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,
© 2003 Frank Kizer
NOTE: Printing hints and
copying limits are at the end of this document.
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 10^{2} ,
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)=x^{2}-y^{2}
.
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 f_{1}
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 x^{2}
+ y^{2} =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 C_{1}
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..
RELEASE DATE:
7/7/14
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