Texas Instruments TI-84 - Plus - Edition Graphing Calculator User manual

TI-83, TI-83 Plus, TI-84 Plus Guide
Guide for Texas Instruments TI-83,
TI-83 Plus, or TI-84 Plus Graphing
Calculator
This Guide is designed to offer step-by-step instruction for using your TI-83, TI-83 Plus, or TI-84
Plus graphing calculator with the fourth edition of Calculus Concepts: An Informal Approach to
the Mathematics of Change. You should utilize the subject index on page 114 for this Guide to
find the location of a specific topic on which you need instruction.
Setup Instructions
Before you begin, check the calculator setup to be certain that the settings described
below are chosen. Whenever you use this Guide, we assume (unless instructed otherwise)
that your calculator settings are as shown in Figures 1, 2, and 3.
• Press MODE and choose the settings shown in Figure 1 for the basic setup.
• Specify the statistical setup with STAT 5 [SetUpEditor] followed by 2ND 1 , 2ND 2
, 2ND 3 , 2ND 4 , 2ND 5 , and 2ND 6. Press ENTER to view the screen in
Figure 2.
• Check the window format by pressing 2ND ZOOM (FORMAT) and choose the
settings shown in Figure 3.
 If you do not have the darkened choices shown in Figure 1 and Figure 3, use the
arrow keys to move the blinking cursor over the setting you want to choose and
press ENTER .
 Press 2ND MODE (QUIT) to return to the home screen.
TI-84 Plus Basic Setup TI-84 Plus Statistical
Setup TI-84 Plus Window Setup
Figure 1 Figure 2 Figure 3
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TI-83, TI-83 Plus, TI-84 Plus Guide
Basic Operation
You should be familiar with the basic operation of your calculator. With your calculator in hand,
go through each of the following.
1. CALCULATING You can type in lengthy expressions; just be certain that you use
parentheses to control the calculator's order of operations. Always enclose in
parentheses any numerator, denominator, quantity to be raised to a power, or
exponent that consists of more than one symbol.
Evaluate 1
415 895
7
*+. Enclose the denominator in
parentheses. It is not necessary to use parentheses around
the fraction 895/7 because there is only one symbol in the
numerator and one symbol in the denominator..
Evaluate (3)
.
4
−−
+
5
81456
. Use (−)for the negative symbol and −
for the subtraction sign. To clear the home screen, press
CLEAR .
NOTE: The numerator, denominator, and quantity to be raised to the 4th power
must be enclosed in parentheses.
Now, evaluate e3*2 ≈40.17. The calculator prints the left
parenthesis when you press 2ND LN (ex). The calculator
assumes that you are inserting a right parenthesis at the end of
what you type unless you insert one elsewhere.
2. USING THE ANS MEMORY Instead of again typing an expression that was just
evaluated, use the answer memory by pressing 2ND (−)(ANS).
Calculate 1
415 895
7
1
*+
F
H
G
GI
K
J
J
−
using this nice shortcut.
Type Ans-1 by pressing 2ND (−)(ANS) x−1.
3. ANSWER DISPLAY When the denominator of a fraction has no more than three
digits, your calculator can provide the answer in the form of a fraction. When an
answer is very large or very small, the calculator displays the result in scientific
notation.
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TI-83, TI-83 Plus, TI-84 Plus Guide
The “to a fraction” key is obtained by pressing MATH 1
[`Frac].
The calculator’s symbol for times 1012 is E12. Thus, 7.945E12
means 7.945*1012 or 7,945,000,000,000.
The result 1.4675E−6means 1.4675*10−6, which is the scientific
notation expression for 0.0000014675.
4. STORING VALUES Sometimes it is beneficial to store numbers or expressions for
later recall. To store a number, type the number, press STO`ALPHA , type the letter
corresponding to the storage location, and then press ENTER . To join several short
commands together, use ALPHA .( :) between the statements. Note that when you
join statements with a colon, only the value of the last statement is shown.
Store 5 in Aand 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 .
• There are 27 memory locations in the calculator: Athrough Zand θ. Whatever
you store in a particular memory location stays there until it is replaced by
something else. Before you use a particular location, 0 is the stored value.
5. ERROR MESSAGES When your input is incorrect, the calculator displays an
error message.
If you have more than one
command on a line without the
commands separated by a colon (:),
an error message results when you
press ENTER .
• When you get an error message, Choose 2 [Goto] to position the cursor to where
the error occurred so that you can correct the mistake or choose 1 [Quit] to
begin a new line on the home screen.
• A common mistake is using the negative symbol (−)instead of the subtraction
sign −or vice-versa. When you choose Goto, the calculator highlights the
position of this error.
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TI-83, TI-83 Plus, TI-84 Plus Guide
If you try to store something to a particular memory location
that is being used for a different type of object, a DATA
TYPE error results. Consult either Troubleshooting the
Calculator in this Guide or your particular calulator
Owner’s Guidebook.
• When you are executing a program that you have transferred to or input in your
calculator, you should always choose the 1: Quit option. Choosing 2: Goto will
call up the program code, and you may inadvertently change the program so that
it will not properly execute.
• Other error messages are referred to throughout this Guide at the points that
they might occur in a specific application.
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TI-83, TI-83 Plus, TI-84 Plus Guide
Chapter 1 Ingredients of Change:
Functions and Linear Models
1.1 Models and Functions
Graphing a function in an appropriate viewing window is one of the many uses for a function
that is entered in the calculator’s graphing list. Because you must enter a function formula on
one line, it is important to use parentheses whenever they are needed.
NOTE: If you are not familiar with the basic operation of the calculator, you should work
through pages A-1 through A-3 of this Guide before proceeding with this material.
1.1.1 ENTERING AN EQUATION IN THE GRAPHING LIST The graphing list contains
space for 10 equations, and the output variables are called by the names Y1, Y2, ..., and Y0.
When you want to graph an equation, you should enter it 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 calculator’s
table function. We illustrate graphing with the equation in Example 3 of Section 1.1: v(t) =
3.5(1.095t).
Press Y= to access the graphing list.
If there are any previously entered equations that you will no
longer use, delete them from the graphing list.
To delete an equation
from the Y= list, position
the cursor on the line with
the equation. Press
CLEAR .
For convenience, we use the first, or Y1, location in the list.
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,T,θ,n ) . Note
that you should press X,T,θ,n for x, not the times sign key, U.
STAT PLOT ERROR:Plot1, Plot2, and Plot3 at the top of the Y= list should not be darkened
when you are graphing an equation and not graphing data points. If any of these is darkened,
use ▲until you are on the darkened plot name. Press ENTER to make the name(s) not dark
(that is, to deselect the plot.
DRAWING A GRAPH As is the case with most applied problems in Calculus Concepts,
the problem description indicates the valid input interval. Consider Example 3 of Section 1.1:
The value of a piece of property between 1985 and 2005 is given by v(t) = 3.5(1.095t)
thousand dollars where tis the number of years since the end of 1985.
The input interval is 1985 (t= 0) to 2005 (t= 20). Before drawing the graph of von this inter-
val, enter the v(t) equation in the Y= list using X as the input variable. We now draw the graph
of the function vfor xbetween 0 and 20.
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Chapter 1
Press 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
because we set their values in the next set of instructions.)
Xmin and Xmax are, respectively, the settings of 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 1 for all applications in this Guide.)
To have the calculator determine the
view for the output, press ZOOM ▲
[ZoomFit]. Now press ENTER to see
the graph of the function v.
Notice that any vertical line drawn on this graph intersects it in only one point, so the graph
does represent a function.
Press WINDOW to see the view set by Zoomfit.
The view has 0 ≤x≤20 and 3.5 ≤y≤21.496... .
(Notice that ZoomFit did not change the x-values that you set
manually.)
MANUALLY CHANGING THE VIEW OF A GRAPH We just saw how to have the
calculator 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, 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 continue using the
function vin Example 3 of Section 1.1, but here assume that you have not yet drawn the graph
of v.
Press 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. (The graph to the right was
drawn with Ymin = −5 and Ymax = 5). Press GRAPH .
Evaluating Outputs on the Graphics Screen: Press TRACE .
Recall that 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. Note that the equation of the function
whose graph you are drawing appears at the top of the screen.
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TI-83, TI-83 Plus, TI-84 Plus Guide
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, 3.5) and (20, 21.496).
Press WINDOW , enter 3.5 for Ymin and 22 for Ymax, and press
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.)
NOTE: Instead of using TRACE with the exact input to evaluate outputs on the graphics
screen, you could use the TABLE function or evaluate the function at 0 and 20 on the home
screen. We next discuss using these features.
TRACING TO ESTIMATE OUTPUTS You can display the coordinates of certain points
on the graph by tracing. Unlike the substitution feature of TRACE that was discussed on the
previous page, 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-values into the equation that is being graphed. We again assume that you
have the function v(x) = 3.5(1.095x) entered in the Y1 location of the Y= list.
With the graph on the screen, press TRACE , press and hold
►to move the trace cursor to the right, and press and hold
◄to move the trace cursor to the left. The equation that you
are tracing appears at the top of the graphing 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 yis approximately 13.9 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.
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Chapter 1
EVALUATING OUTPUTS ON THE HOME SCREEN The input values used in the eval-
uation process are actual values, not estimated values such as those generally obtained by
tracing near a certain value.
We again consider the function v(t) = 3.5(1.095t) .
Using xas the input variable, enter Y1 = 3.5(1.095^X). Return to
the home screen by pressing 2ND MODE (QUIT). Go to the
Y−VARS menu with VARS ►[Y−VARS]. Choose 1: Function
by pressing the number 1 or ENTER .
NOTE: We choose Y1 as the function location most of the time, but you can use any of the ten
available locations. If you do, replace Y1 in the instructions with the location you choose.
Choose 1: Y1 by pressing the number 1
or ENTER . Y1 appears on the home
screen. Press (, type the x-value to be
substituted into the equation, and press
). Then press ENTER .
NOTE: The closing parenthesis on the right is not needed if nothing else follows it.
It is now a simple matter to evaluate the function at other inputs.
For instance, substitute x= 20 into the equation by recalling the
previous entry with 2ND ENTER (ENTRY), change 15 to 20
by pressing ◄◄◄and typing 20, and then press ENTER .
Evaluate Y1 at x= 0 by recalling the previous entry with 2ND
ENTER (ENTRY), change 20 to 0 by pressing ◄◄◄
DEL , and then press ENTER .
EVALUATING OUTPUTS USING THE TABLE Function outputs can be determined by
evaluating on the graphics screen or by evaluating on the home screen. You can also evaluate
functions using the 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) = 3.5(1.095t) at the input t = 15. Even
though you can use any of the function locations, we again choose to use Y1. Press Y= , clear
the function locations, and enter 3.5(1.095^X) in location Y1 of the Y= list.
Choose TABLE SETUP with 2ND WINDOW (TBLSET). To
generate a list of values beginning with 13 such that the table
values differ by 1, enter 13 in the TblStart location, 1in the ∆Tbl
location, and choose AUTO in the Indpnt: and Depend: locations.
(Remember that you “choose” a certain setting by moving the
cursor to that location and pressing ENTER .
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TI-83, TI-83 Plus, TI-84 Plus Guide
Press 2ND GRAPH (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.
If you are interested in evaluating 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.
Return to the table set-up with 2ND
WINDOW (TBLSET), and choose ASK
in the Indpnt: location. Press ENTER .
Press 2ND GRAPH (TABLE) to go to
the table and enter the x-values.
FINDING INPUT VALUES USING THE SOLVER Your calculator solves for the input
values of any equation that is in the form "expression = 0". This means that all terms must be
on one side of the equation and 0 must be on the other side before you enter the equation into
the calculator. The expression can, but does not have to, use xas the input variable.
The calculator offers several methods of solving for input variables. We first illustrate
using the SOLVER. (Solving using graphical methods will be discussed after using the
SOLVER is explored.) You can refer to an equation that you have already entered in the Y=
list or you can enter the equation in the solver.
Return to the home screen with 2ND MODE (QUIT). Access
the solver by pressing MATH 0. If there are no equations
stored in the solver, you will see the screen displayed on the
right −or
−if the solver has been previously used, you will see a screen
similar to the one shown on the right. If this is the case, press
▲and CLEAR to delete the old equation. You should then
have the screen shown in the previous step.
Let’s now use the solver to answer the question in part cof Example 3 in Section 1.1: “When
did the land value reach $20,000?” Because the land value is given by v(t) = 3.5(1.095t)
thousand dollars where tis the number of years after the end of 1985, we are asked to solve the
equation 3.5(1.095t) = 20. We must enter the equation into the solver with a 0 on the left side,
so subtract 20 from each side of 3.5(1.095t) = 20 and exchange the sides to obtain the equation
0 = 3.5(1.095t) – 20 = 0.
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Chapter 1
If you already have Y1 = 3.622(1.093^X) in the graphing list, you
can refer to the function as Y1 in the SOLVER. If not, you can
enter 3.622(1.093^X) instead of Y1 in the eqn: location of the
SOLVER. Press ENTER .
If you need to edit the equation, press ▲until the previous
screen reappears. Edit the equation and then return here. With
the cursor in the Xlocation, enter a guess −say 19. (You could
have also used as a guess*the value that was in the Xlocation
when you accessed the SOLVER.)
*More information on entering a guess appears at the end of this discussion.
CAUTION:You should not change anything in the “bound” location of the SOLVER. The
values in that location are the ones between which the calculator searches1for 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 SOLVER.)
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 ALPHA ENTER (SOLVE).
The answer to the original question is that the land value was
$20,000 about 19.2 years after 1985 – i.e., in the year 2005.
• Notice the black dot that appears next to Xand next to the last line on the above screen.
This is the calculator’s way of telling you that a solution has been found. The bottom line
on the screen that states left −rt = 0 indicates the value found for Xis an exact solution
since both sides of the equation evaluate to the same quantity.
• If a solution continues beyond the edge of the calculator screen, you see “…” to the right
of the value. Be certain that you press and hold ►to scroll to the end of the number. The
value may be given in scientific notation, and the portion that you cannot see determines
the location of the decimal point. (See Basic Operation, #3, in this Guide.)
HOW TO DETERMINE A GUESS TO USE IN THE EQUATION SOLVER What you
use in the solver as a guess tells the calculator 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 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 SOLVER follow. We illustrate these methods using the land value function from Example
3 of Section 1.1 and the equation v(t) = 3.5(1.095t) = 20.
1It 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 your Graphing Calculator Guidebook for details.
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TI-83, TI-83 Plus, TI-84 Plus Guide
1. Enter the function in some location of the graphing list – say
Y1 = 3.5(1.095^X) and draw a graph of the function. Press
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.
Remember that you can use any letter to represent the input in the solver, but to graph a
function, you must use Xas the input variable.
2. Enter the left- and right-hand sides of the equation in two
different locations of the Y= list – say Y1 = 3.5(1.095^X) and
Y2 = 20. With the graph on the screen, press 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 SOLVER.
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 or by using the
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 32 in Section 1.1 of
Calculus Concepts. 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 xsuch 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 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:
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Chapter 1
Press Y= and clear all locations with CLEAR . Enter the func-
tion 3x– 0.8x2+ 4 – 2.3 in Y1. You can enter x2with X,T,θ,n
x2or enter it with X,T,θ,n ^2. Remember to use −, not
(−),for the subtraction signs.
Draw the graph of Y1 with ZOOM 4 [Zdecimal] or ZOOM 6
[ZStandard]. If you use the former, press WINDOW and reset
Xmax to 6 and Ymax to 5 to get a better view the graph. (If you
reset the window, press GRAPH to draw the graph.)
To graphically find an x-intercept, i.e., a value of xat which the
graph crosses the horizontal axis, press 2ND TRACE (CALC)
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 left 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 right bound for the x-intercept.
For your “guess”, press ◄to move the cursor near to where
the graph crosses the horizontal axis. 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
occurs 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.)
INTERSECTION METHOD for solving the equation f(x) = c:
Press 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.
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TI-83, TI-83 Plus, TI-84 Plus Guide
Draw the graphs with ZOOM 4 [ZDecimal] or ZOOM 6
[ZStandard]. If you use the former, press WINDOW and reset
Xmax to 6 and Ymax to 8 to get a better view the graph. (If you
reset the window, press GRAPH to draw the graph.)
To locate where Y1 = Y2, press
2ND TRACE (CALC) 5 [intersect].
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 equation of each function appears on the top left
of the screen as the cursor moves from the curve to the line.
Next, supply a guess for the point of intersection. Use ◄to
move the cursor near the intersection point you want to find – in
this case, the leftmost point. 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
occurs where X= 4.25.
SUMMARY OF ESTIMATING AND SOLVING 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 entered in the Y= list (p. A6)
• Use close values obtained from the TABLE (p. A8)
When you are asked to find or determine an output or an input value, you should:
• Evaluate an output on the graphics screen (p. A5)
• Evaluate an output on the home screen (p. A7)
• Evaluate an output value using the table (p. A8 )
• Find an input using the solver (p. A8)
• Find an input value from the graphic screen (using the x-intercept
method or the intersection method) (p. A11)
1.1.2 FINDING THE SUM, DIFFERENCE, PRODUCT, QUOTIENT, OR COMPOSITE
FUNCTION 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.
Suppose that a function fhas been entered in Y1 and a function ghas been entered in Y2.
Your calculator will evaluate and graph these constructed functions:
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Chapter 1
Enter Y1 + Y2 in Y3 to obtain the sum function (f + g)(x)= f(x) + g(x).
Enter Y1 – Y2 in Y4 to obtain the difference function (f – g)(x)= f(x) – g(x).
Enter Y1*Y2 in Y5 to obtain the product function (f
⋅
g)(x)= f(x)
⋅ g(x).
Enter Y1/Y2 in Y6 to obtain the quotient function (f ÷ g)(x)= fx
gx
()
()
.
Enter Y1(Y2) in Y7 to obtain the composite function (f o g)(x)= f(g(x)).
The functions can be entered in any location in the Y= list. Although the calculator will not
give an algebraic formula for a constructed function, you can check your final answer by
evaluating your constructed function and the calculator-constructed function at several
different points to see if they yield the same outputs.
FINDING A DIFFERENCE FUNCTION We illustrate this technique with the functions for
Cost = C(x) = - 0.32x2 + 6x+ 1030 dollars for the total cost of milk production on the xth day of
last month (found on p. 11 of Calculus Concepts) and Revenue = R(x) =
[]
(
)
0.02 1.90 1.5 0.5 0.8x x⎡ +⋅+
⎣⎤ ⎦
thousand dollars revenue from the sale of milk on the xth day of
last month (found on p. 13 of Calculus Concepts). To construct a profit function, P(x), we must
subtract C(x) from R(x). Because the output of R(x) is thousand dollars, we divide C(x) by by
1000 so that our output units will be thousand dollars. ()
() () 1000
Cx
Px Rx=− thousand dollars on
the xth day of last month.
Press Y= . Clear previously entered equations with CLEAR .
Enter Rin Y1 by pressing ( . 02 X,T, ,n 1. 90 )+
* ( 1. 5 . 5 ( . 8 ^ X,T, ,n ) )+ENTER and enter Cin Y2 by
pressing (-) .32 X,T,θ,n x2
+
6 X,T,θ,n
+
1030
ENTER
The difference function, the profit P(x) = ()
() 1000
Cx
Rx−=
Y1 – Y2/1000, is entered in Y3 with VARS ►1 [Function] 1
[Y1] −VARS ►1 [Function] 2 [Y2]
÷
1000.
To find the profit in on the 2nd day of last month, evaluate Y3
when X = 2. Return to the home screen with 2ND MODE
(QUIT). Press VARS ►1 [Function] 3 [Y3] ( 2 ) ENTER
to see the result. We find that the profit on the 2nd day of last
month was approximately $2.49 thousand.
• You can evaluate on the home screen, graphics screen, or in the table. We choose to use
the home screen, but you should choose the method you prefer.
FINDING A PRODUCT FUNCTION We illustrate this technique with the functions
given in Example 4 of Section 1.1 of Calculus Concepts: Milk price = M(x) = 0.02x+ 1.90
dollars per gallon on the xth day of last month and milk sales = S(x) = 1.5 + 0.5(0.8x) gallons
of milk sold on the xth day of last month.
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TI-83, TI-83 Plus, TI-84 Plus Guide
Press Y= , and clear each previously entered equation with
CLEAR . Enter Min Y1 by pressing .02 X,T,θ,n +1 .
90 ENTER and input Sin Y2 by pressing 1.5
+
.5 ( .8
^X,T,θ,n ) ENTER .
Enter the product function R(x) = M(x) ⋅S(x) = Y1⋅Y2 in Y3
with the keystrokes VARS ►1 [Function] 1 [Y1] VARS
►1 [Function] 2 [Y2].
NOTE: You do not have to, but you can, use Ubetween Y1 and Y2 to indicate a product
function. You cannot use parentheses to indicate the product function because the calculator
will think that you are entering Y3 as a composite function.
To find milk sales on the 5th day of last month, evaluate Y3 at
X= 5. We choose to do this on the home screen. Return to the
home screen with 2ND MODE (QUIT). Press VARS ►1
[Function] 3 [Y3] ( 5 ) ENTER to see the result. We find that
milk sales on the 5th day of last month were R(5) ≈$3.328
thousand.
1.1.3 USING A CONSTRUCTED FUNCTION We illustrate solving a constructed function for
a particular output using the functions from Example 5 in Section 1.1. Suppose that the
monthly total cost for the production of ggallons of milk can be modeled as K(g) = 20.1 +
0.2gthousand dollars, and that during the same time period revenue can be
modeled as T(g) = 2.2gdollars for g gallons of milk. We want to know how much milk need
to be produced/sold in order for the dairy company to break even.
03g≤≤0
Press Y= , and clear each previously entered equation with
CLEAR . Enter Tin Y1 by pressing 2.2X,T,θ,n ENTER
and enter Kin Y2 by pressing 20100 . 2 X,T,θ,n+ENTER .
To graph K(g) and T(g) on the same axes, we need to use input
values from 0 to at least 10,000 since K is never less than
20,100. Set the window from Xmin = 0 to Xmax = 15,000 .
Press ZOOM ▲[ZoomFit]. Now press ENTER to see the
graph of the function v.
Copyright © Houghton Mifflin Company. All rights reserved. 15

Chapter 1
To locate where Y1 = Y2, press 2ND TRACE (CALC) 5
[intersect]. 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. Next, supply a guess for
the point of intersection. Use ◄to move the cursor near the
intersection point you want to find – in this case, the leftmost
point. Press ENTER . The break even point occurs when 10,050
gallons are sold and revenue is $22,110.
CHECKING YOUR ANSWER FOR A COMPOSITE FUNCTION We illustrate this
technique with the functions that are given on page 16 of Section 1.1 of Calculus Concepts:
altitude = F(t) = −2t3+ 1.8t2+ 1.7t+ 4.4 thousand feet above sea level where tis the time into
flight in minutes and air temperature = A(F) = 279(0.85F) – 66 degrees Fahrenheit where Fis
the number of thousand feet above sea level. Remember that when you enter functions in the
Y= list, you must use Xas the input variable.
Press Y= , and clear each previously entered equation with
CLEAR . Enter Fin Y1 by pressing (−).2 X,T,θ,n ^3
+1 .8 X,T,θ,n x2+ 1 .7 X,T,θ,n + 4 .4ENTER
and enter Ain Y2 by pressing 279 ( .85 ^ X,T,θ,n ) −66
ENTER .
Enter the composite function (AoF)(x) = A(F(x)) = Y2(Y1) in Y3
with the keystrokes VARS ►1 [Function] 2 [Y2] ( VARS
►1 [Function] 1 [Y1] ) ENTER .
Next, enter your algebraic answer for the composite function in
Y4. Be certain that you enclose the exponent in Y4 (the func-
tion in Y1) in parentheses.
(The composite function in the text is the one that appears to the
right, but you should enter the function that you have found for
the composite function.)
We now wish to check that the algebraic answer for the composite function is the same as the
calculator’s composite function by evaluating both functions at several different input values.
You can do these evaluations on the home screen, but as seen below, using the table is more
convenient.
TURNING FUNCTIONS OFF AND ON IN THE GRAPHING LIST Note in the prior
illustration that we are interested in the output values for only Y3 and Y4. However, the table
will list values for all functions that are turned on. (A function is turned on when the equals
sign in its location in the graphing list is darkened.) We now wish to turn off Y1 and Y2.
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TI-83, TI-83 Plus, TI-84 Plus Guide
Press Y= and, using one of the arrow keys, move the cursor
until it covers the darkened = in Y1. Press ENTER . Then,
press ▼until the cursor covers the darkened = in Y2. Press
ENTER . Y1 and Y2 are now turned off.
A function is turned off when the equals sign in its location in the graphing list is not dark. To
turn a function back on, simply reverse the above process to make the equal sign for the func-
tion dark. When you draw a graph, the TI-83 graphs of all functions that are turned on. You
may at times wish to keep certain functions entered in the graphing list but not have them graph
and not have their values shown in the table. Such is the case in this illustration.
We now return to checking to see that Y3 and Y4 represent the same function.
Choose the ASK setting in the table setup so that you can check
several different values for both Y3 and Y4. Recall that you
access the table setup with 2ND WINDOW (TBLSET). Move
the cursor to ASK in the Indpnt: location and press ENTER .
Press 2ND GRAPH (TABLE), type in the x-value(s) at which
the function is to be evaluated, and press ENTER after each one.
We see that because all these outputs are the same for each
function, you can be fairly sure that your answer is correct.
Why does ERROR appear in the table when x= 57? Look at the value when x= 20; it is very
large! The computational limits of the calculator have been exceeded when x= 57. This is an
OVERFLOW ERROR.
.
1.2 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.
CAUTION: Be very careful when you enter data in your calculator because your model and
all of your results depend on the values that you enter! Double check your entries.
1.2.1 ENTERING DATA We illustrate data entry using the values in Table 1.11 in Section 1.2 of
Calculus Concepts:
Year 1999 2000 2001 2002 2003 2004
Tax (dollars) 2532 3073 3614 4155 4696 5237
Press STAT 1 [EDIT] to access the 6 lists that hold data. You see only the first 3 lists, (L1, L2,
and L3) but you can access the other 3 lists (L4, L5, and L6) by having the cursor on the list
name and pressing ►several times. If you do not see these list names, return to the statistical
setup instructions on page A-1 of this Guide.
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Chapter 1
In this text, we usually use list L1 for the input data and list L2 for the output data. If there
are any data values already in your lists, first delete any “old” data using the following
procedure. (If your lists are clear, skip these instructions.)
DELETING 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 lists with STAT 1 [EDIT]. (You probably have
different values in your lists if you are deleting “old” data.) Use
▲to move the cursor over the name L1. Press CLEAR
ENTER .
Use ►▲to move the cursor over
the name L2. Press CLEAR ENTER .
Repeat this procedure to clear old data
from any of the other lists you will use.
(Now you are ready to enter new data.)
Position the cursor in the first location in list L1. Enter the input
data into list L1 by typing the years from top to bottom in the L1
column, pressing ENTER after each entry.
After typing the sixth input value, 2004, use ►to cause the
cursor go to the top of list L2. Enter the output data in list L2 by
typing the entries from top to bottom in the L2 column, pressing
ENTER after each tax value.
EDITING DATA If you incorrectly type a data value, use the cursor keys (i.e., the arrow
keys ►, ◄, ▲, and/or ▼)to darken the value you wish to correct and then type the
correct value. Press ENTER .
• To insert a data value, put the cursor over the value that will be directly below the one you
will insert, and press 2ND DEL (INS). The values in the list below the insertion point
move down one location and a 0 is filled in at the insertion point. Type the data value to
be inserted over the 0 and press ENTER . The 0 is replaced with the new value.
• To delete a single data value, move the cursor over the value you wish to delete and press
DEL .The values in the list below the deleted value move up one location.
FINDING FIRST DIFFERENCES When the input values are evenly spaced, you can use
program DIFF to compute first differences in the output values. Program DIFF available to
download at the Calculus Concepts web site:
http://college.hmco.com/mathematics/latorre/calculus_concepts/3e/students/programs.html
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TI-83, TI-83 Plus, TI-84 Plus Guide
Have the data given in Table 1.1 in Section 1.2 of Calculus
Concepts entered in your calculator.
Exit the list menu with 2ND MODE (QUIT).
To run the program, press PRGM followed by the number that
is to the left of the DIFF program location, and press ENTER .
The message on the right appears on your screen. (Note: We
use the information in lists L4 and L5 when considering
exponential or quadratic models.)
Press STAT 1 [EDIT] to view the first differences in list L3.
The first differences in L3 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 (entered in L1) 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 try to use program DIFF with input data that
are not evenly spaced, the message INPUT VALUES NOT EVENLY SPACED appears and the
program stops.
SCATTER PLOT SETUP The first time that you draw a graph of data, you need to set your
calculator to draw the type of graph you want to see. Once you do this, you should not need to
do this set up again (unless for some reason the settings are changed). If you always put input
data in list L1 and output data in list L2, you can turn the scatter plots off and on from the Y=
screen rather than the STAT PLOTS screen after you perform this initial setup.
Press 2ND Y= (STAT PLOT) to display the STAT PLOTS
screen. (Your screen may not look exactly like this one.)
On the STAT PLOTS screen, press 1 to display the Plot1 screen,
press ENTER to turn Plot1 on, and then highlight each option
shown on the right, pressing ENTER to choose it. (You can
choose any of the 3 marks at the bottom of the screen.)
Press Y= and notice that Plot1 at the top of the screen is dark-
ened. This tells you that Plot1 is turned on and ready to graph
data. Press 2ND MODE (QUIT) to return to the home screen.
• A scatter plot is turned on when its name on the Y= screen is darkened. To turn Plot1 off,
use ▲to move the cursor to the Plot1 location, and press ENTER .Reverse the process
to turn Plot1 back on.
Copyright © Houghton Mifflin Company. All rights reserved. 19

Chapter 1
• Lists can be named and stored in the calculator’s memory for later recall and use. Refer to
p.23 of this Guide for instructions on storing data lists and later recalling them for use.
CAUTION: Any time that you enter the name of a numbered list (for instance: L1, L2, and
so forth), you should use the calculator symbol for the name, not a name that you type with
the alphabetic and numeric keys. For example, to enter the list name L1 on the calculator,
press 2ND 1 (L1). If you instead entered L1 with the keystrokes ALPHA ) (L) 1, you
would create a new list called L1 that the calculator would not recognize as the list containing
data that you entered while in the STAT [EDIT] mode.
The screen to the right contains three entries marked L1. The
first is the statistical list L1 that contains input data. See what is
in your L1 list by pressing 2ND 1 (L1) ENTER .
The second entry is obtained by pressing ALPHA )(L) 1
ENTER .Whatever constant is stored in memory location L
multiplied by 1 is what you see printed on the screen.
Your list L1 may not
contain the same values as
the one shown above.
DRAWING 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 1.2 of Calculus Concepts.
Year 1999 2000 2001 2002 2003 2004
Tax (in dollars) 2541 3081 3615 4157 4703 5242
Press Y= . If any entered function is no longer needed, clear it by moving the cursor to its
location and pressing CLEAR . If you want the function(s) to remain but not graph when you
draw the scatter plot, refer to page 16 of this Guide for instructions on how to turn function(s)
off. Also be sure that Plot 1 on the top left of the Y= screen is darkened.
Press STAT 1 [EDIT]. Using the table given above, enter the
year data in L1 and the modified tax data in L2.
(You can either leave values in the other lists or clear them.)
Press ZOOM 9 [ZoomStat] to have the calculator set an auto-
scaled view of the data and draw the scatter plot.
Note that ZoomStat does not reset the x- and y-axis tick marks.
You should do this manually with the WINDOW settings if you
want different spacing between the 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
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20