Texas Instruments Titanium TI-89 User manual

Copyright © Houghton Mifflin Company. All rights reserved. TI89-1
GuideforTexasInstrumentsTI-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 Instructions
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.
WARNING: 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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• 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 andreturntothehomescreenwith HOME .
• Beginbypressing 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
415
895
7
*+
. Enclose the denominator in parentheses so
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
what you want to delete and then press
 . The entry line is always in insert mode. You do
not need to clear the entry line before beginning to type a new expression.
Evaluate
(3)
.
4
−
−
+5 8 1456
.Use ( − ) for the negative symbol and − for
the subtraction sign. The numerator and denominator must be
enclosed in parentheses and
−
3
4
≠(
−
3)
4
.

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Now, evaluate e
3
*0.027 and e
3*0.027
. The TI-89 prints the left
parenthesis when you press
 X (e
x
). 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 on the entry line and press
► . The cursor appears on
the right-hand side of the entry line expression. If you press
◄ , the
cursor appears on the left-hand side of the entry line expression. Try
editing e
3
*0.027 to type e
3*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
insert the entry or answer. Second, move the cursor, repeatedly using
▲ , until what you are
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 typ-
ing an expression that was just evaluated, use the answer memory by pressing
2nd ( − ) (ANS).
Find
1
415
895
7
1
*+
F
H
G
GI
K
J
J
−
using this shortcut.
Enter
Ans with 2nd ( − )(ANS) .
NOTE: Thelast-calculatedanswerisreferredtobytheTI-89ontheentrylineas 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,
press
 ENTER . We also illustrate how to retrieve a previous entry that is in the history area.
Type
2
51 3
+ , press ENTER , and then
press
 ENTER . Press ▲ until
7
1315
is highlighted. Press ENTER .
Use
◄ and insert a decimal point after
the 7. Press
ENTER .

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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, 1.4675 E
−
6 means 1.4675*10
−
6
,whichis
scientific notation for 0.0000014675. Enter
“ E ” by pressing EE .
The result 1.5 E 28 means 15,000,000,000,000,000,000,000,000,000 =
1.5(10
28
). In the history area, an arrowhead (  ) that points to the
right at the end of a number means that the number or expression
continues. Highlight the number or expression and press
► to
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 stor-
age 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 aand 3 in b,andthencalculate4a–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
thesamenamethattheTI-89alreadyusesforabuilt-invariable(suchas
LOG or ans ).
• Whatever you store in a particular memory location stays there until it is replaced by some-
thing 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 yas undefined variables.
WARNING: 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 aby the
variable x, you must use
 between the letters. Also, because 5 has been stored to a, the TI-89

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substitutes that value when the expression is evaluated. All user-defined variables are stored in a
folder
1
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] keyonthe VAR-LINK screen. We illustrate by deleting the variable a:
Deleting a single variable: Suppose we want to delete the variable
a. Gotothe
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. Ac-
cess this operator by pressing the key directly under
= .
Evaluate 5.3r–2.1h
2
for for r=6andthenforr=2.9andh=7by
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-Shoot-
ing 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.

Copyright © Houghton Mifflin Company. All rights reserved. TI89-6
Chapter 1 Ingredients 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(1.093
t
).
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
3.622 ( 1 .093 ^ 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
points. If any of these has a check mark, use
▲ until the cursor is on the plot name and press
F4 [ b ] to make the check mark go away. Otherwise, you may receive an error message.
DEFINING A FUNCTION ON THEHOME 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(1.093
t
)with F4 [Other] 1 [Define] alpha
0(V) ( T ) = 3 .622 ( 1 .093 ^ 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 xinstead
of tas the input variable. (The TI-89 now knows this function
by two names:
y1 and v.)

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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(1.093
t
)
thousand dollars where tis 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 von this inter-
val, have the function venteredinthe
Y= list using xas the input variable. (See Section 1.1.1 of
this Guide.) We now draw the graph of vfor xbetween 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-andy-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 func-
tion. Press
 F1 (Y = ) and enter v(x)
in the y2 location. Press ▲ and F4
[ b ] to turn off y1 .
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.
WARNING: You must use xas the input variable when in the
Y = list. If you use tor another
letter inside the parentheses following
v , you will likely get an “undefined variable” message.

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1.1.3 MANUALLY CHANGI NG THE VI EW OF A GRAPH We just saw how to have the TI-
89 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 vin 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 ,20for 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
drawn with
ymin =
−
5and ymax =7). Press  F3 (GRAPH).
NOTE: If you see nothing on the screen, press  F1 (Y = ) and highlight y1 . If there is no
checkmarknextto
y1, press F4 [ b ] and  F3 (GRAPH).
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
graphonthescreen.)
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 ,
and press
 F3 (GRAPH). If the graph you obtain is not a
good view of the function, repeat the above process using x-
values 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 x-
values 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 x-

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values into the equation that is being graphed. We again assume that you have the function v(x)
= 3.622(1.093
x
) 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-andy-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 yis approximately 13.7 when xis 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 EVALUATING OUTPUTS ON THE HOMESCREEN The input values used in the eval-
uation 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(1.093
t
) that is in Example 4 of Section 1.1.
Using xas 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 vwith input t(see the bottom of page
C-6), it can easily be used to find the function outputs.

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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(1.093
t
) at the input t= 15. Even
though you can use any of the function locations, we again choose to use
y1 .Press  F1
(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 highlight-
ing 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
within the table or
 F4 (TblSet).
Choose ASK in the Indpnt: location. To
enter the x-values 15, 0, and 20, type
each value, press
ENTER ▼ , type
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 FINDING INPUT VALUES USING 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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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 eof 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(1.093
t
)
thousand dollars where tis the number of years after the end of 1980, we are asked to solve the
equation 3.622(1.093
t
) = 20. That is, we are asked to find the input value tthat 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 .
WARNING:Ifyourefertoanequationbyitslocationinthe Y= list, you must use the com-
pletenameoftheequation(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
the value. Be certain that you press
▲ to highlight the solution and then press and hold
► to scroll to the end of the answer.
USING THENUMERIC SOLVER Thesolveinstructionisveryeasytouseandoffersa
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.

TI89-12 Chapter 1
Copyright © Houghton Mifflin Company. All rights reserved.
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 searches
2
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 DETERMINE AGUESS TO USE IN THE NUMERICSOLVER
This sec-
tion 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(1.093
t
) = 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
hold down either
► or ◄ until you get an estimate of the
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.

TI-89 Guide TI89-13
Copyright © Houghton Mifflin Company. All rights reserved.
1.1.9 GRAPHICALLY FINDING INTERCEPTS 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 x-
intercept 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(1.093
x
) = 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–0.8x
2
+ 4 that corresponds to the
output f(x) = 2.3. That is, we are asked to find xsuch that 3x–0.8x
2
+4=2.3. Becausethis
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 x-
intercept method or the intersection method. We present both, and you should use the one you
prefer.
X -INTERCEPT METHOD for solving the equation f(x)–c=0:
Press
 F1 (Y = ) and clear all locations with CLEAR . Enter
the function 3x–0.8x
2
+4–2.3in y1 .(Typex
2
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
 F3 (GRAPH)
to redraw the graph.)
To graphically find an x-intercept, i.e.,avalueofxat 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
mark the left bound. Now press and hold
► until you are to
the right of this x-intercept. Press
ENTER to mark the location
of the upper bound for the x-interceptandtofindit.

TI89-14 Chapter 1
Copyright © Houghton Mifflin Company. All rights reserved.
Thevalueoftheleftmostx-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.)
INTERSECTI 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–0.8x
2
+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.
(If you reset the window,
 F3 (GRAPH) redraws the graph.)
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
from the curve to the line. Next, press
◄ to move the cursor to
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 .
Thevalueoftheleftmostx-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.

TI-89 Guide TI89-15
Copyright © Houghton Mifflin Company. All rights reserved.
1.1.10 SUMMARY OF ESTIMATING AND SOLVINGMETHODS 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 fhas been entered in
y1 in the Y= list or defined as y1(x)
and that a function ghas 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)=
fx
gx
()
()
.
Enter
y1 ( y2(x) ) to obtain the composite function (f
o
g)(x)=f(g(x)).
1.3.2 FINDING A DIFFERENCE FUNCTION 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(1.105
t
)
million dollars and costs = C(t)=
−
39.2t
2
+ 540.1t+ 1061.0 thousand dollars tyears 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., aformularather
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 Swith
F4 [Other] 1 [Define] alpha
3(S) ( T ) = 3 .570 ( 1 .105 ^ T )ENTER and
define Cwith
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 tbecause it is given as the input variable in the text illustration. However, if
you prefer x, replace every tby xin the instructions.

TI89-16 Chapter 1
Copyright © Houghton Mifflin Company. All rights reserved.
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.
WARNING: 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 tby 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)
andthenfind p(2). We illustrate the copy and paste feature of the
TI-89todothistask. Press
▲ until s(t) − 0.001c(t) in the history
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
[Define] alpha STO
 (P) ( 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 FINDING A PRODUCT FUNCTION 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(0.921
x
)gal-
lons of milk sold on the xth day of last month.
You can, if you wish, define the functions sand g. (Note that the new definition for swill
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 )=(.007 X + 1 .492 )  ( 31 − 6
.332 ( .921 ^ X ) )ENTER .

TI-89 Guide TI89-17
Copyright © Houghton Mifflin Company. All rights reserved.
NOTE: You do not have to, but you can, use the times sign  between Sand Gto indicate the
product function. We enclosed each of Sand Gin parentheses to indicate a product when they
arewrittennexttoeachother,sothetimessignisnotnecessary.
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)”
statement and paste it into the entry line, press
◄ and use ►
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 find-
ing a composite function with those functions given on page 33 of Section 1.3 of Calculus
Concepts: altitude = F(t)=
−
222.22t
3
+ 1755.95t
2
+ 1680.56t+ 4416.67 feet above sea level
where tis the time into flight in minutes and air temperature = A(F) = 277.897(0.99984
F
)–66
degrees Fahrenheit where Fis 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 Fin
y1 by pressing ( − ) 222 .22 X ^3+1755 .95
X ^2+1680 .56 X + 4416 . 67 ENTER and enter Ain
y2 by pressing 277 .897 ( .99984 ^ X )−66 ENTER .
Enter the composite function (A
o
F)(x)=
A(F(x)) =
y2 ( y1(x) ) in y3.
Press HOME . Enter y3(x) on the entry
line to display the composite function.
WARNING: 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 eraised 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 TURNINGFUNCTIONSINTHEGRAPHINGLISTOFFANDON There are times
when you would like to keep the equations of certain functions in the
Y= list but you do not
wantthemtographortobeshowninthetable. Anyfunctionthatisturnedonwillgraphand
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.

TI89-18 Chapter 1
Copyright © Houghton Mifflin Company. All rights reserved.
On the screen shown to the right, y1 and y3 areturnedon(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 TheTI-89candrawthegraphoftheinverseofa
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
dis the distance traveled in miles.
Press
 F1 (Y = ), clear any previously entered functions, and
enter the function Fin 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 dis 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
for the view you choose with
 F3
(GRAPH).
Press  F1 (Y = ) and enter in y2 your answer to part aof
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.

TI-89 Guide TI89-19
Copyright © Houghton Mifflin Company. All rights reserved.
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 band cof 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 COMPOSITIONPROPERTYOFINVERSEFUNCTIONS This concept, involved in
part dof 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 GRAPHING A PIECEWISE CONTINUOUS FUNCTION Piecewise continuous func-
tions are used throughout the text. You will need to use your calculator to graph and evaluate
outputs of piecewise continuous functions. Several methods
3
canbeusedtodrawthegraphofa
piecewise function. One of these is presented below using the function that appears in Example
6ofSection1.3inCalculus Concepts:
The population of West Virginia from 1985 through 1993 can be modeled by
Pt tt
tt
() thousand people when
thousand people when
=−+≤<
+≤≤
R
S
T
23514 3903667 85 90
77 1098 7 90 93
..
..
where tis 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
letter. Then use
▼ to find what you are looking for and enter it in what you are typing.
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.

TI89-20 Chapter 1
Copyright © Houghton Mifflin Company. All rights reserved.
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 xas 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
yourcalculatorto
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: Style
4
]2[Dot].
Next, set the window. The function Pis 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. Usetheentrylineon
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.