Calculus & Diff Eqs forTI-89 Titanium

Content:  This document covers special expression, graphing, graphing in 3D, complex numbers, list arithmetic, tables, special
functions, calculus, matrices, sequences, differential equations, complex numbers, Dot-Product and cross-product of vectors
as performed on the TI-89 Titanium calculator.
 DATE LAST REVISED:  7/7/14

 INDEX:

To facilitate lookup, the instructions are divided into the following categories:

         I.   Basic Operating Information - Friendly values using TRACE, navigating the APPS
              features,  how to change MODE, scientific notation,logarithms,
       II.   Special Expressions – Greatest Integer, etc.,  finding
              primes of a number,
       III.  Graphing and Evaluating Functions – Graphing a function, obtain standard size window, Change to custom size
              window, use of TRACE, finding max and min of a graph, function value at specific x, using box zoom, intersection
              points, finding zeros, finding coordinates to graph parabola by hand, graphing inequalities, marking points on a graph,
              3D graphing, etc.
        IV.  Special Functions – Changing from radians to degrees,  graphing piecewise functions, etc.  Using [TABLE] to find points
             for graphing a parabola by hand.
        V. Polynomials - Expand an expression, Simplifying an expression, Solving a polynomial, Factor a polynomial, Solving a
             quadratic equation, Solving inequalities, Solving polynomials with degree higher than 2, graphing polynomials from the
                polynomial application,
        VI. Derivatives, Antiderivatives, & Tangents – Finding the derivative of a function, finding the implicit derivative, finding the numerical derivative
                of a function, determining the value of the derivative at a point on a graph, drawing a tangent line to a graph,
                finding the anti-derivative of a function, calculating the value of a definite integral, finding the limit.
       VII. Matrices – Entering and editing matrices, multiplying matrices, other matrix math, doing rref, solving a system of linear
              equations,
        VII. Differential Equations - Solutions to first order ODE, solutions to second order ODE, Draw slope fields, Draw slope fields
              and solution, Draw solution without slope fields, draw a solution with slope fields,
         IX.  Sequences - Finding several terms of a sequence, finding a specific term of a sequence, summing sequences, cumulative
               sum of sequence, graphing a sequence.
        XI.  Complex Numbers - Solving a polynomial with complex coefficients,
       XII.  Vectors - Cross product of two vectors,
      

 GENERAL:   Important Note: It is expected that those who use this document will be familiar with the basics of the calculator
      Consequently, the basic operation of the calculator will not be repeated in this document. For basic user information click here
       to go to the user's guide.

 I.  BAISC OPERATING INFORMATION
     1)  Friendly Values on Graphs Using TRACE:
       Many times when you use the TRACE function, you will get an x-value such as 2.784532.  If you change the x-min
       and x-max in the WINDOW function to be multiples of 7.9 and the y-min and y-max to multiples of 3.8 , the displayed 
       values will be "friendlier."  That is, they will be integers or numbers with one or two decimal places.  You can always
       set the values by hand, but the easiest method is to use the ZDecimal function of ZOOM.  Just press ZOOM; then
       4, for ZDecimal. 
          It may be that the display is now partially off the screen.  If you want the entire graph on the screen, use the Zoom Out  
       function. To do that when in the graph mode,  press F2; then 3, ENTER.
       If you’re trying to find the value at a specific point, a zero for example, and the cursor still does not fall on the x-axis,
       you could try different strategies such as ZBox, but I usually prefer to use the zero function.  To do that, press F5, 2.
      That will set you up for finding a root.  Remember that any time you want to get back to the standard window just
      press F2, 6.

    2.  Using the APPS screen:
         a)  Press APPS to go to the screen that has icons for all of the applications. 
         b)  If the applications are not on the first screen,  press the first letter of the title of the icon you want to  
             select.  For example, if you want to select Matrices D, the first letter of the title Data/Matrices.  For Lists 
             you would press  S for statistics. 
         c)  Press 2ND, APPS to toggle back and forth between the present screen and the previous APPS screen.  Note
              that not all non-APPS screens are recalled by this method. 
          d)  Another way to organize the APPS functions is to place them in folders.  As for myself, I have placed the
               six APPS that I use most often in the MATH folder.  That way when I press APPS, I can then press
               F1,2, 4 and my most-often-used APPS will immediately be displayed on the screen.  To organize the APPS, do
               this:
               1)  Press APPS, F1, right cursor arrow, 3.  A dialog box with a list of APPS will appear on the screen.  Scroll
                    up or down the list with the cursor arrows and press the cursor right arrow to select one of the APPS.

   3. MODE SETTING: Some users can save themselves some possible frustrations by setting the
      Exact/Approximate function on the MODE menu before doing operations such as logs, roots, trigonometric
      functions, and fractions. The situation is that on Exact and Auto, the calculator is looking for an exact answer, and
      if there is not an exact answer the calculator will merely return the entry with no results, except for extracting
      the exponent for a perfect square in some operations. Example are as follows: For Exact or Auto sin (2) will
      return sin (2), whereas, if the MODE is set for Approximate, the return will be 9.09297....So, for beginning users
      it is probably best to set the MODE to Approximate. The Exact or Auto does have its important uses, however. For example,
      Tan (π/3) will return 1.73...in Approximate MODE, while if Auto or Exact are used, one gets √3. Those results are
        especially useful when working with trigonometric functions.   

 4. Scientific Notation:
     To enter a number such as 3 x 10² , press 3,  EE (left col. row 7), 3. If you want the results of an operation in
     scientific notation, "Exponential Format" in the MODE menu must be set to scientific.

 6.  Logarithms:
     a. To find the common log, press ♦; then 7. The default is the common log.
     b.  Enter the number, press ), and then press ENTER.
     c. For logs to other bases, the syntax is log(number, base). For example, 256 to base 2 would be entered as log(256,2).
     d.  The natural logarithm is on the keyboard. Just press 2ND, the X-key; close the parentheses and press ENTER. Note that
          depending on how you have MODE set, you may need to enter a decimal after the number. For example, you might need
          ln(3.). I usually just enter it ln(3.0).
        

 II.  SPECIAL EXPRESSIONS:
  1. Finding the greatest integer function of a number.
         a) From the Home screen, press the CATALOG key; then press the letter I.
         b) Move the cursor down to int( and press ENTER.
         c) The expression int( will be pasted to the Home screen.
          d) Enter your number and close the parentheses.
          e) Press ENTER. (Note that this also works for negative numbers.)

 2.  Finding the primes of a number.
      a.  From the home screen press F2, 2, and factor( will be pasted to the screen.
      b.  Enter the number you choose, close the parentheses,  and press ENTER to display the results.

III.  GRAPHING & EVALUATING FUNCTIONS:
   
 1. Graphing a function.
         a) Press ♦, F1 to display the "Y=" screen. All equations must be in the slope-intercept form, y=mx+b,
             before entry.  Note that only F1 and F2 are activated in this mode.
         b) Enter the function(s) using the appropriate keys to enter the variable.
         c) Press ♦, GRAPH to graph the function.  (If you don’t see your graph, press
            TRACE and use the arrows to find the maximum or minimum value of your 
            function.  Press ENTER and the graph should appear.)
         d) To leave the screen without graphing, press Home.
         e) Press CLEAR while the cursor is on the same line as the function to erase the
             function.

 2.  To obtain the standard size viewing window:
           a) Press the F2 to go to the Zoom menu.
           b) Press 6 to execute Z Standard.

  3.   To change the viewing window to a custom size:
           a) Press ♦, WINDOW.
           b) Use the cursor keys to move the to the value to be changed.
           c) Enter the new value.
           d) Press ♦, Graph to see the new graph, or press ♦, F1 to return to the Y= screen.
           e) ZSquare keeps the y-scale the same and adjusts the x-scale so that one unit on the x-axis equals one
               unit on the y-axis.
            f)  ZDecimal makes each movement of the cursor equivalent to one-tenth of a
             unit.
            g) ZInteger makes each movement of the cursor equivalent to one unit.

  4.  Evaluating a function.
         a) Press ♦, F1 and enter a function, y1 for example.
         b) Press ENTER.
         c) At the Home screen, enter y1(value) or for a series of values y1({value, value, value}). For
             example, y1({1, 2, 3})
         d) Press ENTER and the values will be displayed.

 5.  To change or erase a function:
         a) Press ♦, F1.
         b) Use the arrow keys to move the cursor to the desired location and make
             changes by inserting, deleting, or changing the desired characters.
         c) To erase a function, with the cursor on the same line as the function, press the
             CLEAR key.

  6. To use the trace function:
         a) From the Graph screen, press F3.
          b) Use the right and left arrow keys to move the cursor along the graph. The
              coordinates of the cursor location are shown at the bottom of the screen.
         c) If more than one graph is on the screen, you can press the up or down arrows
            to jump from one graph to another.
         d)  To end the trace operation, press ESC.

  7.  Finding the maximum and minimum points.
         a) Enter the function and graph it.
         b) Press F5. Press 3 for minimum or 4 for maximum.
         c) Move the cursor to the left of the point where a minimum or maximum appears to be and press ENTER.
         d) Move the cursor to the right of the point and press ENTER.
         e) The coordinates of the maximum or minimum will appear at the bottom of the screen.
          f)  Press ESC to end this operation.

  8.  Finding the value of a function at a given value of x.
         a) Enter the function and the graph it  .
         b) Press F5.
         c) Press 1 to select value.
         d) Enter the x-value and press ENTER. The x- and y-values will appear at the bottom of the screen. 
             Press ESC to terminate this operation.

  9.  To zoom in using a box.
          a) Enter the function and graph it.
          b) Press the F2.
          c) Press 1 to select ZBox.
          d) Move the cursor above and to the left of the location you want enlarged and
              press ENTER.
          e) Move the cursor below and to the right of the location you want enlarged and
              press ENTER.
          f) The box is then enlarged to fill the screen. Press F2, 6 to return to the Standard screen.

10.  Finding the intersection point of two graphs.
          a) Enter two functions on separate "Y=" lines and press ♦, GRAPH to graph the equations. Equations must
             be in slope-intercept form.
          b) Press F5 to select the list of operations
          c) Press 5 to select intersection.  The cursor will appear on one of the curves.  Move the cursor
              to a position so that the curves are clearly separated and to the left of the first intersection point.
              press ENTER.  Press ENTER again if the cursor is on the second graph; then press ENTER again
              if the cursor is to the left of the intersection point. 
          d) The calculator will request the Upper Bound.  Move the cursor to the right of the first intersection
              point but to the left of the second intersection point if there is more than one intersection point.
          e) Press ENTER and the coordinates for that particular intersection point will appear at the bottom
              of the screen.  Press ESC to get out of this procedure. 
          f)  If there is more than one intersection point, evaluate the other points in the same way, except the
              cursor will initially be placed to the left of the point you are evaluating.

11.  Solving an equation in one variable. (Also known as finding the roots or x-axis intercepts.)
          a) Enter the function and graph.
          b) Press F5, 2..
          c) Move the cursor to the left of the first x-axis intercept and press ENTER.
          d) Move the cursor to the right of that intercept and press ENTER.
          f) The coordinates for the root (zero) will appear at the bottom of the screen.
          g)  Additional zeros can be found by repeating this method for other x-axis intercepts.

12.  Finding coordinates to graph a parabola by hand.
          a)  Enter the graph in your calculator as described above.
          b)  Next locate the vertex by pressing F5, and pressing either 3 or 4, depending on
               whether the vertex is a minimum or maximum for the parabola.
          c)  Move the cursor slightly to the left of the vertex and press [ENTER].
          d)  Move the cursor slightly to the right of the vertex and press [ENTER].  The x- and y-values
               for the vertex will appear at the bottom of the screen  
          e)  To find additional poinst, press [♦],[TABLE] and a table of (x,y) values will appear.   (You may want to set your
               independent variable  for Ask so that you can enter x-values of your own choosing.  If it is not set for Ask,
               press  [♦], [TBLSET] and press the right arrow key and press 2 for  "Ask," and a table will appear if you started
               the change from the table screen .
           f)  Enter two more values for "x" in the table and the corresponding values for "y" will appear.
           g)  Use these coordinates and the symmetry property of a parabola to graph the parabola on a
                sheet of paper. (NOTE:  If the vertex is at an integer value, you can find the vertex from the
                table rather than using steps a through d to find the vertex.)

13. Graphing Inequalities.
           a)  Write each equation in the y =mx + b format and enter them into  the "Y=" positions.   
               (Remember  that you may need to change the direction of the inequality sign if you have to
               multiply by -1 during the rearranging of the equation.)
           b)  Select the graph to be shaded and press 2ND, F1 (to select F6). 
           c)   Press 7 (Above) or 8 (Below) as appropriate and press ENTER.  The shaded graph will appear.
           d)  Press ♦, GRAPH and the shaded graph will appear.

14.  Marking points on a graph.
           A few students who capture screens like to have marks on graphs.  Although some would
           consider this more trouble than it's worth, I'll include it for those brave souls who feel they must
           have them.
            a)  First, if you are a little picky about having the marks line up exactly on the graph, you  
                should press F2, 4 to select ZoomDec.  Then graph your function and see if the x-
                and y-values that you are interested in appear on the screen.  If not, press F2 (Zoom), ALPHA, C to
                select Set Factors.  A dialog box will appear for setting xfact and yfact. Make the xFact and yFact
                integers larger than 1.  Press ENTER to save the new zoom factors.
            b)  Go to the Home screen to start the procedure.  The syntax is PtOff x,y.  So, press 
                 CATALOG, p, and cursor to PtOff.
            c)  Press ENTER and PtOff will appear on the Home screen. 
            d)  Enter the x-coordinate, y-coordinate, and press ENTER.  A barely visible, for me, clear spot
                 will appear on the graph at the coordinates you selected.
            e)  To erase all points, press 2ND, F1,1, ENTER.

15. Graphing in 3D:
      Let's do the function z(x,y)=x²-y² .
      a) First set up the MODE by pressing MODE; then press the right arrow followed by pressing 5 for 3D.
      b) Press♦,Y= and key in the equation opposite z1.
      c) Press ♦,F3 (Graph) after some brief calculaions by the calculator, the graph similar to the one below
          will appear.
      NOTE: You can rotate or tilt the graph with the arrow keys.
     

 IV. SPECIAL FUNCTIONS
 1.  Changing from radian measure to the degree mode:
          a) Press the MODE key
          b) Move the cursor to Angle and press the right cursor arrow to select what you want.
          c) Press ENTER to return to the previous screen.

 2. Graphing piecewise functions or  functions on an interval:
           Let's look at the following piecewise function:
                     |   x    for x> 0
          f(x) =   |    -x  for x <0
      Method 1: Some of you may prefer to merely enter the following: (x)*(x>=0)+(-x)*(x<0). It's more flexible, in that it will
          work with the TI-83Plus/TI-84 as well as the TI-89 series.
       Method 2: If you only use a TI-89 series calculator, you might prefer a method that, to me at least, is somewhat less
           prone to typing errors. That method is outlined below.
         The piecewise expression has the form f₁ if condition, else f_2.  Let's graph this:
          a)  Press ♦, F1 to display the Y= screen.
          b)  Enter "when" as as follows:  Press CATALOG, w, select "when(" and press ENTER.
          c)  Press x; ♦, >(the decimal point key) to enter ≥.
          d)  Press 0, [,], x, [,], (-), x,).  You should have "when(x≥ 0, x, -x)" on the the entry bar.
          e)  Press ♦, F3 (GRAPH) and the graph will be displayed. 
              Note: One may expect that entry of the first method is faster than the last, but, actually, I did a quick time check and
                did the last faster. .
             

V.  POLYNOMIALS
1.  Expanding an Expression (can also be used for partial fractions):
      a)  From the home screen, press F2, 3.  The expression expand( will be pasted to the home screen.
      b)  Enter the number you want to expand, for example (x²-1)/(x^3 +2x²),  and close the parentheses. Your expression
           should look like this:  expand((x^2-2)/(x^3+2x^2))
      c)  Press ENTER and the answer will be displayed on the Home screen.

 2.  Simplifying Algebraic Expressions:

     General: There are two different functions that can be used in simplifications. I'll describe them both below.  I
frankly don't use the calculator for simplifications since I find it easier to do them by hand.  There are a couple things
you should be aware of if you choose to do simplifications with the calculator.  First, the expression given by the
calculator may not be formatted in the same way that most of us are accustomed to.  Second, the calculator can sometimes
interpret the expression incorrectly. I find that happens most often when I have an expression such as 7xy^2 +28x^2y^3. 
The calculator will reproduce the expression as 7*xy^2+28*x^2*y^3 and give an incorrect factorization.  The key is to
force multiplication by rewriting the expression as follows:  7x*y^2+28x^2y^3.  So, a good practice is to always force
multiplication when a first degree term is involved and check the computer duplication of your expression against
your actual entry.
Using propFrac
a.  From the HOME screen, press F2, 7 to select propFrac(.
b. Enter your expression being careful to get the parentheses correct. Example: profFrac((x^2 +3x +2)/(x+1)).
c. Press ENTER and the result, x+2,  will be displayed.
Using factor(
This can also be simplified using the function factor(.
a. From the HOME screen, press F2, 2 to select factor(.
b. Enter your expression being careful to get the parentheses correct. Example: factor((x^2 +3x +2)/(x+1)).
c. Press ENTER and the result will, x+2, will be displayed.
Note that factor( can also be used to simplify such expressions as 12x^2y^3 + 6x^2y + 4xy.
a. From the HOME screen, press F2, 2 to select factor(.
b. Enter your expression being careful to get the parentheses correct. Example: factor(12x^2y^3 + 6x^2y + 4x*y)
c. Press ENTER and the, somewhat confusing to some, expression 2x(3x(2y^2+1) +2)y will be displayed. If you're fairly
good at algebra, you can easily change this to 2xy(6xy^2 +3x +2), which is the more standard way of expressing
the simplification. Note carefully that the first degree variables xy are connected by a multiplication sign to prevent
an error in factorization.

3.  Factor a Polynomial:
     a)  Press F2, 2 to select factor(.
     b)  Enter your expression being sure to get the parentheses correct.  Example: factor(x^2 +5x +6).
     c)  Press ENTER and the result, (x+2)(x+3) will be displayed.

4. Solving a Quadratic Equation:
    a)  From Home press F2, 1 to select solve(.
    b)  Enter your equation. Example:  solve(x^2+2x +3=11, x).  Note that you must have the variable, in this case x,
         that you are finding the value for.
    c)  Press ENTER and the answer, x=-4 or x=2, will be displayed.

5.  Solving Inequalities:
     a)  From Home press F2, 1 to select solve(.
     b)  Enter you equation and close the parentheses. Example:  solve(x^2 +2>3, x).  The > symbol is obtained
          by pressing 2ND, and the key below 2.
     c)  Press ENTER and the result, x<-1 or x>1, will be displayed.

6.  Solving polynomials including those with degree higher than 2:
     a)  If you are not sure that all of the roots are real, press MODE, cursor to Complex Format and press the right
          cursor arrow; then press 2 to select RECTANGULAR unless you specifically want polar. Press ENTER to
          save any change.
     b)   Press APPS, select the Polynomial icon and press ENTER.
     c)  Press 3 for New.  On the next screen, enter the degree of the polynomial you are solving, for example 3, and
          press ENTER.
     d)  Enter the coefficients for the terms starting with the highest degree term. Press ENTER after each entry.
     e)  Press F5 and the solution will be displayed.  To enter a new polynomial, press ESC.

7.  Graphing a polynomial from the polynomial application:  
     a)  Enter the polynomial coefficients according to the item directly above. 
          1)  Press F6 (that's 2ND, F1), 1, press the right arrow key.
          2)  Select Full for Split Screen, select y1 for  Store Poly To, and select whatever you want for ZOOM. If you
               don't have a choice, start with ZoomStd.
          3)  Press ENTER and the graph will be displayed. 

VI. Derivatives, Antiderivatives, & Tangents
1.   Finding the Derivative of an Function:
          a) Press Home if not already at the Home Screen.
          b) Press F3, 1 to select d(   for  differentiate.
          c) Enter the function.  For example, enter (x-y) ³/(x+y) ² as follows:
              d ( (, x, -, y, ), ^, 3, ÷, (,x, +, y, )^,2, comma, x,) so that d((x-y)^3/(x+y)^2, x) is displayed on the entry bar.
          d) Press ENTER to display the result.
              Math professors are inclined to say, "With the quotient formula and a little algebra, you'll come up with the
             calculator answer of  [(x-y)^2(x+5y)/(x+y)^3. I'll resist that because I want to give the reader an idea of how
             much work is required for even this simple problem to get the calculator answer. So, using the Santa Clause
              routine for the derivative of a quotient, that is, ho,d(hi); hi, d(ho); ho, ho, we get as a first result the following:
             [ (x+y)^2*3(x-y)^2 -[(x-y)^3*2(x+y)]]/((x+y)^4, and then we start simplifying.
             [3(x+Y)(x-y)^2 -2(x-y)^3]/(x+y)^3 (Cancel (x+y).
             [(x-y)^2[3(x+y)-2(x-y)]/(x-y)^3 (Factor out (x-y)^2)
             [(x-y^2)(3x+3y-2x+2y)]/(x+y)^3 and simplify to get
             [(x-y)^2(x+5y)/(x+y)^3 which is what the calculator gives us.
 

2.   Finding the Implicit Derivative:
       I believe that this is a new capability that does not appear on the TI-89 (not Titanium), but, at the end of this
     discussion, I will suggest a method for finding the derivative implicitly on the TI-89.  Here is how to do it with
     the TI-89 Titanium:
         a)  From the Home screen, press F-3, ALPHA, D, and impDif( will be pasted to the
              Home screen.
         b)  Enter information according to this syntax:  impDif(x^2 + y^2 = 5, x,y).  Press ENTER and the implicit
              derivative, -x/y, (1st derivative)  will be displayed.   If you want the second, or even the third derivative, you
              can add that after the variable, y, as in the following:  impDif(x^2 + y^2 = 5, x,y,2).  The 2 is for the
              2nd derivative.  For the first derivative, omit the number.
         c)  Press ENTER and the derivative will be displayed. 
     Now, how to do this with the TI-89 (not Titanium):
       Remember some time back in the past you learned that if z is a function of x and y,  then you can write the
       relationship below:
       dy/dx = -(∂z/∂x)/(∂z/∂y)  (Eq 1)
       So, let's just take the derivative treating y as a constant; then the derivative treating x as a constant. 
       Then we'll just plug them into Eq 1 above.  We may need to do a little simplification, but that should get
        us close.  Don't forget the negative sign in Eq. 1.
        So, here are the steps to do x² + y² =5:
         a)  Press 2ND, MATH, ALPHA, B, 1 and d( will be pasted to the home screen.
         b)  Enter terms so that your entry looks like this:  d(x^2+y^2 = 5, x). 
         c)  Press ENTER and the answer, -2x, will be displayed. That's the numerator.
         d)  Replace the "x" for the variable, the last x, with "y."
         e)  Press ENTER and 2y will be displayed.  That's the denominator. 
         g)  Now, plug those into Eq 1 and you'll have dy/dx= (-2x/2y) = -x/y, which is the same answer we obtained
              with the TI-89 Titanium.  Of course, you could define a variable, but I'll leave that to the calculator geeks.         

 3.   Finding the numerical derivative of a function:
       a) From the Home screen, press the F3, ALPHA, A. .  The expression nDeriv( will be displayed
       b)  Enter the expression you want to find the derivative of.  For example, d(x^2)dx would be nDeriv(x^2,x).
       c)  Press ENTER and the result will be displayed.
          

 4.  Determining the value of the derivative from points on a graph:
          a) First enter the function and graph it.  The press the F5, 6,1.  A flashing cursor will be displayed on the graph. 
          b) Move the cursor to the point on the graph where you want to find the derivative and press
              ENTER.
          c)  The value of the derivative at that point will be displayed.

 5.  Drawing a tangent line at a point.
          a) Enter the function and graph it.
          b) From the graphing screen, press F5, ALPHA, A. 
          c) Move the flashing cursor that appears to the point where you want the tangent and press ENTER.
          d) The tangent line will be drawn at that point.
          e) To clear the tangent line, press ESC, F4.
         

 6.  Finding the Antiderivative of a Function: 
          a) From the Home screen, press 2ND, ∫, (the number 7 key).
          b) Enter the expression of interest.  For example ∫ x^2 dx would be ∫ (x^2, x)
          e) Press the ENTER and the antiderivative will be displayed.

  7.  Calculating the value of a definite integral:
          a) Graph the function of interest and press F5, 7.
          b) Move the flashing cursor to the lower limit you desire and press ENTER; then move the
              cursor to the upper limit you desire and press ENTER
          c) The area will be shaded and the value of the are will be displayed.

  8.  Finding the limit of a function:
       a)  From the Home screen, press F3, 3 to paste limit( to the screen.
       b)  The syntax is "limit( expression, var, point, direction)," where direction from the left is -1 and from
            right is 1.  Lets do the limit of 1/x as x approaches 5.
       c)  Enter information so that you have "limit(1/x, x, 5).
       d)  Press ENTER and the answer, .2 will appear.
       e)  To find the limit of the same expression as x →∞, erase 5 and press ♦, CATALOG to paste ∞ to the
                screen.
          f)  Make sure the parentheses are closed;  then press ENTER and the answer, 0, will be displayed.

 VII.  MATRICES:
 General:  When it comes to doing matrices, there are different ways of setting them up and editing them,
but I am going to stick with one way.  If you want to try another, read your user manual (good luck.).  I'm first
going to tell you how to set up a matrix using the Data/Matrix editor; then I'll show you how to edit a matrix
that has already been set up.  Finally, I will tell you how a teacher might set up his/her calculator to save lecture
time with the details.

1)  Entering and Editing a Matrix:
       A)  Entering a Matrix:
         a)  Press [APPS] if you're not already at the APPS menu; then select the Data/Matrix icon and press
              [ENTER].  Press 3 to select New.  Note:  When you go to the APPS screen, the Data/Matrix icon
              may not be displayed on the screen.  To display a group which has the Data/Matrix icon, press D. 
         b)  With the cursor opposite Type, press the right cursor arrow and press 2 for Matrix.
         c)  Move the cursor down to the space for Variable and type in whatever variable you want.  Type in whatever
              unrestricted variable you choose.  If you intend to retain this matrix, I suggest you not use the
              variables a-z because sometimes it is convenient to clear all of these at the same time.
         d)  After you have entered the variable move down and enter the number of rows and columns; then
              press ENTER and the matrix table will be displayed.
         e)  You can now either enter data or press Home and edit the matrix when you need it. To enter data, enter
              each value and press ENTER.  The cursor will automatically move to the next position for the next entry.
         f)  After you are finished, press Home to go to the home screen.       
      B)  Editing a Matrix:
            a)  Press [APPS] if you're not already at the APPS menu; then select the Data/Matrix icon and press
              [ENTER].  Press 2 to select Open
            b)  Press the right cursor arrow and press 2 to select Matrix; then move the cursor down to Variable, press
                 the right arrow, select the matrix you want and press ENTER, ENTER.  The matrix will be displayed for
                 editing.
            c)  Enter data and press ENTER after each entry. 
            d)  After you've entered all of the data, press Home to go to the home screen for matrix operations.
             Note:  If you want to display a matrix from the home screen you can Press 2ND, VAR-LINK (the minus key),
                 F2, cursor to Var Type, press right cursor, 4, ENTER.  Then highlight the matrix you want and press
                 ENTER.  The matrix name will be displayed.  Press ENTER and the matrix will be displayed on the Home
                 Home screen.
        C)  Comments:
              a)  As I go through the operations you will notice that I have variable names for my matrices that may seem
                   to you to waste memory.  I have done that because I mostly teach others how to do matrices, and my
                   method save time for me.  I have chosen to store permanently matrices of several different dimensions
                   and to give them variable names so that I can immediately identify what dimensions they have.  For
                   example, Matrix m134 would be the first of the 3x4 matrices; m234 would be the second.

 2)  Multiplying two matrices [m133] * [m233]:
         a)  Enter the data into matrices [m133] and [m233] according to "Editing a Matrix" above, and press Home
              to go to the home screen.
         b)  From the home screen, enter the name of the matrix, e.g., m133.
         c)  Press x (the multiply sign) and enter the name of the second matrix, e.g., m233. 
         d)  Press ENTER to perform the multiplication step.
         e)  Remember that the numbers of columns in [m133] must equal the number of
                  rows in [m233] or you will get a dimension error.

3)  Doing other matrix math:
         a)  Press 2nd, MATH, 4.  On a pop-up list you will see a list of operations that you can do.    To find the
              transpose, use T; to  find the determinant, use det(, to find the reduced row-echelon form use rref(, and
              so on with the other operations.  .
         b)  After you select the operation you want, press ENTER and the operation will be pasted to the home screen..
         c)  Enter the variable name of the matrix you want and close the parentheses.
         d)  Press ENTER again to get your answer.    

NOTE:  You can do any of the elementary row operations, swap rows, multiply a row by a constant, add two rows,
multiply one row by a constant and add it to another row, by pressing 2ND, MATH, 4, ALPHA, J, pressing the right arrow
key and selecting the operation you want to do.  They are very useful for doing the arithmetic for Gauss or Gauss-Jordan
elimination, but considerable time is required to get the hang of doing row operations. So, since most students don’t take the time
to use those functions, I’m not going to include them.  Instead, I’ll give you my Website as a reference for doing those operations if
you want to do them.  If there are enough request, I will include that.

 4)  Doing rref and ref:
      Since rref and ref are used extensively in our college, I'm going to include them as separate items.
         a)  First enter the data in your matrix (First set up the matrix if you do not already have one set up.)
              as outlined above and press Home to go to the home screen.
         b)  Press 2ND, MATH, 4; then 3 for ref( or 4 for rref(.  The chosen operation will be pasted to the
              Home screen.
         c)  Enter the name of the matrix where you entered your data, e.g., m134, and close the parentheses.
         d)  Press ENTER and the new matrix will be displayed on the home screen.  Of course, to clear the screen,
             press F1, 8. 

5)  Solving a system of linear equations:
      Let's take the following set of simple equations:
         3x -3y = -2
         2x +y = 1
       Entering Data in the matrix:
            a)  Press [APPS] if you're not already at the APPS menu; then select the Data/Matrix icon and press
              [ENTER].  Press 2 to select Open.  (Note: If you do not have a 2x2 matrix stored in your calculator,
              follow "Entering a Matrix" above to set up your matrix.
            b)  On the dialog box that appears, press the right cursor arrow and press 2 to select Matrix; then move the
                 cursor down to Variable, press the right arrow, select the matrix you want and press ENTER, ENTER. 
                 The matrix will be displayed.
            c)  Enter each value of the matrix and press ENTER after each value.  Enter only the coefficients of the
              variables and the constants.  Do NOT enter variables, or plus signs, but do enter negative signs (not minus
              signs.). Enter the numbers 3, -3, -2, 2, 1, 1 and press ENTER after each number.
            d)  When your are finished entering data, press Home to go to the home screen.
                 Note:  If you want to display a matrix from the home screen you can Press 2ND, VVAR-LINK (the minus key),
                 F2, cursor to Var Type, press right cursor, 4, ENTER.  Then highlight the matrix you want and press
                 ENTER.  The matrix name will be displayed.  Press ENTER to display the matrix.
         Solving the system of equations using the rref operation:
            e)  Press 2ND, MATH, 4; then press 4 for rref(.  The chosen operation will be pasted to the
                 Home screen.
            f)  Enter the name of the matrix where you entered your data, e.g., m134, and close the parentheses.
            g)  Press ENTER and the new matrix will be displayed on the home screen.  Of course, to clear the screen,
                 press F1, 8. 

VIII.  DIFFERENTIAL EQUATIONS:
       1)  Find the general and particular solutions to first order and ODE:
           General solution of dy/dx = 2x.
             a)  From the Home screen, press F3, ALPHA, C to paste deSolve( to the screen.
             b)  Enter information so that you have deSolve(y' =2x,x,y).  The prime symbol is obtained by pressing
                 2ND, = or by pressing 2ND, CHAR ( the + key), 3, 7.
             c)  Press ENTER and the answer will be given.  Note that @1 would be C₁ in a calculus book.
                Now, find the particular solution when y = 6 and x=2.
             d)  Press 2ND, ans (the minus key).
             e)  Press 2ND, MATH, 8,8 to past "and" to the screen.
              f) Enter y(2) =6, so that you have "ans(1) and y(2)=6.
             g) Press ENTER and the answer will be displayed.
                 Note that if you wanted that particular solution form the beginning, you could have entered
                 deSolve(y'=2x and y(2)=6, x, y), and then pressed ENTER.
                 Note that 'and" is entered by using ALPHA, (-) for a blank space, and then
                 typing "and" with ALPHA and the letter keys.

        2)  Find the general and particular solutions of a second order differential equation:
               Solve y" +5y' +4y = 0
               a)  From the Home screen, press F3, ALPHA, C to paste deSolve( to the Home screen.
               b)  Enter the equation so that the entry looks like the following:  deSolve(y" +5y' +4y=0, x,y).
                    Notice that the prime symbol is entered by pressing 2ND; then the = key.  For the
                    double prime, you will need to press 2ND, = for each of the two primes.
               c)  Press ENTER and the solution will be displayed.
               d)  Solve for the particular solution as in the procedure above except that you will add "and
                    y'(2)= 3 or some such value as required by the problem. 

       3)  Draw the slope field for a differential equation:
            Let's take the very simple example y' = 2x .  Notice that to draw a slope field or a differential equation, you must change
            GRAPH on the MODE menu from FUNCTION to DIFF EQUATIONS and if you want to graph the slope fields, you must turn on SLPFLD. The
             first few steps below describe how to do that. 
             a)  Press MODE, press the right cursor arrow.  Press 6, ENTER to set up the Y= screen for differential equations.
             b)  Press ♦, F1 to display the y= screen.
             c)  Press ♦, | (the key above EE) to display GRAPH FORMATS dialog box.  The fields there
                  should be as follows:  Axes = ON, Labels = ON, Solution Method = RK, and Fields= SLPFLD .
             d)  Press ENTER to return to the Y= screen.
             e)  Enter  2t and press ENTER.  Do not put a initial value for yi1 at this time.
             f)   Press ♦, F3 and the slope field will be drawn.  Note that with some equations, considerably
                  time may be required for a solution.  "Busy" will appear on the right bottom of the screen while the solution
                  is being arrived at.

          4)  Draw the slope field and graph for a differential equation:
               a)  Do the same as in item 1 immediately above, but opposite yi1 enter a constant such as -1.   
               b)  Press ♦, F-3 and the slope field and the graph will be drawn.

          5)  Draw graph for a differential equation without slope field:
               Do the same as in item 1 immediately above except for the following:
                a)  Opposite yi1 enter a constant such as -1.   
                b)  During setup in step "c" in item 1 above, turn the slope fields off by pressing the right arrow until
                     FLDOFF appears.  If you have already done the equation with a slope field and want to eliminate
                     it, press ♦, | (key above EE), cursor to Fields, press the right cursor arrow, and press 3. 
                c)  Press ENTER to return to the Y= screen.
                d)  Press ♦, F3 to graph the solution to the differential equation.  Note that if you don't have a value
                     entered in Yi1, the graph will not be drawn.

            Important Note: When you're finished with differential equations, don't forget to change the mode back to
              function with the strokes MODE, right cursor arrow, 1, ENTER.

   IX.  SEQUENCES:
         1)  Find the first four terms of the sequence a_n =3n-2.
              a)  From the Home screen, press 2ND, MATH, 3, 1.  The term seq( will be pasted to the Home
                   screen.
              b)  Enter 3; ALPHA; n; -,2, ALPHA; n,  1,4, ). You now should have seq(3n-2, n, 1,4) on the
                   home screen. 
              c)  Press ENTER and {1  4  7  10} will be displayed.

         2)  Find the sum of the sequence above. 
              This type problem will usually be written using the summation symbol, Σ.
              a)  From the Home screen, press 2ND, MATH,3, 6 and sum( will be pasted to the Home
                   screen.
              b)  Press 2ND, MATH, 3, 1 and seq( will be pasted to the screen. You should now have sum(seq( on the Home
                   screen.
              c)  Enter 3; ALPHA; n; -,2,ALPHA; n, 1,4),)  You now should have sum(seq(3n-2, n, 1,4 ))
                   displayed on the home screen. 
              d)  Press ENTER and 22 will be displayed.

         3)  Find the cumulative sum of the above sequence.
              a)  From the Home screen, press 2ND, MATH,3, 7 and cumsum( will be pasted to the Home
                   screen.
              b)  Press 2ND, MATH, 3, 1, ) and you should have: cumsum(seq( on the home screen.
              c)  Enter 3; ALPHA; n; -, 2,ALPHA; n,1,4,),) and  you should have cumSum(seq(3n-2, n, 1,4))
                   on the Home Screen.   
              d)  Press ENTER and {1  5  12  22} will be displayed. Note that this method gives the sum after each
                   increment of the variable n.
         4)  Find the 5th term of the above sequence. 
              Although this is easily done by hand, some students like to check their results. So here's how to
              do it with your calculator.
              a)  Press 2ND, MATH, 3, 1, ENTER and seq( will be pasted to the Home screen.
              b)  Enter 3; ALPHA; n; -,2,ALPHA; n, 5,5,). You now should have seq(3n-2, n, 5,5) on the
                   home screen.   (Note that the same number is entered for the beginning and end.)
              c)  Press ENTER and {13} will be displayed.

         5)  Graphing a sequence: 
              a)  Press MODE, press the right cursor arrow and press 4 for sequence.  Press ENTER.
              b)  Press ♦, F1 and enter the sequence. 
              c)  Enter the sequence, for example n^2-n and press ENTER; then enter an initial value in ui1
                   if you want one and press ENTER.
              d)  Press ♦, F3 and the values will be plotted.  You may want to change the WINDOW to get
                   more values on the screen. 
    Don't forget to change the MODE back to Function when you're finished with the sequences.  To do
    that, press MODE, right arrow, 1.

 X.  Complex Numbers: 

        1.  Solving a Polynomial with Complex Coefficients:
              a) Press APPS, select the Polynomial icon, and press ENTER. 
              b) On the screen that appears, enter the degree, for example 2 and press ENTER.
              c) On the screen that appears, press 3 for New.
              d) Enter the complex coefficients and press F5.  The complex values for "x" will be displayed.

XI. Vectors:
       1) Dot-Product of Two Vectors:
             Let's do the vectors <-2,-4,4> and <5,4,1>.
              a) Press HOME; then press 2nd, MATH (the 5 key), 4 (Matrix), ALPHA, L (Vector Ops),  3(dotP). The expression
                  dotP( will be displayed at the entry bar of the HOME screen.
              b) Enter information so that the entry looks as follows: dotP([-2,-4,4],[5,4,1]). The square brackets are
                  the 2nd function of the comma and divide keys.
              c) Press ENTER and the answer -22 will be displayed.
        2) Cross-Product of Two Vectors:
             Method 1:
             Let's do the vectors <-2,-4,4> and <5,4,1>.
              a) Press HOME; then press 2nd, MATH, 4 (Matrix), ALPHA, L (Vector Ops), 2 (crossP). The expression
                  crossP( will be displayed at the entry bar of the HOME screen.
              b) Enter information so that the entry looks as follows: crossP([-2,-4,4],[5,4,1]). The square brackets are
                  the 2nd function of the comma and divide keys.
              c) Press ENTER and the answer [-20 22 12] will be displayed.
            Method II:
                For some reason, some prefer to use matrices. Although I prefer Method I, the matrix method can be
                cone as follows:
            a) Highlight the Data/Matrix icon and press ENTER..
            b) On the menu that appears, select "New" if you don't already have a matrix set up that is suitable for vectors
                or select "Open" if you already have set up.
            c) Opposite "Variable" enter a new variable name or select one already set up.
            d) Enter 3 for the number of rows and 3 for the number of columns; then press ENTER to go to
                the matrix table.
            e) In the first row enter I, j, and k. Press ALPHA before each entry; then enter the numbers for
                x, y, and z components of the two vectors in the second and third columns. Be sure that you
                enter then in the correct order. That is, for u x v, be sure to enter u first.
             f) Press HOME and then 2nd, MATH (the number 5 key). Move the cursor to "Matrix," press the
                right arrow key, and finally press 2 to enter det(.
            g) Enter the variable name of your matrix, say it's m1, so that you have det(m1), and press ENTER
                to display the answer, -2(10i-11j-6k), which is a factored form of method I
                Note: The calculator tends to factor the answer. If it has long decimals, you might go to MODE,
                F2 and set the exact/approx to EXACT. Try again to see if your answer looks better..