the matrix editor, enter the rows and columns of your matrix. Symbolic expressions can be used.
- After entering your matrix, press the HOME key.
- Now, enter the following into the command line:
Augment(A,identity(n))→B
Recall that the right arrow is entered using the STO key. This action augments the matrix A with the identity matrix and stores the augmented matrix in a variable called B. The number n represents the dimension of the matrix A.
- Now enter
rref(B)
from the command line. The reduced matrix should have the identity in the left half and the inverse of A in the right half.