Guide for Texas I nstruments TI

Guide for Texas I nstruments TI
Guide for Texas I nstruments TI -89
Graphics Calculator
This Guide is designed to offer step-by-step instruction for using your TI-89 graphing calculator with
the text Calculus Concepts: An Informal Approach to the Mathematics of Change. A technology icon
next to a particular example or discussion in the text directs you to a specific portion of this Guide.
You should also utilize the table of contents in this Guide to find specific topics on which you need
instruction.
Setup I nstructions
Generally, the TI-89 will show the home screen when it is turned on. If not press 2nd ESC (QUIT)
or press HOME . Before you begin, check the TI-89 setup and make sure the settings described below
are chosen. Whenever you use this Guide, we assume (unless instructed otherwise) that your
calculator settings are as shown in the figures below.
•
Press MODE and choose the settings shown in Figures 1, 2, and 3 for the basic setup. Pressing
F1 , F2 , and F3 open up the three pages of the MODE menu that are shown below.
TI-89 Basic Setup
Figure 1
Figure 2
Figure 3
• If you do not have the darkened choices shown in any figure, use the arrow keys to move the
blinking cursor over the setting you want to choose and press ENTER . When “→” appears next
to a word, press
►
to see the available choices and, if necessary, use
▼
to select the one you
need. Then press ENTER to choose that selection and return to the mode screen.
WARNI NG: To save all changes when exiting from all menu boxes (such as the MODE screen), you
must press ENTER to leave that box. If you exit the box by using ESC or by pressing any other key,
the changes you made will disappear.
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TI 89-1
TI 89-2
•
Chapter 1
Check the graph format by pressing APPS 2 [Y= Editor] (or use
▼
to highlight the second
choice and press ENTER ). Press F1 [Tools] 9 [Format] and choose the settings in Figure 4.
Press ENTER and return to the home screen with HOME .
•
Begin by pressing 2nd F1 [F6: Clean Up] 2 [NewProb] ENTER which clears single-character
variable names, turns off functions, clears graphs, and so forth. You can also clear the history
area (the main portion of the screen showing previous entries) with F1 8 [Clear Home] and clear
the entry line at the bottom of the screen with CLEAR . You may need to press CLEAR more
than once. If you clear the history area and entry line, your screen looks like the one in Figure 6.
TI-89 Graph Format
Begin a New Problem
Cleared Screen
Figure 4
Figure 5
Figure 6
Basic Operation
You should be familiar with the basic operation of your calculator. With your calculator in hand, go
through each of the following. We illustrate several methods – choose the ones you prefer.
1. CALCULATI NG You can type in lengthy expressions; just make sure that you use parentheses
when you are not sure of the calculator's order of operations. Always enclose in parentheses any
numerators and denominators of fractions and powers that consist of more than one term.
Evaluate
1
4* 15 +
895
. Enclose the denominator in parentheses so
7
that the addition is performed before the division into 1. It is not
necessary to use parentheses around the fraction 895/7.
•
The TI-89 prints the same way you should have the expression written on your paper. Always
check the left side of the screen and compare what you entered with what you have on paper.
•
To make a correction in the entry line, use
►
or
◄
to position the cursor to the right of
. The entry line is always in insert mode. You do
what you want to delete and then press
not need to clear the entry line before beginning to type a new expression.
4
Evaluate
( −3) − 5
. Use (−) for the negative symbol and − for
8 + 1.456
the subtraction sign. The numerator and denominator must be
enclosed in parentheses and −34 ≠ (−3)4.
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TI 89-3
TI-89 Guide
Now, evaluate e3*0.027 and e3*0.027. The TI-89 prints the left
parenthesis when you press
X (ex). If you do not include a
right parenthesis or insert an extra parenthesis, the TI-89 gives an
error message. Press ESC , fix the error, and press ENTER .
If you want to edit a previous expression to a new expression, have
. The cursor appears on
the cursor on the entry line and press
►
◄
, the
the right-hand side of the entry line expression. If you press
cursor appears on the left-hand side of the entry line expression. Try
editing e3*0.027 to type e3*0.027.
•
To recall an expression prior to the current one (before the history area is cleared), press 2nd
ENTER (ENTRY). You can also select any entry or answer from the history area and “auto-
paste” a duplicate of it on the entry line. This allows you to insert something previously typed or
calculated into a new expression. First, place the cursor in the entry line where you want to
, until what you are
insert the entry or answer. Second, move the cursor, repeatedly using
▲
inserting is highlighted. Third, press ENTER .
2. USI NG THE ANS MEMORY Next we explore recalling previous expressions and answers to
use in new calculations. We also see how to use the TI-89 answer memory. Instead of again typing an expression that was just evaluated, use the answer memory by pressing 2nd (−) (ANS).
Find
F
I
1
895 J
G
G 4 * 15 +
J
H
7 K
−1
using this shortcut.
Enter Ans with 2nd (−) (ANS).
NOTE: The last-calculated answer is referred to by the TI-89 on the entry line as ans(1) when you
enter each expression. The expression for the answer is substituted when the new expression
appears in the history area. You can also use auto-paste to enter the previous results.
3. ANSWER DI SPLAY We have MODE set to AUTO which means that an exact answer is given
whenever possible. If you have a decimal in an expression, a decimal value is returned for the
answer. If you want a decimal approximation for the expression that appears on the entry line,
ENTER . We also illustrate how to retrieve a previous entry that is in the history area.
press
Type 25 + 13 , press ENTER , and then
ENTER . Press
until
press
▲
7
1315
Use
is highlighted. Press ENTER .
◄
and insert a decimal point after
the 7. Press ENTER .
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TI 89-4
Chapter 1
The TI-89 can switch to scientific notation when the number is too
large for the display area in a table or graph. The TI-89’s symbol for
“times 10−6” is E−6. So, 1.4675E−6 means 1.4675*10−6, which is
scientific notation for 0.0000014675. Enter “E” by pressing EE .
The result 1.5E28 means 15,000,000,000,000,000,000,000,000,000 =
1.5(1028). In the history area, an arrowhead ( ) that points to the
right at the end of a number means that the number or expression
to
continues. Highlight the number or expression and press
scroll and see the rest of it. Press
▼
►
to return to the entry line.
4. STORI NG VALUES It often is beneficial to store numbers or expressions for later recall. To
store a number, type the number, press STO alpha , type the letter(s) corresponding to the storage location, and then press ENTER . Join several short commands with a colon between the
statements. Note that when you join statements with a colon by pressing 2nd 4 ( ), only the value
of the last statement is shown as a result.
:
Store 5 in a and 3 in b, and then calculate 4a – 2b.
To recall a stored value, press alpha , type the letter in which the
expression or value is stored, and then press ENTER .
•
Storage location names on the TI-89 can be from one to eight characters long and use letters
and numbers, but they must begin with a letter. You cannot name what you are storing with
the same name that the TI-89 already uses for a built-in variable (such as LOG or ans).
•
Whatever you store in a particular memory location stays there until it is replaced by something else either by you or by executing a program containing that name. It is advisable to
use single-letter names so that the values will be cleared when you begin with NewProb.
NOTE: The TI-89 allows you to enter upper and lower case letters, but it does not distinguish
between them. For instance, VOL, Vol, VOl, vol, voL, and so forth all name the same variable. To
type a lower-case letter, press alpha before typing a letter key (note that a lower-case a appears
under the entry line). To type an uppercase-letter, press alpha (note that an upper-case A
appears under the entry line) before typing the key corresponding to the letter.
If a variable is undefined (i.e., you have not stored a value in it), it
is treated as an algebraic symbol. If a variable is defined, its value
replaces the variable when you enter an expression containing that
variable. It is best, as we see in later chapters, to leave the names x
and y as undefined variables.
WARNI NG: You must be very careful when entering expressions containing variables. In the
next-to-last expression shown on the screen above, the TI-89 assumes “ax” refers to the name of a
single undefined variable. To tell the calculator that you want to multiply the variable a by the
variable x, you must use between the letters. Also, because 5 has been stored to a, the TI-89
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TI 89-5
TI-89 Guide
substitutes that value when the expression is evaluated. All user-defined variables are stored in a
folder1 called MAIN unless you create other folders to hold them. If you cannot remember the
name of a variable, press 2nd − (VAR-LINK) and the names of the variables are listed. After
using
▼
to move the cursor to the name of the variable, you can delete, copy, rename, etc. using
the F1 [Manage] key on the VAR-LINK screen. We illustrate by deleting the variable a:
Deleting a single variable: Suppose we want to delete the variable
a. Go to the VAR-LINK screen with 2nd − (VAR-LINK). Delete a
by first highlighting it and then pressing F1 [Manage] 1 [Delete]
ENTER and ESC . (Note that your MAIN folder may contain more
or different variables from what is shown to the right.)
The “with” operator, which prints as “|”, gives a way to temporarily store values into a variable in
order to evaluate expressions so that you do not have to delete the variable when you finish. Access this operator by pressing the key directly under = .
Evaluate 5.3r – 2.1h2 for for r = 6 and then for r = 2.9 and h = 7 by
typing in the instructions shown to the right. Access “and” with
CATALOG = (A)
ENTER . Press 2nd − (VAR-LINK) and
see that r and h are not defined variables.
▼
5. ERROR MESSAGES
When your input is incorrect, the TI-89 displays an error message.
If you have more than one command on a line without using a colon
(:) to separate them, an error message results when you press
ENTER . Press ESC and correct the error.
If you try to store something to a particular memory location that is
being used for a different type of object, an error results. (In the
screen shown to the right, the variable abc was defined as a data list,
so a number cannot be stored to it.) Consult either Trouble-Shooting the TI-89 in this Guide or your TI-89 Owner’s Guidebook.
•
The error message shown directly above usually occurs when you are executing a program
that is trying to store a value to a variable that you have previously defined as some other
type of object. Rename the other type of object and rerun the program. Remember, store
only numbers to single-letter names to avoid problems such as these.
A common mistake is using the negative symbol (−) instead of the
subtraction sign − or vice-versa. The TI-89 does not give an error
message, but a wrong answer results. The negative sign is shorter
and raised slightly more than the subtraction sign.
1 We work only in the MAIN folder. If you wish to create different folders, see pages 88-90 of the TI-89 Guidebook.
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Chapter 1
I ngredients of Change:
Functions and Linear Models
1.1 Models, Functions, and Graphs
There are many uses for a function that is entered in the calculator. One of the ways we often
use functions is to graph them in an appropriate viewing windows. Because you must enter a
function on one line (that is, you cannot type fractions and exponents the same way you write
them on paper even though the TI-89 displays them in this form after they are entered), it is very
important to have a good understanding of the calculator’s order of operations and to correctly
use parentheses whenever they are needed.
1.1.1 ENTERI NG AN EQUATI ON I N THE GRAPHI NG LI ST The graphing list contains
space for 99 equations, and the output variables are called by the names y1(x)=, y2(x)=, ..., and
y99(x)=. When you want to graph an equation, first enter in the graphing list. You must use x
as the input variable if you intend to draw the graph of the equation or use the TI-89 table.
We illustrate graphing with the equation in Example 4 of Section 1.1: v(t) = 3.622(1.093t).
Press either
F1 (Y=) or APPS 2 [Y= Editor] to access the
graphing list. If there are any previously entered equations that
you will no longer use, delete them from the graphing list. For
convenience, we use the first, or y1, location in the graphing list.
To delete an equation from
the Y= list, position the
cursor on the line with the
equation. Press CLEAR .
We intend to graph this equation, so the input variable must be
called x, not t. Enter the right-hand side of the equation,
3.622(1.093x), with 3 . 6 2 2 ( 1 . 0 9 3 ^ X ) ENTER .
Note that you must use the X key for x (under the HOME
key), not the times sign key,
.
▲
CAUTI ON: Press
and notice the names Plot 1, Plot 2, …, Plot 9 at the top of the Y= list.
These are the data plots that we will not use until Section 1.5. None of these plots should have a
check mark to the left of the name when you are graphing an equation and not graphing data
until the cursor is on the plot name and press
points. If any of these has a check mark, use
▲
F4 [b] to make the check mark go away. Otherwise, you may receive an error message.
DEFI NI NG A FUNCTI ON ON THE HOME SCREEN Functions can be entered in the
graphing list or defined on the home screen. We illustrate both of these methods, and you can
use the one you prefer. Press 2nd ESC (QUIT) or HOME to return to the home screen.
Type in v(t) = 3.622(1.093t) with F4 [Other] 1 [Define] alpha
0 (V) (
T ) = 3 . 6 2 2 ( 1 . 0 9 3 ^ T ) ENTER .
On the home screen, any letter can be used for the variable.
Note that if you prefer, you can enter the equation with x instead
of t as the input variable. (The TI-89 now knows this function
by two names: y1 and v.)
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TI-89 Guide
1.1.2 DRAWI NG A GRAPH As is the case with most applied problems in Calculus Concepts,
the problem description indicates the valid input interval. Consider Example 4 of Section 1.1:
The value of a piece of property between 1980 and 2000 is given by v(t) = 3.622(1.093t)
thousand dollars where t is the number of years since the end of 1980.
The input interval is 1980 (t = 0) to 2000 (t = 20). Before drawing the graph of v on this interval, have the function v entered in the Y= list using x as the input variable. (See Section 1.1.1 of
this Guide.) We now draw the graph of v for x between 0 and 20.
Press
F2 (WINDOW) to set the view for the graph. Enter 0
for xmin and 20 for xmax. (For 10 tick marks between 0 and 20,
enter 2 for xscl. If you want 20 tick marks, enter 1 for xscl, etc.;
xscl does not affect the shape of the graph. Ignore the other
numbers − we set their values in the next set of instructions.)
The numbers xmin and xmax are, respectively, the settings for the left and right edges of the
viewing screen, and ymin and ymax are, respectively, the settings for the lower and upper edges
of the viewing screen. xscl and yscl set the spacing between the tick marks on the x- and y-axes.
(Leave xres set to the default value of 2 for all applications in this Guide.)
To have the TI-89 determine the view for
the output, press F2 [Zoom] alpha =
(A) [ZoomFit] or use
▼
to highlight
ZoomFit and press ENTER .
Note that any vertical line drawn on this graph intersects it in only one point, so the graph does
represent a function.
Press
F2 (WINDOW) to see the view set by ZoomFit.
The view has 0 ≤ x ≤ 20 and 3.622 ≤ y ≤ 21.446... .
(Note that ZoomFit did not change the x-values that you set
manually.)
Let’s now explore how to graph with the function defined on the home screen as v(t).
Press HOME and type v(t) ENTER to
be sure you are using the correct funcF1 (Y=) and enter v(x)
tion. Press
in the y2 location. Press
[b] to turn off y1.
▲
and F4
Draw the graph of v = y2 using exactly the same procedure as that for drawing the graph of y1.
If you need to edit a function, press ENTER or F3 [Edit] to move the cursor to the entry line
from the graphing list. Press F4 [b] to deactivate y2 after you draw its graph.
WARNI NG: You must use x as the input variable when in the Y= list. If you use t or another
letter inside the parentheses following v, you will likely get an “undefined variable” message.
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TI 89-8
Chapter 1
1.1.3 MANUALLY CHANGI NG THE VI EW OF A GRAPH We just saw how to have the TI89 set the view for the output variable. Whenever you draw a graph, you can also manually set
or change the view for the output variable. If for some reason you do not have an acceptable
view of a graph or if you do not see a graph, change the view for the output variable with one of
the zoom options or manually set the WINDOW until you see a good graph. (We will later
discuss other zoom options.) We continue using the function v in Example 4 of Section 1.1, but
assume here that you have not yet drawn the graph of v.
Press
F2 (WINDOW), enter 0 for xmin, 20 for xmax, and
(assuming we do not know what to use for the vertical view),
enter some arbitrary values for ymin and ymax. (This graph was
F3 (GRAPH).
drawn with ymin = −5 and ymax = 7). Press
NOTE: If you see nothing on the screen, press F1 (Y=) and highlight y1.
check mark next to y1, press F4 [b] and F3 (GRAPH).
If there is no
Evaluating Outputs on the Graphics Screen: Press F3
[Trace].* Recall we are given in this application that the input
variable is between 0 and 20. If you now type the number that
you want to substitute into the function whose graph is drawn,
say 0, you see the screen to the right. A 1 appears at the top
right of the screen because the equation of the function whose
graph you are drawing is in y1.
Press ENTER and the input value is substituted in the function.
The input and output values are shown at the bottom of the
screen. (This method works even if you do not see any of the
graph on the screen.)
Substitute the right endpoint of the input interval into the
function by pressing 20 ENTER . We see that two points on
this function are approximately (0, 3.622) and (20, 21.446).
Press
F2 (WINDOW), enter 3.5 for ymin and 22 for ymax,
F3 (GRAPH). If the graph you obtain is not a
and press
good view of the function, repeat the above process using xvalues in between the two endpoints to see if the output range
should be extended in either direction. (Note that the choice of
the values 3.5 and 22 was arbitrary. Any values close to the
outputs in the points you find are also acceptable.)
*Instead of using TRACE, you could use the TI-89 TABLE or evaluate the function at 0 and 20 on the
home screen to find the range of values for the output value. We later discuss using these features.
1.1.4 TRACI NG TO ESTI MATE OUTPUTS You can display the coordinates of certain points on
the graph by tracing. Unlike the substitution feature of TRACE that was just discussed, the xvalues that you see when tracing the graph depend on the horizontal view that you choose. The
output values that are displayed at the bottom of the screen are calculated by substituting the xCopyright © Houghton Mifflin Company. All rights reserved.
TI 89-9
TI-89 Guide
values into the equation that is being graphed. We again assume that you have the function v(x)
= 3.622(1.093x) entered in the y1 location of the Y= list.
With the graph on the screen, press F3 [Trace], press and hold
►
◄
to move the trace cursor to the right, and press and hold
to move the trace cursor to the left. Again, note that the number
corresponding to the location of the equation (in the Y= list) that
you are tracing appears at the top right of the graphics screen.
Trace past one edge of the screen and notice that even though
you cannot see the trace cursor, the x- and y-values of points on
the line are still displayed at the bottom of the screen. Also note
that the graph scrolls to the left or right as you move the cursor
past the edge of the current viewing screen.
►
◄
Use either
or
to move the cursor near x = 15. We
estimate that y is approximately 13.7 when x is about 15.
It is important to realize that trace outputs should never be given
as answers to a problem unless the displayed x-value is
identically the same as the value of the input variable.
1.1.5 EVALUATI NG OUTPUTS ON THE HOME SCREEN The input values used in the evaluation process are actual function values, not estimated values such as those generally obtained
by tracing near a certain value. Actual values are also obtained when you evaluate outputs from
the graphing screen using the process that was discussed in Section 1.1.3 of this Guide.
We again consider the function v(t) = 3.622(1.093t) that is in Example 4 of Section 1.1.
Using x as the input variable, have 3.622(1.093)^ x entered in y1.
Return to the home screen by pressing HOME . Substitute 15
into the function with Y 1 ( 15 ) and find the function value
by pressing ENTER .
NOTE: We choose y1 as the function location most of the time, but you can use any of the
available locations. If you do, replace y1 in the instructions with the function you choose.
It is now a simple matter to evaluate the function at other inputs.
For instance, substitute x = 20 into the equation by editing the
entry line using
, changing 15 to 20 by pressing
►
and typing 20 ) , and then pressing ENTER . Evaluate y1 at 0
using the same procedure.
If you defined the function v with input t (see the bottom of page
C-6), it can easily be used to find the function outputs.
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TI 89-10
Chapter 1
1.1.6 EVALUATI NG OUTPUTS USI NG THE TABLE Function outputs can be determined by
evaluating on the graphics screen, as discussed in Section 1.1.3, or by evaluating on the home
screen as discussed in Section 1.1.5 of this Guide. You can also evaluate functions using the
TI-89 TABLE. When you use the table, you can either enter specific input values and find the
outputs or generate a list of input and output values in which the inputs begin with tblStart and
differ by ∆tbl.
Let’s use the TABLE to evaluate the function v(t) = 3.622(1.093t) at the input t = 15. Even
F1
though you can use any of the function locations, we again choose to use y1. Press
(Y=), clear the function locations, and enter 3.622(1 .093)^ x in location y1.
Choose TblSet with
F4 (TblSet). To generate a list of values
beginning with 13 such that the table values differ by 1, enter 13
in the tblStart location, 1 in the ∆tbl location, have Graph Table
set to OFF, and choose AUTO in the Indpnt: location. (Remember
that you “choose” a certain setting by having it high-lighted and
pressing ENTER .) Exit the setup with ENTER .
Press
F5 (TABLE), and observe the list of input and
output values. Notice that you can scroll through the table with
,
,
, and/or
. The table values may be rounded in
the table display. You can see more of the output by highlighting a particular value and viewing the bottom of the screen.
▼ ▲ ◄
►
NOTE: If you wish, while in the table, you can change the cell width using the F1 [Tools] menu
and option 9 [Format]. The cell widths can vary between 3 and 12, which result in as few as 2
columns and as many as 7 columns. All functions that are checked in the Y= list display in the
table, so we suggest leaving the width at its default setting.
If you want to evaluate a function at inputs that are not evenly spaced and/or you only need a
few outputs, you should use the ASK feature of the table instead of using AUTO. Note that
when using ASK, the settings for tblStart and ∆tbl do not matter and are dimmed on screen.
Choose TblSet with F2 [Setup] from
F4 (TblSet).
within the table or
Choose ASK in the Indpnt: location. To
enter the x-values 15, 0, and 20, type
, type
each value, press ENTER
▼
the next value, and so forth.
NOTE: Unwanted entries or values in the table can be cleared by highlighting the value in the x
column and pressing
or by using F1 [Tools] 8 [Clear Table] to delete all previous entries.
1.1.7 FI NDI NG I NPUT VALUES USI NG THE SOLVER Your calculator solves for the input
values of all the equations we use in this course. The expression can, but does not have to, use x
as the input variable. The TI-89 offers several methods of solving for input variables. We first
illustrate using the solve instruction. (Solving using graphical methods will be discussed after
using the solve instruction and the TI-89 numeric solver are explored.) You can refer to an equa-
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TI 89-11
TI-89 Guide
tion that you have already entered in the Y= list or you can enter the equation directly in the
solver. After we finish exploring all the methods, you should choose the one you prefer.
Return to the home screen with HOME . (If you want to clear
the history area, remember that F1 8 [Clear home] does it.) Enter
solve( on the entry line by typing “solve(” using the keyboard, or
by pressing F2 1 [solve(].
Let’s now use the solver to answer the question in part e of Example 4 in Section 1.1: “When
did the land value reach $20,000?” Because the land value is given by v(t) = 3.622(1.093t)
thousand dollars where t is the number of years after the end of 1980, we are asked to solve the
equation 3.622(1.093t) = 20. That is, we are asked to find the input value t that makes this
equation a true statement.
If you already have y1 = 3.622(1.093^ x) in the graphing list, you
can refer to the function as y1(x) in the entry line. If not, you can
enter 3.622(1.093^ x) instead of y1. You must also tell the TI-89
the name of the input variable. Type either “y1(x)=20, x)” or
“3.622(1.093^ x)=20 , x)” behind “solve(“ and press ENTER .
WARNI NG: If you refer to an equation by its location in the Y= list, you must use the complete name of the equation (i.e., y1(x) instead of y1) when you enter it in the solve instruction.
•
If a solution continues beyond the edge of the calculator screen, you see “ ” to the right of
to highlight the solution and then press and hold
the value. Be certain that you press
►
▲
to scroll to the end of the answer.
USI NG THE NUMERI C SOLVER The solve instruction is very easy to use and offers a
fairly quick method of finding an answer. However, if you have previously used other model TI
calculators, you may be more familiar with the numeric equation solver.
Access the numeric solver with APPS 9 [Numeric Solver]. If
there are no equations in the solver, you will see the screen
displayed on the right. (If there is an equation entered after eqn:,
press CLEAR to delete it.)
If you already have y1 = 3.622(1.093^ x) in the graphing list, you
can refer to the function as y1(x) in the numeric solver. If not,
enter 3.622(1.093^ x) = 20 instead of y1(x) = 20 in the eqn: location.
Press ENTER . Enter a guess* for x − say 19.
▲
If you need to edit the equation, press
until the previous
screen reappears. Edit the equation and then return here. Next,
be certain that the cursor is on the line corresponding to the input
variable for which you are solving (in this example, x). Solve for
the input by pressing F2 [Solve].
*More information on entering a guess appears in the next section of this Guide.
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TI 89-12
Chapter 1
CAUTI ON: You should not change anything in the “bound” location of the SOLVER. The
values in that location are the ones between which the TI-89 searches2 for a solution. If you
should accidentally change anything in this location, exit the solver, and begin the entire process
again. (The bound is automatically reset when you exit the numeric solver.)
The answer to the original question is that the land value was $20,000 about 19.2 years after
1980 – i.e., in the year 2000.
•
Notice the black dot that appears next to x and next to the last line on the above screen.
This is the TI-89’s way of telling you that a solution has been found. When the bottom line
on the screen that states left − rt ≈ 0, the value found for x is an exact solution.
1.1.8 HOW TO DETERMI NE A GUESS TO USE I N THE NUMERI C SOLVER This section of the Guide applies only if you are using the numeric solver described directly above. If
you have decided to use only the solve instruction in the entry line, skip this discussion.
What you use in the solver as a guess tells the TI-89 where to start looking for the answer.
How close your guess is to the actual answer is not very important unless there is more than one
solution to the equation. If the equation has more than one answer, the numeric solver will
return the solution that is closest to the guess you supply. In such cases, you need to know
how many answers you should search for and their approximate locations.
Three of the methods that you can use to estimate the value of a guess for an answer from
the numeric solver follow. We illustrate these methods using the land value function from
Example 4 of Section 1.1 and the equation v(t) = 3.622(1.093t) = 20.
1. Enter the function in some location of the graphing list – say
y1 = 3.622(1.093^ x) and draw a graph of the function. Press
F3 [Trace] and hold down either
or
until you have
an estimate of where the output is 20. Use this x-value, 19 or
19.4 or 19.36, as your guess in the SOLVER.
►
◄
2. Enter the left- and right-hand sides of the equation in two
different locations of the Y= list – say y1 = 3.622(1.093^ x) and
y2 = 20. With the graph on the screen, press F3 [Trace] and
►
◄
or
until you get an estimate of the
hold down either
x-value where the curve crosses the horizontal line
representing 20.
▲
▼
3. Use the AUTO setting in the Table, and with
or
scroll
through the table until a value near the estimated out-put is
found. Use this x-value or a number near it as your guess in
the numeric solver. (Refer to Section 1.1.6 of this Guide to
review the instructions for using the table)
•
You may find it helpful to employ a combination of the above methods by using the split
screen feature of the TI-89. Using a split screen, you can view the graph and table or the
numeric solver and a graph, and so forth, at the same time. See the TI-89 Guidebook for
details.
2 It is possible to change the bound if the calculator has trouble finding a solution to a particular equation. This, however,
should happen rarely. Refer to the TI-89 Guidebook for details.
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TI 89-13
TI-89 Guide
1.1.9 GRAPHI CALLY FI NDI NG I NTERCEPTS Finding the input value at which the graph
of a function crosses the vertical and/or horizontal axis can be found graphically, by using the
solve instruction on the entry line, or by using the numeric solver. Remember the process by
which we find intercepts:
•
To find the y-intercept of a function y = f(x), set x = 0 and solve the resulting equation.
•
To find the x-intercept of a function y = f(x), set y = 0 and solve the resulting equation.
An intercept is the where the graph crosses or touches an axis. Also remember that the xintercept of the function y = f(x) has the same value as the root or solution of the equation
f(x) = 0. Thus, finding the x-intercept of the graph of f(x) – c = 0 is the same as solving the
equation f(x) = c.
We illustrate this method with a problem similar to the one in Activity 40 in Section 1.1 of
Calculus Concepts. You should practice by solving the equation 3.622(1.093x) = 20 using this
graphical method and by solving the equation that follows using one of the solvers.
Suppose we are asked to find the input value of f(x) = 3x – 0.8x2 + 4 that corresponds to the
output f(x) = 2.3. That is, we are asked to find x such that 3x – 0.8x2 + 4 = 2.3. Because this
function is not given in a context, we have no indication of an interval of input values to use
when drawing the graph. We will use the zoom features to set an initial view and then manually
set the WINDOW until we see a graph that shows the important points of the func-tion (in this
case, the intercept or intercepts.) You can solve this equation graphically using either the xintercept method or the intersection method. We present both, and you should use the one you
prefer.
X-I NTERCEPT METHOD for solving the equation f(x) – c = 0:
Press
F1 (Y=) and clear all locations with CLEAR . Enter
the function 3x – 0.8x2 + 4 – 2.3 in y1. (Type x2 with X ^ 2.
Remember to use − , not (−) , for the subtraction signs.)
Draw the graph of y1 with F2 [Zoom] 4 [ZoomDec] or F2
[Zoom] 6 [ZoomStd]. If you use the former, press
F2
(WINDOW) and reset ymax to 5.5 to get a better view of the top
of the graph. (If you reset the window, press
to redraw the graph.)
F3 (GRAPH)
To graphically find an x-intercept, i.e., a value of x at which the
graph crosses the horizontal axis, press F5 [Math] 2 [Zero].
Press and hold
◄
until you are near, but to the left of, the
leftmost x-intercept. Press ENTER to mark the location of the
lower bound for the x-intercept.
Notice the small arrowhead () that appears above the location to
until you are to
mark the left bound. Now press and hold
►
the right of this x-intercept. Press ENTER to mark the location
of the upper bound for the x-intercept and to find it.
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TI 89-14
Chapter 1
The value of the leftmost x-intercept has
the x-coordinate x = −0.5.
Repeat the above procedure to find the
rightmost x-intercept. Confirm that it is
where x = 4.25.
NOTE: If this process does not return the correct value for the intercept that you are trying to
find, you have probably not included the place where the graph crosses the axis between the two
bounds (i.e., between the and marks on the graph.)
I NTERSECTI ON METHOD for solving the equation f(x) = c:
Press
F1 (Y=) and clear all locations with CLEAR . Enter
one side of the equation, 3x – 0.8x2 + 4, in y1 and the other side
of the equation, 2.3, in y2.
Draw the graphs with F2 [Zoom] 4 [ZoomDec] or F2 [Zoom] 6
[ZoomStd]. If you use the former, press
F2 (WINDOW) and
reset ymax to 8 to get a better view of the top of the graph.
F3 (GRAPH) redraws the graph.)
(If you reset the window,
To locate where y1 = y2, press F5
[Math] 5 [Intersection]. Press ENTER
to mark the first curve. The cursor
jumps to the other function – here, the
line. Next, press ENTER to mark the
second curve.
Note that the number corresponding to the location of each
function appears on the top left of the screen as the cursor moves
to move the cursor to
from the curve to the line. Next, press
the left of the intersection point you want to find – in this case,
the leftmost point. Press ENTER .
◄
►
Use
to move the cursor to the right of the intersection point
you are finding to supply the upper bound for that point. The
point of intersection must lie between the two markers for the
lower and upper bounds. Press ENTER .
The value of the leftmost x-intercept has
the x-coordinate x = −0.5.
Repeat the above procedure to find the
rightmost x-intercept. Confirm that it is
where x = 4.25.
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TI 89-15
TI-89 Guide
1.1.10 SUMMARY OF ESTI MATI NG AND SOLVI NG METHODS Use the method you prefer.
When you are asked to estimate or approximate an output or an input value, you can:
• Trace a graph
(Section 1.1.4)
• Use close values obtained from the TI-89 table
(Sections 1.1.3, 1.1.5, 1.1.7)
When you are asked to find or determine an output or an input value, you should:
• Evaluate an output on the graphics screen
(Section 1.1.3)
• Evaluate an output on the home screen
(Section 1.1.5)
• Evaluate an output value using the table
(Section 1.1.6)
• Find an input using the solve instruction or the numeric solver
(Sections 1.1.7, 1.1.8)
• Find an input value from the graphic screen (using the x-intercept
method or the intersection method)
(Section 1.1.9)
1.3 Constructed Functions
Your calculator can find output values of and graph combinations of functions in the same way
that you do these things for a single function. The only additional information you need is how
to enter constructed functions in the graphing list or on the home screen.
1.3.1 FI NDI NG THE SUM, DI FFERENCE, PRODUCT, QUOTI ENT OR COMPOSI TE
FUNCTI ON Suppose that a function f has been entered in y1 in the Y= list or defined as y1(x)
and that a function g has been entered in y2 in the Y= list or defined as y2(x). Your TI-89 will
evaluate, graph, and actually find the symbolic form of these constructed functions:
Enter y1(x) + y2(x) to obtain the sum function (f + g)(x) = f(x) + g(x).
Enter y1(x) – y2(x) to obtain the difference function (f – g)(x) = f(x) – g(x).
Enter y1(x)*y2(x) to obtain the product function (f ⋅ g)(x) = f(x) ⋅ g(x).
Enter y1(x)/ y2(x) to obtain the quotient function (f ÷ g)(x) =
f (x)
g( x )
.
Enter y1(y2(x)) to obtain the composite function (f o g)(x) = f(g(x)).
1.3.2 FI NDI NG A DI FFERENCE FUNCTI ON We illustrate this technique with the functions
that are given on page 31 of Section 1.3 of Calculus Concepts: Sales = S(t) = 3.570(1.105t)
million dollars and costs = C(t) = −39.2t2 + 540.1t + 1061.0 thousand dollars t years after 1996.
We use the functions on the home screen, but you can also use the Y= list locations. However, if
you are in the Y= list, the symbolic form of the constructed function is not displayed.
NOTE: Before you start a problem in which you want a symbolic result (i.e., a formula rather
than a number), clear all individual letter variable names with 2nd F1 [F6] 2 [NewProb]
ENTER or 2nd
F1 [F6] 1 [Clear a-z] ENTER . (See the warning message on the next page.)
On the home screen, define S with F4 [Other] 1 [Define] alpha
3 (S) (
T
)
= 3 . 570 ( 1 . 105 ^
T
)
ENTER and
define C with F4 1 [Define] alpha ) (C) ( T ) = (−) 39
. 2 T
•
^ 2 + 540 . 1 T
+ 1061 ENTER .
As we previously mentioned, you can use any input variable on the home screen. We
choose to use t because it is given as the input variable in the text illustration. However, if
you prefer x, replace every t by x in the instructions.
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TI 89-16
Chapter 1
Type on the entry line the difference
function, which is the profit function
S(t) – 0.001C(t). Press ENTER . Press
▲
►
and press and hold
to scroll to
the right to see the entire expression.
WARNI NG: If you do not get a symbolic result, it is because the variable you are using as the
input variable has a number stored in it. That is, the variable is a defined rather than an
undefined variable. The variable you use for the input variable must be an undefined variable
in order to obtain a symbolic result. Refer to the information at the top of page C-5 of this
Guide for more information.
To find the profit in 1998, evaluate the profit function at t = 2.
You can edit the entry line and replace t by 2 or you can enter
s(x) − 0.001c(x) in the Y= list and use the table. We find that the
profit in 1998 was P(2) ≈ 2.375 million dollars.
If you need to use the profit function for other calculations, you
may find it easier to define a new function, p(t) = s(t) − 0.001c(t)
and then find p(2). We illustrate the copy and paste feature of the
until s(t) − 0.001c(t) in the history
TI-89 to do this task. Press
area is darkened.
▲
▼
Press F1 [Tools] 5 [Copy]. Use
to move the cursor to the
entry line and press F1 [Tools] 6 [Paste]. Use
◄
to move the
cursor to the far left position in the entry line. Type F4 [Other] 1
STO (P) (
[Define] alpha
T
)
= and press ENTER .
You can now find p(2) and/or enter p(x) in the Y= list so that you
can use the table to find other values, graph the function, and so
forth.
1.3.3 FI NDI NG A PRODUCT FUNCTI ON We illustrate this technique with the functions that
are given on page 32 of Section 1.3 of Calculus Concepts: Milk price = S(x) = 0.007x + 1.492
dollars per gallon on the xth day of last month and milk sales = G(x) = 31 – 6.332(0.921x) gallons of milk sold on the xth day of last month.
You can, if you wish, define the functions s and g. (Note that the new definition for s will
replace the one defined in Section 1.3.3.) However, because we are only finding the product
function and only one output of it, we choose to name only the product function.
Find and define the product function with F4 [Other] 1 [Define]
T
(
X
. 332 (
)
=
(
. 921 ^
. 007 X
X
)
+ 1 . 492 )
( 31 − 6
) ENTER .
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TI 89-17
TI-89 Guide
NOTE: You do not have to, but you can, use the times sign between S and G to indicate the
product function. We enclosed each of S and G in parentheses to indicate a product when they
are written next to each other, so the times sign is not necessary.
To find milk sales on the 5th day of last month, evaluate t(x) at
x = 5. On the entry line, type in T ( 5 ) and press ENTER .
We find that milk sales were T(5) ≈ $40.93.
If you want to find the product function, copy the “Define t(x)”
and use
statement and paste it into the entry line, press
◄
until the cursor is just to the right of the = sign, and press
►
to
delete “Define t(x) =”. Press ENTER .
1.3.4 CHECKI NG YOUR ANSWER FOR A COMPOSI TE FUNCTI ON We illustrate finding a composite function with those functions given on page 33 of Section 1.3 of Calculus
Concepts: altitude = F(t) = −222.22t3 + 1755.95t2 + 1680.56t + 4416.67 feet above sea level
where t is the time into flight in minutes and air temperature = A(F) = 277.897(0.99984F) – 66
degrees Fahrenheit where F is the number of feet above sea level. While the same technique that
we used to find the product function also can be used to find the composite function, for variety
we use a slightly different procedure.
Press
F1 (Y=) and clear any previously entered functions.
Enter F in y1 by pressing (−) 222 . 22 X ^ 3 + 1755 . 95
X
y2
^ 2 + 1680 . 56 X
+ 4416 . 67 ENTER and enter A in
by pressing 277 . 897 ( . 99984 ^ X ) − 66 ENTER .
Enter the composite function (AoF)(x) =
A(F(x)) = y2(y1(x)) in y3.
Press HOME . Enter y3(x) on the entry
line to display the composite function.
WARNI NG: The TI-89 requires that you give the output symbol, not the function symbol,
when you refer to the formula that computes the output of a function. If you do not type x in
parentheses following y3, an argument error results.
•
Note that the symbolic form of the composite function contains e raised to a power, not
0.99984 raised to a power (as in y2). Converting between these two exponential forms is
discussed in Section 2.2 of Calculus Concepts.
1.3.5 TURNI NG FUNCTI ONS I N THE GRAPHI NG LI ST OFF AND ON There are times
when you would like to keep the equations of certain functions in the Y= list but you do not
want them to graph or to be shown in the table. Any function that is turned on will graph and
will show in the TI-89 table. A function is turned on when a check mark appears next to it in
the Y= list and is turned off when there is no check mark next to it in the Y= list.
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TI 89-18
Chapter 1
On the screen shown to the right, y1 and y3 are turned on (i.e.,
activated) and y2 is turned off (i.e., deactivated).
With the function highlighted, use F4 [b] to toggle the check
(and graph) on and off. F5 [All] turns all functions on or off.
1.3.6 GRAPHI NG AN I NVERSE FUNCTI ON The TI-89 can draw the graph of the inverse of a
function. Using the calculator graph, you can check your algebraic answer. We illustrate this
idea using the function in Example 4 of Section 1.3 of Calculus Concepts:
The fares for a cab company are determined by the function F(d) = 1.8d + 2.5 dollars where
d is the distance traveled in miles.
Press
F1 (Y=), clear any previously entered functions, and
enter the function F in y1. Remember to use x as the input
variable. (Refer to Sections 1.1.2 and 1.1.3 of this Guide for
hints on how to set the window to graph this function.)
This problem does not state an interval of values for which the input variable is defined, so we
begin by guessing one that makes sense in context. Because d is the distance traveled, we
know that d ≥ 0. We choose a maximum value of 50. (Your guess is as good as the one that
is given below.)
Press
F2 (WINDOW) and enter
appropriate values for the input and
output. Draw the graph of the function
F3
for the view you choose with
(GRAPH).
Press
F1 (Y=) and enter in y2 your answer to part a of
Example 4 – the inverse function for F. (Your answer to part a
may or may not be the same as what is shown to the right.)
NOTE: Remember that the TI-89 requires that you use x as the input variable and that you use
parentheses around any numerator and/or denominator that consists of more than one symbol in
a fraction.
Press
F3 (GRAPH). If you do not
have a good view of both functions,
reset the window. Draw the TI-89’s
inverse function with 2nd F1 [F6:
Draw] 3 [DrawInv] and type in y1(x).
Press ENTER . If you do see a third graph, your inverse
function formula is not correct.
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TI 89-19
TI-89 Guide
CAUTI ON: If you do not see a third graph, as is the case in the graph in the last picture, you
may have computed the formula correctly. However, you might have an incorrect inverse
function formula whose graph does not show in the current viewing window. We suggest that if
you do not see a third graph that you turn off the graphing location where you have entered your
inverse function formula − here y2. (Remember that you turn off a function by having the check
mark to the left of the name not showing.) Then, on the home screen, press ENTER to redraw
the inverse function graph and visually check that it is the same graph as the one for y2. Realize
that these methods provide only a visual check on your answer and are not exact.
The answers to parts b and c of Example 4 are found using the
distance (inverse) function in y2 and the fare (original) function
in y1, respectively.
Don’t forget to include units of measure with the answers.
1.3.7 COMPOSI TI ON PROPERTY OF I NVERSE FUNCTI ONS This concept, involved in
part d of Example 4, provides another check on your answer for the inverse function.
Form the compositions of the inverse function and the original
function according to the Composition Property of Inverse
Functions and enter them in y3 and y4. Turn off the functions in
y1 and y2. (Section 1.3.3 of this Guide.)
Have the TABLE set to ASK mode (See Section 1.1.6) and press
F5 (TABLE). Enter several values for x to see if both
composite functions return that value of x. If so, your answer
for the inverse function is very likely correct.
1.3.8 GRAPHI NG A PI ECEWI SE CONTI NUOUS FUNCTI ON Piecewise continuous functions are used throughout the text. You will need to use your calculator to graph and evaluate
outputs of piecewise continuous functions. Several methods3 can be used to draw the graph of a
piecewise function. One of these is presented below using the function that appears in Example
6 of Section 1.3 in Calculus Concepts:
The population of West Virginia from 1985 through 1993 can be modeled by
R−23.514t + 3903.667 thousand people when 85 ≤ t < 90
P( t ) = S
T7.7t + 1098.7 thousand people when 90 ≤ t ≤ 93
where t is the number of years since 1900.
The TI-89 syntax for drawing a piecewise function consisting of two pieces is
when(condition, true expression, false expression)
The TI-89 CATALOG contains all the TI-89 commands. If you press the first letter of the word
you are trying to find in the catalog, it automatically scrolls to the first word that begins with that
to find what you are looking for and enter it in what you are typing.
letter. Then use
▼
3 Instructions for drawing a piecewise function consisting of three pieces and more than three pieces are given on pages
194-195 of the TI-89 Guidebook.
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TI 89-20
Chapter 1
Be on the home screen and press CATALOG . Because when
begins with w, press . (W). ENTER copies the instruction to
the entry line. Also notice that the general syntax for each
instruction is printed at the bottom of the screen in the catalog.
We intend to graph the population function P, so we need to enter it in the Y= list with x as the
input variable. Because the break point is at x = 90, the piecewise function syntax becomes
when(x < 90, −23.514x + 3903.667, 7.7x + 1098.7)
Clear any functions that are in the Y=
list. Type the above when statement in
the y1 location. The < symbol is
accessed with 2nd 0 (<).
•
Note that x < 90 prints at the end of the first piece of the function in the y1 position. The
word else that appears with the second piece means that this part of the function should
be used everywhere else; that is, when x ≥ 90.
Your calculator draws graphs by connecting function outputs wherever the function is defined.
However, this piecewise function breaks at x = 90. The TI-89 will connect the two pieces of P
unless you tell it not to do. Whenever you draw graphs of piecewise functions, you should set
your calculator to Dot mode as described below so that it will not connect the different pieces of
the function. (When a particular style is chosen, a check mark appears by it in the style list.)
Highlight y1 and press 2nd F1 [F6: Style4] 2 [Dot].
Next, set the window. The function P is defined only when the
input is between 85 and 93. So, on the home screen we evaluate
P(85), P(93), and P(90) to help when setting the vertical view.
NOTE: Instead of finding the outputs as shown above, you can set the window with ZoomFit as
described in Section 1.1.2 of this Guide. If you do this, reset ymin to a smaller value so that
you can better view the break point.
Press
F2 (WINDOW), set xmin =
85, xmax = 93, ymin ≈ 1780, and ymax ≈
1905. Press
F3 (GRAPH).
Take a closer look at the break point
with the window given below.
Set xmin = 89, xmax = 91, ymin = 1780,
and ymax = 1810. Use the entry line on
the home screen to evaluate the other
outputs needed in this example.
4 The different graph styles you can draw from this location are described in more detail on page 100 in your TI-89
Guidebook.
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TI 89-21
TI-89 Guide
1.4 Limits: Describing Function Behavior
The TI-89 table is an essential tool when you are estimating limits numerically. Even though
rounded values are shown in the table due to space limitations, the TI-89 displays at the bot-tom
of the screen many more decimal places for a particular output when you highlight that output.
Your calculator also finds limits algebraically, but you should use the numerical and graphical
methods to gain insight as to the meaning of the algebraic answers.
1.4.1 ESTI MATI NG LI MI TS NUMERI CALLY Whenever you use the TI-89 to estimate limits
numerically, you will find it easiest to set the TABLE to ASK mode. We illustrate using the
function u that appears in Example 2 of Section 1.4 in Calculus Concepts:
Press
F1 (Y=) and use CLEAR to delete all previously-
entered functions. Enter u(x) =
3x 2 + 3 x
9 x 2 + 11x + 2
. Be certain to
enclose the numerator and denominator of the fraction in
parentheses.
Press
F4 (TblSet). If ASK is not selected, choose ASK by
moving the cursor to the in the Independent: location, press
►,
choose ASK, and press ENTER . Press ENTER again to exit
the screen.
Press
F5 (TABLE). Delete any values that are there with
or with F1 [Tools] 8 [Clear Table]. To numerically estimate
lim u(x), enter values to the left of, and becoming closer and
x→−1−
closer to, −1.
NOTE: The values you enter do not have to be those shown in the text or these shown in this
table. The cell width in the table has been increased in order to show you the values entered. If
you want to do this, use F1 [Tools] 9 [Format] and choose a larger cell width.
CAUTI ON: Your instructor will very likely have you write the table you construct on paper. Even if
you increase the cell width, the TI-89 eventually displays rounded values (such as the one that is
highlighted in the table above) because of space limitations. When this happens, be certain to highlight the value and look on the bottom of the screen to see what the value actually is. The inputs that
were entered in the table above are −1.5, −1.05, −1.005, −1.0005, −1.00005, and −1.000005. The outputs also have been rounded off.
ROUNDI NG OFF: Recall that rounded off means that if one digit past the digit of interest if less
than 5, other digits past the digit of interest are dropped. If one digit past the one of interest is 5 or
over, the digit of interest is increased by 1 and the remaining digits are dropped. (For instance, the
highlighted number in the last screen shown above, −1.000005, has been rounded off by the TI-89 to 5
decimal places. Because the 5 in the sixth decimal place location is in the “5 or over” category, the 0
before this 5 is increased by 1 with the digits past that point dropped to give −1.00001.)
RULE OF THUMB FOR DETERMI NI NG LI MI TS FROM TABLES: Suppose that we want
lim u(x) accurate to 3 decimal places. Watch the table until you see that the output is the same
−
x→−1
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TI 89-22
Chapter 1
value to one more decimal place (here, to 4 decimal places) for three consecutive outputs. Then, round
that common value off to the requested 3 places for the desired limit. Your instructor may establish a
different rule from this one, so be sure to ask.
Using this Rule of Thumb and the results that are shown on the last calculator screen, we estimate that
lim u(x) = 0.429. We now need to estimate the limit from the right of −1.
−
x→−1
Delete the values currently in the table. To numerically estimate
lim u(x), enter values to the right of, and becoming closer and
+
x→−1
closer to, −1. (Note: Again, the values that you enter do not
have to be those shown in the text or these shown to the right.)
Because the output 0.4285… appears three times in a row, we estimate that lim + u(x) =
x→−1
0.429. Then, because lim − u(x) = lim + u(x) = 0.429, we estimate that lim u(x) = 0.429.
x→−1
x→−1
x→−1
We now illustrate finding the limit in part b of Example 2 in Section 1.4 of Calculus Concepts:
Delete the values currently in the table. To numerically estimate
lim u(x), enter values to the left of, and becoming closer and
−
x→−2/9
closer to, −2/9 = −0.222222…. Because the output values appear
to become larger and larger, we estimate that lim − u(x) → ∞.
x→−2/9
Delete the values currently in the table. To numerically estimate
lim u(x), enter values to the right of, and becoming closer
+
x→−2/9
and closer to, −2/9. Because the output values appear to become
larger and larger, we estimate that lim − u(x) → −∞. Thus,
x→−2/9
lim
x→−2/9
u(x) does not exist.
1.4.2 CONFI RMI NG LI MI TS GRAPHI CALLY – ZOOMI NG I N AND OUT A graph can
be used to confirm a limit that you estimated numerically. You also can zoom in or zoom out
on the graph to obtain a better view of the limit you are estimating. We again illustrate using
the function u that appears in Example 2 of Section 1.4 in Calculus Concepts.
Have the function u(x) =
3x 2 + 3 x
9 x 2 + 11x + 2
entered in the y1 location
of the Y= list. A graph drawn with F2 [Zoom] 4 [ZoomDec] or
F2 [Zoom] 6 [ZoomStd] is not very helpful.
To confirm that lim u(x) = 0.429, we are only interested in
x→−1
values of u that are near −1. So, choose values very near to −1
for the x-view and evaluate the function at those x-values to help
determine the y-view. We manually set the window to values
such as those shown to the right and draw the graph.
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TI-89 Guide
If you look closely, you can actually see the “hole” in the graph
to trace to where x = −1,
at x = −1. Press F3 [Trace], use
►
and the TI-89 confirms that u is not defined at x = −1. Now
press
►
and observe the y-value. Press
◄
several times to
go to the left of x = −1 and confirm from the y-values that the
limit is approximately what we had determined numerically.
Note that we confirmed that the limit exists by seeing that the two parts of the graph (to the left
and right of x = −1) move toward the same point. Tracing around x = −1 provides a check on the
numerical value of the limit.
The previous instructions show how to zoom in by manually setting the window. You can
also zoom in with the zoom menu of the calculator. We next describe this method.
Return to the graphing screen set with F2 [Zoom] 4 [ZoomDec] or
F2 [Zoom] 6 [ZoomStd] or any screen where you can see the
portion of the graph around x = −1.) Press F2 [Zoom] 2 [Zoom
In] and use
◄
and
▲
to move the blinking cursor until you are
near the point on the graph where x = −1. Press ENTER .
Depending on the horizontal view, you may or may not be able
to see the hole in the graph at x = −1. If your view is not magnified enough to see what is happening around x = −1, repeat the
zoom-in process. To check your numerical estimate of the limit,
press F3 [Trace], use the arrow keys to move to either side of x
= −1, and observe the y-values.
Now let’s consider the limit in part b of Example 2 of Section 1.4. We want to confirm with a
graph what was found numerically; that is, we wish to confirm that lim u(x) does not exist.
x→−2/9
We want to zoom out on the graph to confirm that the limit of
u(x) as x approaches −2/9 does not exist. To do this, we set a
small x-view and a larger y-view. (Note that these values are
arbitrary – any small x-view that includes −2/9 and any y-view in
which the graph can be seen clearly will do.)
Draw the graph with
F3 (GRAPH). Before continuing with
the limit investigation, we need to eliminate the “extra” vertical
line that appears on the above graph at x = −2/9. The line appears
because we are graphing in Line mode, which tells the TI-89 to
connect points on the graph. (You may or may not have the line
on your graph.)
Set y1 set to draw in Dot mode by highlighting y1 and pressing
2nd F1 [F6: Style] 2 [Dot]. Redraw the graph with
F3
(GRAPH).
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TI 89-24
Chapter 1
You can zoom out more by resetting ymin and ymax to larger
values. Then, press F3 [Trace] and go left and right on the
graph to confirm that it “splits” near x = −2/9 ≈ −0.222222….
It appears that as x approaches −2/9 from the left that the
outputs become larger and larger and that as x approaches −2/9
from the right that the outputs become smaller and smaller.
1.4.3 I NVESTI GATI NG END BEHAVI OR NUMERI CALLY Investigating end behavior using
the TI-89 table is very similar to numerically estimating the limit at a point. We illustrate this
using the function that appears in Example 4 of Section 1.4 in Calculus Concepts.
Have u(x) =
3x 2 + x
3x + 11x 2 + 2
in the y1 location of the Y= list. Be
certain that you remember to enclose both the numerator and
denominator of the fraction in parentheses.
Have TblSet set to ASK and press
F5 (TABLE). Delete any
values that are in the table. In order to numerically estimate
lim u(x), enter values of x that get larger and larger. (Note:
x→∞
The values that you enter do not have to be those shown in the
text or these shown to the right.)
We assume that we want the limit accurate to 3 decimal places.
According to our Rule of Thumb for Determining Limits from
Tables (page C-21), once we see the same value to 4 decimal
positions 3 times in a row, we can estimate the limit by rounding
off the answer to 3 decimal places.
lim u(x) ≈ 0.273
x→∞
Note that the last entry
in the table (x = 10,000)
was not necessary.
Part b of Example 4 asks for lim u(x). Delete or type over the
x→−∞
values that are currently in the table. Then, enter values of x that
get smaller and smaller. We estimate lim u(x) ≈ 0.273.
x→−∞
CAUTI ON: It is not the final value, but a sequence of several values, that is important when
determining limits. If you enter a very large or very small value, you may exceed the limits of
the TI-89’s capability and obtain an incorrect number. Always look at the sequence of values
obtained to make sure that all values found make sense.
1.4.4 I NVESTI GATI NG END BEHAVI OR GRAPHI CALLY As was the case with limits at a
point, a graph of the function can be used to confirm a numerically-estimated limit. We again
illustrate with the function that appears in Example 4 of Section 1.4 in Calculus Concepts.
Have u(x) =
3x 2 + x
3x + 11x 2 + 2
in the y1 location of the Y= list. (Be
certain that you remember to enclose both the numerator and
denominator of the fraction in parentheses.) A graph drawn with
F2 [Zoom] 4 [ZoomDec] is a starting point.
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TI 89-25
TI-89 Guide
We estimated the limit as x gets very large or very small to be
0.273. Now, u(0) = 0, and it does appears from the graph that u
is never negative. Set a window with values such as xmin = −10,
xmax = 10, ymin = 0, and ymax = 0.35. Press
F3 (GRAPH).
To examine the limit as x gets larger and larger, change the winF3 (GRAPH),
dow so that xmax = 100, view the graph with
change the window so that xmax = 1000, view the graph with
F3 (GRAPH), and so forth. The graph to the right was drawn
with xmax = 10,000. Use F3 [Trace] with each graph.
Repeat the process as x gets smaller and smaller, but change xmin
rather than xmax after drawing each graph. The graph to the right
was drawn with xmin = −9000, xmax = 10, ymin = 0, and ymax =
0.35. Again, press F3 [Trace] on each graph screen to view some
of the outputs and confirm the numerical estimates.
1.4.5 FI NDI NG LI MI TS ALGEBRAI CALLY As previously mentioned, the TI-89 finds limits
algebraically. You can use this feature to confirm numerical and graphical estimates or use it as
a method of finding limits. The TI-89 syntax is limit (function, input variable, point, direction).
For a limit from the left, direction = any negative number; for a limit from the right, direction =
any positive number; for a limit from both sides, direction = 0 or direction is omitted from the
instruction. We illustrate using the function u that appears in Example 2 of Section 1.4.
Press HOME F4 1 [Define] and type
u(x) =
3x 2 + 3 x
9 x 2 + 11x + 2
. Find lim − u(x).
x→−1
Access the limit instruction with F3
[Calc] 3 [limit(].
NOTE: We used −3 to indicate the direction in the limit statement. The TI-89 considers only
the negative sign, so any negative number gives the same result – try it! Also remember that if
ENTER gives it.
you want a decimal approximation for the answer,
Find lim + u(x), lim u(x),
x→−1
lim
x→−2/9+
x→−1
lim
x→−2/9−
u(x),
u(x), and lim u(x).
x→−2/9
(See the note below for a shortcut.)
NOTE: Instead of retyping the limit statement each time to find the above limits, remember
to put the cursor on the right-hand
that when the entry line is highlighted, you can press
side of the statement and press
◄
►
to put the cursor on the left-hand side of the statement. The
statement is ready for editing using the arrow keys to move to the correct location and using
to delete unwanted symbols.
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TI 89-26
Chapter 1
The results of our numerical and graphical investigations are confirmed by the algebraic
answers. Note that the TI-89 prints undef to mean undefined or does not exist.
1.5 Linear Functions and Models
This portion of the Guide gives instructions for entering real-world data into the calculator and
finding familiar function curves to fit that data. You will use the beginning material in this
section throughout all the chapters in Calculus Concepts.
CAUTI ON: You need to name the data variable in the TI-89, and you are instructed below to
use the name calcc. Do not change or vary this name. If you do, the programs that you use
later will not properly execute. Also, be careful when you enter data in the TI-89 because the
model and all results depend on the values that are entered! Always check your entries.
1.5.1 ENTERI NG DATA
Year
Tax (in dollars)
We illustrate data entry using the values in Table 1.46 in Section 1.5.
1996
2532
1997
3073
1998
3614
1999
4155
2000
4696
2001
5237
Before beginning, clear out previous work with HOME 2nd F1 [F6] 2 [NewProb] ENTER .
Press APPS 6 [Data/Matrix Editor] and choose 3 [New]. Choose
the settings shown to the right for the first two positions. Name
the variable by pressing 2nd alpha (a-lock) and type c a l c c.
(We use this variable name for all data in this Guide. See the
CAUTION note above.)
Press ENTER until you see the screen
shown to the right. These are the lists
that hold data. You can access many
other lists by highlighting the list name
.
and using
►
In this Guide, we usually use list c1 for the input data and list c2 for the output data. If there are
any data values already in your lists, see Section 1.5.3 of this Guide and first delete any “old”
data. To enter data in the lists, do the following:
Position the cursor in the first location in list c1. Enter the input
data into list c1 by typing the years from top to bottom in the c1
column, pressing ENTER or
after each entry. (If a letter
▼
rather than a number prints, alpha-lock is still on. Press alpha to
release it.) Use the shortcut in the NOTE below to enter the tax
values in c2.
NOTE: After typing the sixth input value, 2001, use
of list c1 and then press
►
▲
to have the cursor go to the top
to be in the first row of the c2 column. Enter the output data into
list c2 by typing the entries from top to bottom in the c2 column, pressing ENTER or
▼
▼
after
causes the cursor to jump to the bottom of the current column. See
each tax value. (
page 232 of the TI-89 Guidebook for other key combinations that scroll the data lists.)
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TI 89-27
TI-89 Guide
1.5.2 EDI TI NG DATA
If you incorrectly type a data value, use the cursor keys (i.e., the arrow
► , ◄ , ▲ , and/or ▼ ) to highlight the value you wish to correct and then type the
correct value. Press ENTER or ▼ to enter the corrected value.
keys
•
To insert a data value, put the cursor over the value that will be directly below the one you
will insert, and press 2nd F1 [F6: Util] 1 [Insert] 1 [cell]. The values in the list below the
insertion point move down one location and undef is filled in at the insertion point. Type the
data value to be inserted and press ENTER . The “undef” is replaced with the inserted data
value.
•
To delete a single data value, highlight the value you wish to delete, and press
or press
F1 [F6: Util] 2 [Delete] 1 [cell]. The values in the list below the deleted value move up
2nd
one location.
1.5.3 DELETI NG OLD DATA Whenever you enter new data in your calculator, you should first
delete any previously-entered data. There are several ways to do this, and the most convenient
method is illustrated below.
Access the data with APPS 6 [Data/Matrix Editor] and choose 1
[Current]. (You probably have different values in your lists if
you are deleting “old” data.)
Put the cursor on any entry in the column to be cleared.
Press 2nd F1 [F6: Util] 5 [Clear column]. Use
►
to move the cursor to
any entry in the next column and repeat
this procedure to clear old data from
any other lists you will use.
1.5.4 FI NDI NG FI RST DI FFERENCES When the input values are evenly spaced, you can use
program DIFF to compute first differences in the output values. Program DIFF is given in the
TI-89 Program Appendix at the Calculus Concepts Website. Consult the Programs category in
TroubleShooting the TI-89 in this Guide if you have questions about obtaining the programs.
Have the data given in Table 1.46 in Section 1.5 of Calculus
Concepts entered in your calculator. (See Section 1.5.1 of this
Guide.)
Exit the data editor and go to the home screen with HOME .
To run the program, type the name of the program in the entry
line and put “( )” behind the name. It is easier to type the program name if you lock alpha mode with 2nd alpha [a-lock]. If
you prefer, get the program name with 2nd − (VAR-LINK),
press
▼
until diff is highlighted, and press ENTER . Type ) .
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TI 89-28
Chapter 1
Press ENTER . The message on the
right appears on your screen. Press
ENTER again. If you have not entered
the data in the correct location, press 2
[no]. If you are ready to continue, press
1 [yes] or ENTER .
Press F1 [Choice?] 1 [first differences].
(Options 2 and 3 are used in the next
chapter of this Guide.)
The first differences are displayed.
Press F1 [Choice?] 4 [Quit program].
To return to the home screen, press
F5 [PrgmIO] or HOME .
If you need to scroll the list to see the
rest of the first differences or recall
them, enter d1 in the entry line.
•
The first differences are constant at 541, so a linear function gives a perfect fit to these tax
data.
Note: Program DIFF should not be used for data with input values (c1) that are not evenly
spaced. First differences give no information about a possible linear fit to data with inputs that
are not the same distance apart. If you run program DIFF with input data that are not evenly
spaced, the message INPUT VALUES NOT EVENLY SPACED appears and the program stops.
1.5.5 SCATTER PLOT SETUP The first time that you draw a graph of data, you need to set the
TI-89 to draw the type of graph you want to see. Once you do this, you never need to do this
set up again (unless for some reason the settings are changed). If you always put input data in
list c1 and output data in list c2, you can turn the scatter plots off and on from the Y= screen
rather than the Plot Setup screen after you perform this initial setup.
Access the data with APPS 6 [Data/Matrix Editor] and choose 1
[Current]. Press F2 [Plot Setup]. (Your screen may not look
exactly like this one.)
On the Plot Setup screen, have Plot 1 highlighted and press F1
[Define]. Choose the options shown on the right. (You can
choose any of the 5 available marks.) Type in c1 for x and c2 for
y.
Press ENTER to save the settings. Note that Plot 1 is turned on
because there is a check mark to the left of the name. Press
ENTER to return to the data lists.
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TI 89-29
TI-89 Guide
▲
Press
F1 (Y=) and
. Notice that Plot1 at the top of the
screen has a check mark to the left of the name. This tells you
that Plot1 is turned on and ready to be graphed. You can also see
the settings you chose for the plot.
•
A scatter plot is turned on when there is a check mark to the left of its name on the Y=
screen. From now on, you can turn the scatter plot off and on by having Plot1 either
checked or not checked. To turn Plot1 off, press F4 [b] to remove the check mark.
Reverse the process to turn Plot1 back on.
•
TI-89 data lists can be named and stored in the calculator’s memory for later recall and use.
Refer to Section 1.5.13 and 1.5.14 of this Guide for instructions on storing data lists and
later recalling them for use.
1.5.6 DRAWI NG A SCATTER PLOT OF DATA Any functions that are turned on in the Y= list
will graph when you plot data. Therefore, you should clear or turn them off before you draw a
scatter plot. We illustrate how to graph data using the modified tax data that follows Example
2 in Section 1.5 of Calculus Concepts.
Year
Tax (in dollars)
1996
2541
1997
3081
1998
3615
1999
4157
2000
4703
2001
5242
Access the Y= graphing list. If any entered function is no longer needed, clear it by pressing
CLEAR . If you want the function(s) to remain but not graph when you draw the scatter plot,
remove the check mark next to that function with F4 [b]. Also be sure that Plot 1 at the top of
the Y= screen is turned on (i.e., checked).
Access the data with APPS 6 [Data/Matrix Editor] and choose 1
[Current]. Using the table given above, enter the year data in c1
and the modified tax data in c2 according to the instructions
given in Section 1.5.1 of this Guide.
(You can either leave values in the other lists or clear them.)
Press
F1 (Y=) and then F2 [Zoom] 9 [ZoomData] to have
the calculator set an autoscaled view of the data and draw the
scatter plot. (Note that ZoomData also resets the x- and y-axis
tick marks.)
Recall that if the data are perfectly linear (that is, every data point falls on the graph of a line),
the first differences in the output values are constant. The first differences for the original tax
data were constant at $541, so a linear function fit the data perfectly. What information is
given by the first differences for these modified tax data?
Run program DIFF. (See Section 1.5.4
of this Guide.) Recall that the program
stores the first differences in list d1 if
you want to recall them on the home
screen.
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Chapter 1
These first differences are close to being constant. This information, together with the linear
pattern shown by the scatter plot, are a good indication that a linear function is likely to give a
good fit to the data.
1.5.7 FI NDI NG A LI NEAR FUNCTI ON TO MODEL DATA You will often have your TI-89
find the linear function that best fits a set of data. Other functions are fit using the same steps.
Access the data with APPS 6 [Data/
Matrix Editor] and choose 1 [Current].
Press F5 [Calc]. On the first line
(Calculation Type…), press
►
and 5
[LinReg]. Type in c1 for x and c2 for y.
On the fourth line (Store RegEQ to…),
and choose where you want
press
to paste the linear function. Press
►
ENTER .
CAUTI ON: The best-fit function found by the calculator is also called a regression function.
The coefficients of the regression function never should be rounded! This is not a problem
because the calculator pastes the entire equation it finds into the Y= list at the same time the
function is found if you follow the instructions given above.
Use
▼
to move between the various
options in the Calculate screen. When
you have entered all the information,
press ENTER to exit the screen. Press
F1 (Y=) and see that the function
has been pasted in the chosen location.
CAUTI ON: The number that is labeled corr on the STAT VARS screen is called the correlation coefficient. This value and the one labeled R2, which is the coefficient of determination,
are numbers that you will learn about in a statistics course. It is not appropriate5 to use these
values in a calculus course.
Graphing the Line of Best Fit: After finding a best-fit equation, you should always draw the
graph of the function on a scatter plot to verify that the function provides a good fit to the data.
Press
F3 (GRAPH) or F2 [Zoom] 9 [ZoomData] to over-
draw the function on the scatter plot of the data.
(As we suspected from looking at the scatter plot and the first
differences, this function provides a very good fit to the data.)
5Unfortunately, there is no single number that can be used to tell whether one function better fits data than another. The
correlation coefficient only compares linear fits and should not be used to compare the fits of different types of functions.
For the statistical reasoning behind this statement, read the references in footnote 8 on page C-35.
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TI-89 Guide
1.5.8 USI NG A MODEL FOR PREDI CTI ONS You can use one of the methods described in
Sections 1.1.5 or 1.1.6 of this Guide to evaluate the linear function at the indicated input value.
Predict the tax owed in 2002 where the tax is found using the
linear model computed in the last section of this Guide: Tax =
540.371t − 1,076,042.467 dollars where t is the year. Remember
that you should always use the full model, i.e., the function you
pasted in y1, not the rounded equation given above, for all
computations.
The 2002 tax, predicted by evaluating an output on the home
screen, is about $5781. We now predict the tax in 2003 using
the TI-89 table. As seen to the right, the predicted tax is
approximately $6322.
1.5.9 COPYI NG A GRAPH TO PAPER Your instructor may ask you to copy what is on your
graphics screen to paper. If so, use the following ideas to more accurately perform this task.
After using a ruler to place and label a scale (i.e., tick marks) on your paper, use the trace
values (as shown below) or the data to graph a scatter plot on your paper.
Press
F3 (GRAPH) to return the modified tax data graph
►
.
found in Section 1.5.7 to the screen. Press F3 [Trace] and
The symbol P1 in the upper right-hand corner of the screen
indicates that you are tracing the scatter plot (Plot 1) of the data.
▼
Press
to move the trace cursor to the linear function graph.
The number in the top right of the screen tells you the location of
the function that you are tracing (in this case, y1). Use
►
◄
and/or
to locate values that are as “nice” as possible and
mark those points on your paper. Use a ruler to connect these
points and draw the line.
•
If you are copying the graph of a continuous curve rather than a straight line, you need to
trace as many points as necessary to see the shape of the curve while marking the points on
your paper. Connect the points with a smooth curve.
1.5.10 ALI GNI NG DATA We return to the modified tax data entered in Section 1.5.6. If you
want c1 to contain the number of years after a certain year instead of the actual year, you need
to align the input data. In this illustration, we shift all of the data points to three different positions to the left of where the original values are located.
Press APPS 6 [Data/Matrix Editor] 1 [Current] to access the data
lists. To align the input data as the number of years past 1996,
and/or
) so that c1 is highfirst press the arrow keys (
lighted. Tell the TI-89 to subtract 1996 from each number in c1
with alpha ) (c) 1 − 1996 ENTER . Instead of an actual
◄
▲
year, the input now represents the number of years since 1996.
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TI 89-32
Chapter 1
NOTE: Another way to tell the TI-89 the alignment formula is to have the cursor on any cell in
c1 and press F4 [Header]. Type alpha ) (c) 1 − 1996 and press ENTER .
•
If you did not get the values 0, 1, 2, 3, 4, and 5 in c1, press F1 [Tools] 9 [Format] and select
ON in the Auto-calculate position. Press ENTER to return to the data editor.
Once the data have been aligned, clear the alignment definition
from the column header. To do this, have c1 highlighted and
press ENTER (or press F4 [Header]) to highlight the alignment
definition at the bottom of the screen. Press CLEAR ENTER .
WARNI NG: If you do not clear the alignment definition from the column header, the Autocalculate feature will keep subtracting 1996 from the data every time you have c1 highlighted
and ENTER is pressed.
Find the linear function by pressing F5 [Calc]. On the first line
(Calculation Type…), press
►
and 5 [LinReg]. Type in c1 for x
and c2 for y. On the fourth line (Store RegEQ to…), press
choose y2(x).
►
and
Press ENTER . Press
F1 (Y=)
and see that the function has been
pasted in the chosen location. (Recall
that you can highlight the function
location and press
in order to see
the complete function.)
►
Highlight the y1 line and uncheck the function with F4 [b] to
turn y1 off. Press F2 [Zoom] 9 [ZoomData] to overdraw the
function y2(x) on the scatter plot of the data.
If you now want to find the linear function that best fits the modified tax data using the input
data aligned another way, say as the number of years after 1900, first return to the data lists
with APPS 6 [Data/Matrix Editor] 1 [Current] and highlight c1.
Add 96 to each number currently in c1
with alpha ) (c) 1 + 96. Press
ENTER . The input now represents
the number of years since 1900. Press
F4 [Header] CLEAR
•
ENTER .
The last step above was to clear the alignment definition once the data have been aligned.
Remember to do this or the input data may change without you realizing it when you
perform other calculations in the lists.
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TI-89 Guide
Press F5 [Calc]. On the first line, press
►
and 5 [LinReg].
Type in c1 for x and c2 for y. On the fourth line, press
►
and
store the linear function to y3(x). Press ENTER .
Press
F1 (Y=) and see that the
function is pasted in y3. Uncheck y1
and y2 and press F2 [Zoom] 9 [ZoomData] to overdraw the function on the
scatter plot.
•
Remember, if you have aligned the data, the input value at which you evaluate the function
may not be the value given in the question you are asked. You can use any of the equations
to evaluate function values.
Using the function in y1 (the input is the year), in y2 (the input is
the number of years after 1996), or in y3 (the input is the number
of years after 1900), we predict that the tax owed in 2003 is
approximately $6322.
1.5.11 NAMI NG AND STORI NG DATA You can name data and store it in the TI-89 memory for
later recall. You may or may not want to use this feature; it would be helpful if you plan to use
a large data set several times and don’t want to reenter it each time you use it. To illustrate the
procedure, let’s name the current data.
Be in the data editor ( APPS 6 [Data/Matrix Editor] 1 [Current]).
Press F1 [Tools] 2 [Save Copy As …]. In the Variable box, enter
a name for the data, say tax. Press ENTER until this screen
disappears.
Return to the home screen with HOME . You can verify that
the data have been stored to a data variable by pressing 2nd
− (VAR-LINK).
(Note that your list of variables may not be
exactly the same as that shown on the screen to the right.)
CAUTI ON: The list of variables is in alphabetical order, so you may need to scroll to find the
name of the data variable you stored. It is very important that you not store data to a name that
is routinely used by the TI-89. Such names are c1, c2, …, c99, MATH, Log, MODE, A, B, and so
forth. Also note that if you use a single letter as a name, this might cause one or more of the
programs to not execute properly.
1.5.12 RECALLI NG STORED DATA The data you have stored remains in the memory of the TI89 until you delete it using the instructions given in Section 1.5.15 of this Guide. When you
wish to use the stored data, recall it to the data editor as we next illustrate with the tax data.
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Chapter 1
Press APPS 6 [Data/Matrix Editor] 2
[Open]. Press
►
and choose tax.
Press ENTER ENTER . The data
editor now contains the tax data.
1.5.13 DELETI NG STORED DATA You do not need to delete any data unless your memory is
getting low, the name interferes with a program execution, or you just want to do it. To illustrate, we delete the tax data. Note that this procedure works to delete any variable.
From the home screen, press 2nd −
(VAR-LINK). Use
▼
to scroll down
to and highlight the name of the variable you want to delete. Press F1
[Manage] 1 [Delete]. Press ENTER .
WARNI NG: When the data editor is open to a particular data set, that data variable does not
appear in the list of variables. You should be on the home screen before accessing the list of
variables with VAR-LINK.
1.5.16 WHAT I S “BEST FI T”? It is important to understand the method of least squares and the
conditions necessary for its use if you intend to find the equation that best-fits a set of data. You
can explore6 the process of finding the line of best fit with program LSLINE. (Program LSLINE
is given in the TI-89 Program Appendix.) For your investigations of the least-squares process
with this program, it is better to use data that is not perfectly linear and data for which you do
not know the best-fitting line.
Before using program LSLINE, clear all functions from the Y= list, turn on Plot1 by having
its name checked on the Y= screen, and enter your data in lists c1 and c2 in the calcc data variable. (If Plot 1 is not turned on and the data is not stored in calcc, the program will not run.)
F1 (Y=) F2 [Zoom] 9 [ZoomData]. Press WINDOW and
Next, draw a scatter plot with
reset xscl and yscl so that you can use the tick marks to help identify points when you are asked to
F3 (GRAPH), view the scatter plot, and then
give the equation of a line to fit the data. Press
return to the home screen.
To activate program LSLINE, type lsline() on the entry line of the
home screen or press 2nd − (VAR-LINK) followed by the key
of to first letter of the program name, here 4 (L), and ENTER
)
ENTER . The program displays this message.
NOTE: While the program is calculating, the indicator in the bottom right-hand corner of the
screen says BUSY. When the program pauses, this indicator says PAUSE. Program LSLINE
pauses several times during execution for you to view the screen. Whenever this happens, you
should press ENTER to resume execution after you have viewed the screen.
6 This program works well with approximately 5 data points. Interesting data to use in this illustration are the height and
weight, the arm span length and the distance from the floor to the navel, or the age of the oldest child and the number of
years the children’s parents have been married for 5 randomly selected persons.
Copyright © Houghton Mifflin Company. All rights reserved.
TI-89 Guide
TI 89-35
Press ENTER to continue the program
and it next tells you what the tick mark
settings are. After pressing ENTER ,
you see a graph of the scatter plot.
The program next asks you to find the y-intercept and slope of
some line you estimate will go “through” the data. (You should
not expect to guess the best-fit line on your first try!) Use the
tick marks to estimate rise divided by run and note a possible yintercept. Enter your guess for the slope and y-intercept.
After pressing ENTER again, your line
is drawn and the errors are shown as
vertical line segments on the graph.
Next the sum of squared errors, SSE, is
displayed for your line.
Decide whether you want to move the y-intercept of the line up or down or change the slope of
the line to improve the fit to the data.
Press ENTER type 1 to choose the Try Again? option. You are
then shown the scatter plot to view. After deciding on a new
guess for the slope and/or y-intercept, press ENTER . Input the
new values when requested.
The process of viewing your line, the
errors, and display of SSE is repeated.
If the new value of SSE is smaller than
the SSE for your first guess (as is the
case here), you have improved the fit.
When you are satisfied with your line,
enter 2 [no] at the Try Again? prompt.
The program shows the linear fit calculation. Press ENTER and the line of
best fit and your last line are drawn.
Press ENTER and the line of best fit
and its errors display. The SSE for the
best-fit line is shown and then all the
parameters of the best-fit line are given.
Press ENTER to end the program. Use program LSLINE to explore7 the method of least
squares8 that the TI-89 uses to find the line of best fit.
7 Program LSLINE is for illustration purposes only. Actually finding the line of best fit for a set of data should be done
according to the instructions in Section 1.5.7 of this Guide.
8 Two articles that further explain “best-fit” are H. Skala, “Will the Real Best Fit Curve Please Stand Up?” Classroom
Computer Capsule, The College Mathematics Journal, vol. 27, no. 3, May 1996 and Bradley Efron, “Computer-Intensive
Methods in Statistical Regression,” SIAM Review, vol. 30, no. 3, September 1988.
Copyright © Houghton Mifflin Company. All rights reserved.
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