When you include scientific- or engineering-notation numbers in an expression, the TI-86 does not necessarily display answers in scientific or engineering notation. The mode settings (page 34) and the size of the number determine the notation of displayed answers. Entering Complex Numbers On the TI-86, the complex number a+bi is entered as (a,b) in rectangular complex-number form or as (rq ) in polar complex-number form. For more information about complex numbers, read Chapter 4.
Entering Other Characters
This is the 2nd key This is the ALPHA key
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The 2nd Key The - key is yellow. When you press -, the cursor becomes (the 2nd cursor). When you press the next key, the yellow character, abbreviation, or word printed above that key is activated, instead of the keys primary function. The ALPHA Key The 1 key is blue. When you press 1, the cursor becomes (the uppercase ALPHA cursor). When you press the next key, the blue uppercase character printed above that key is pasted to the cursor location. When you press - n, the cursor becomes (the lowercase alpha cursor). When you press the next key, the lowercase version of the blue character is pasted to the cursor location.

- returns the STAT menu

To enter a space within text, press 1. Spaces are not valid within variable names. For convenience, you can press 2 instead of n x to enter the commonly used x variable.

1 X returns an X

- n X returns an x
The Name= prompt and store symbol () set ALPHA-lock automatically.
ALPHA-lock and alpha-lock To enter more than one uppercase or lowercase alpha character consecutively, set ALPHAlock (for uppercase letters) or alpha-lock (for lowercase letters). To set ALPHA-lock when the entry cursor is displayed, press 1 1. To cancel ALPHA-lock, press 1. To switch from ALPHA-lock to alpha-lock, press - n. To set alpha-lock when the entry cursor is displayed, press - n 1. To cancel alpha-lock, press 1 1.
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To switch from alpha-lock to ALPHA-lock, press 1.

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Using Functions in Expressions A function returns a value. Some examples of functions are , L , + , , and log. To use functions, you usually must enter one or more valid arguments.
In this guidebook, optional arguments are shown in brackets ( and ). Do not include these brackets when you enter the arguments.
When this guidebook describes the syntax of a function or instruction, each argument is in italics. For example: sin angle. Press = to enter sin, and then enter a valid angle measurement (or an expression that resolves to angle). For functions or instructions with more than one argument, you must separate each argument from the other with a comma. Some functions require the arguments to be in parentheses. When you are unsure of the evaluation order, use parentheses to clarify a functions place within an expression. Using an Instruction An instruction initiates an action. For example, ClDrw is an instruction that, when executed, clears all drawn elements from a graph. You cannot use an instruction in an expression. Generally, the first letter of each instruction name is uppercase on the TI-86. Some instructions take more than one argument, as indicated by an open parenthesis ( ( ) at the end of the name. For example, Circl( requires three arguments, Circl(x,y,radius). Entering Functions, Instructions, and Operators You can enter a function, instruction, or operator in any of three ways (log 45, for example). Paste it to the cursor location from the keyboard or a menu (< 45). Paste it to the cursor location from the CATALOG (- w & L & & b 45). Enter it letter by letter ( - n 1 L O G 45). As you can see in the example, using the built-in function or instruction typically is easier.
The A to Z Reference describes all TI-86 functions and instructions, including their required and optional arguments.
In the CATALOG, to move to the first item beginning with a letter, press that letter (as in L in the example).
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When you select a function, instruction, or operator, a symbol comprising one or more characters is pasted to the cursor location. Once the symbol is pasted to the cursor location, you can edit individual characters. For example, assume that you pressed - w / / * & & b to paste yMin to the cursor location as part of an expression. Then you realized you wanted xMin. Instead of pressing nine keys to select xMin, you can simply press ! ! ! ! 2. Entering Consecutive Entries To enter two or more expressions or instructions consecutively, separate each from the next with a colon (- ). When you press b, the TI-86 executes each entry from left to right and displays the result of the last expression or instruction. The entire group entry is stored in last entry (page 28). The Busy Indicator When the TI-86 is calculating or graphing, a moving vertical line is displayed as the busy indicator in the top-right corner of the screen. When you pause a graph or a program, the busy indicator is replaced by the pause indicator, a moving vertical dotted line. Interrupting a Calculation or Graph To interrupt a calculation or graph in progress, press ^. When you interrupt a calculation, the ERROR 06 BREAK message and menu are displayed. To return to the home screen, select QUIT (press *). To go to the beginning of the expression, select GOTO (press &). Press b to recalculate the expression.

In the example, the symbol indicates that the value before it is to be stored to the variable after it (Chapter 2). To paste to the screen, press X.
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Chapter 5: Function Graphing introduces graphing.
When you interrupt a graph, a partial graph and the GRAPH menu are displayed. To return to the home screen, press : : or any non-graphing key. To restart graphing, select an instruction that displays the graph.

Diagnosing an Error

If a syntax error occurs within a stored equation during program execution, select GOTO to return to the equation editor, not to the program (Chapter 5).
When the TI-86 detects an error, it returns an error message, such as ERROR 04 DOMAIN or ERROR 07 SYNTAX. The Appendix describes each error type and possible reasons for the error. If you select QUIT (or press - l or : :), the home screen is displayed. If you select GOTO, the previous screen is displayed with the cursor on or near the error. Correcting an Error
Note the error type (ERROR ## errorType). Select GOTO, if available. The previous screen is displayed with the cursor on or near the error. Determine the cause for the error. If you cannot, refer to the Appendix for possible causes. Correct the error and continue.
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Reusing Previous Entries and the Last Answer
Retrieving the Last Entry When you press b on the home screen to evaluate an expression or to execute an instruction, the entire expression or instruction is placed in a storage area called ENTRY (last entry). When you turn off the TI-86, ENTRY is retained in memory. To retrieve the last entry, press -. The current line is cleared and the entry is pasted to the line. Retrieving and Editing the Last Entry
On the home screen, retrieve the previous entry. Edit the retrieved entry. Re-execute the edited entry. - ! ! ! ! ! 32 b
Retrieving Previous Entries The TI-86 retains as many previous entries as possible in ENTRY, up to a capacity of 128 bytes. To scroll from the newest to the older previous entries stored to ENTRY, repeat -. If you press - after displaying the oldest stored entry, the newest stored entry is displayed again; continuing to press - repeats the order.
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Selecting Deltal from the menu pastes Deltalst( to the cursor location.
Deltalst(list) Sortx ListName,ListName,

frequencyListName

For Sortx and Sorty, both lists must have the same number of elements.
Sorty xListName,ListName,
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Select(xListName, yListName)
Selecting SetLE from the menu pastes SetLEdit to the cursor location. You can create new list names as SetLEdit arguments.
Selects one or more specific data points from a scatter plot or xyLine plot (only), then stores the selected data points to xListName and yListName (Chapter 14) Sets up the list editor; SetLEdit with one to 20 ListNames loads them in the specified order; SetLEdit with no arguments removes all current list names from the list editor and enters the default lists xStat, yStat, and fStat to columns 1, 2, and 3 Attaches formula to listName; formula resolves to a list, which is dynamically stored and updated in listName (page 162)
SetLEdit column1ListName, column2ListName,.,

column20ListName

Form("formula",listName)
Using Mathematical Functions with Lists
You can use a list as a single argument for many TI-86 functions; the result is a list. The function must be valid for every element in the list; however, when graphing, undefined points do not result in an error. When you use lists for two or more arguments in the same function, all lists must have the same number of elements (equal dimension). Here are some examples of a list as a single argument.
{1,2,3}+10 returns {13} {5,10,15}{2,4,6} returns {90} 3+{1,7,(2,1)} returns {(4,0) (10,0) (5,1)} {4,16,36,64} returns {6 8} sin {7,5} returns {.656986598719 L.958924274663} {1,15,36}<19 returns {0}
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Attaching a Formula to a List Name
You cannot edit an element of a list created from an attached formula unless you first detach the formula from the list name. When you include more than one list name in an attached formula, each list must have the same dimension.
You can attach a formula to a list name so that the formula generates a list that is stored and dynamically updated in the list name. When you edit an element of a list that is referenced in the formula, the corresponding element in the list to which the formula is attached is updated. When you edit the formula itself, all elements in the list to which the formula is attached are updated. To attach a formula to a list name on the home screen or in the program editor, the syntax is: Form("formula",listName) When you enter a new list name as the second argument for Form( , the list name is created and stored in the LIST NAMES menu and VARS LIST screen upon execution.

*LCust(item#,"title" ,item#,"title",.)
A command line that is longer than the screen is wide automatically continues at the beginning of the next line.
Entering a Command Line You can enter on a command line any instruction or expression that you could execute on the home screen. In the program editor, each new command line begins with a colon. To enter more than one instruction or expression on a single command line, separate each with a colon. To move the cursor down to the next new command line, press b. You cannot move to the next new command line by pressing #. However, you can return to existing command lines to edit them by pressing $. Menus and Screens in the Program Editor TI-86 menus and screens may be altered when displayed in the program editor. Menu items that are invalid for a program are omitted from menus. Menus that are not valid in a program, such as the LINK menu or MEM menu, are not displayed at all. When you select a setting from a screen such as the mode screen or graph format screen, the setting you select is pasted to the cursor location on the command line.
All CATALOG items are valid in the program editor.
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Variables to which you typically store values from an editor, such as the window variables, become items on program-only menus, such as the GRAPH WIND menu. When you select them, they are pasted to the cursor location on the command line.

Running a Program

Paste the program name to the home screen. Either select it from the PRGM NAMES menu (8 &) or enter individual characters.
To resume the program after a pause, press b.
Press b. The program begins to run.
Each result updates the last-answer variable Ans (Chapter 1). The TI-86 reports errors as the program runs. Commands executed during a program do not update the previous-entry storage area ENTRY (Chapter 1). The example program below is shown as it would appear on a TI-86 screen. The program: Creates a table by evaluating a function, its first derivative, and its second derivative at intervals in the graphing window Displays the graph of the function and its derivatives in three different graph styles, activates the trace cursor, and pauses to allow you to trace the function
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PROGRAM:FUNCTABL :Func:Fix 2:FnOff:PlO ff :y1=.6 x cos x :ClLCD :Eq4St(y1,STRING) :Outpt(1,1,"y1=") :Outpt(1,4,STRING) :Outpt(8,1,"PRESS ENT ER") :Pause :ClLCD :y2=der1(y1,x,x) :y3=der2(y1,x,x) :DispT :GrStl(1,1):GrStl(2,2 ):GrStl(3,7) :2xRes :ZTrig :Trace

{1,5,9}{1,L6,9} b {0} 2+23+2 b 2+(23)+2 b [1,2][3N2,L1+3] b "A""a" b 0 1

Not equal to:

items nPr number Returns the number of permutations of items (n) taken number (r) at a time. Both arguments must be real nonnegative integers. integer

5 nPr 2 b

In Dec number base mode: 10 b 10+10 b 8 18
Designates a real integer as octal, regardless of the number base mode setting.
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In Oct number base mode:

Sets octal number base mode. Results are displayed with the suffix. In any number base mode, you can designate an appropriate value as binary, decimal, hexadecimal, or octal by using the , , , or designator, respectively, from the BASE TYPE menu. number 4Oct list 4Oct matrix 4Oct vector 4Oct Returns the octal equivalent of the real or complex argument.
In Dec number base mode: 28 b Ans4Oct b 16 20

{7,8,9,10}4Oct b {11 12}

OneVar
STAT CALC menu (OneVa shows on menu)
Performs one-variable statistical analysis using real data points in xList and frequencies in frequencyList. The values used for xList and frequencyList are stored automatically to built-in variables xStat and fStat, respectively.

OneVar xList

{0,1,2,3,4,5,6}XL b {6} OneVar XL b
Scroll down to see more results.
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Uses xStat and fStat for xList and frequencyList. These built-in variables must contain valid data of the same dimension; otherwise, an error occurs.
integerA or integerB Compares two real integers bit by bit. Internally, both integers are converted to binary. When corresponding bits are compared, the result is 1 if either bit is 1; the result is 0 only if both bits are 0. The returned value is the sum of the bit results. For example, 78 or 23 = 95. 78 = = = 95 You can enter real numbers instead of integers, but they are truncated automatically before the comparison.

Deselects the specified stat plot numbers.

PlOff b Done

Deselects all stat plot numbers.

PlOn [1,2,3]

PlOn 2,3 b
Selects the specified stat plot numbers, in addition to any plot numbers that are already selected.

PlOn b Done

Selects all stat plot numbers.
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Plot1( Plot2( Plot3(

STAT PLOT menu
The syntax and descriptions to the right refer to Plot1(, but they apply as well to Plot2( and Plot3(.
Scatter plot Plot1(1,xListName,yListName,mark) Plot1(1,xListName,yListName) Defines and selects the plot using real data pairs in xListName and yListName. The optional mark specifies the character used to plot the points. If you omit mark, a box is used. mark:
1 = box () 2 = cross (+) 3 = dot ()
{L9,L6,L4,L1,2,5,7,10}L1 b {L9 L6 L4 L1 {L7,L6,L2,1,3,6,7,9}L2 b {L7 L6 L7 9} Plot1(1,L1,L2) b Done ZStd b
xyLine plot Plot1(2,xListName,yListName,mark) Plot1(2,xListName,yListName) Modified box plot Plot1(3,xListName,1 or frequencyListName,mark) Plot1(3,xListName,1 or frequencyListName) Plot1(3,xListName) Defines and selects the plot using real data points in xListName with the specified frequencies. If you omit 1 or frequencyListName, frequencies of 1 are used. Histogram Plot1(4,xListName,1 or frequencyListName) Plot1(4,xListName) Box plot Plot1(5,xListName,1 or frequencyListName) Plot1(5,xListName)
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Sets polar graphing mode. complexNumber 4Pol Displays complexNumber in polar form (magnitudeangle), regardless of the complex number mode. list 4Pol matrix 4Pol vector 4Pol Returns a list, matrix, or vector in which each element of the argument is displayed in polar form.
In RectC complex number mode: L2 b Ans4Pol b (0,1.41421356237) (1.414213562371.570 {1,L2} b {(1,0) (0,1.141421356 Ans4Pol b {(10) (1.4142135623

PolarC

In PolarC complex number mode: L2 b (1.414213562371.570
Sets polar complex number mode (magnitudeangle). magnitudeangle Used to enter complex numbers in polar form. The angle is interpreted according to the current angle mode.

Polar complex:

In Radian angle mode and PolarC complex number mode: (1,2)+(3p/4) b (5.16990542093.9226

"The answer is:"STR b The answer is: sub(STR,5,6) b answer

Subtraction: N

6N2 b 10NL4.5 b {10,9,8}N4 b In RectC complex number mode:

4 14.5 {4}

{8,1,(5,2)}N3 b {(5,0) (L2,0) (2,2)}
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listA N listB matrixA N matrixB vectorA N vectorB Returns a list, matrix, or vector that is the result of each element in the second argument subtracted from the corresponding element in the first argument. The two real or complex arguments must have the same dimension.

{5,7,9}N{4,5,6} b {3}

[[5,7,9][11,13,15]]N[[4,5,6][7,8, 9]] b [[3] [6]] [5,7,9]N[1,2,3] b [6]
MATH MISC menu LIST OPS menu

sum list

sum {1,2,4,8} b sum {2,7,L8,0} b In Radian angle mode: tan p/4 b tan (p/4) b tan 45 b In Degree angle mode: tan 45 b tan (p/4) r b
Returns the sum of all real or complex elements in list.
tan angle or tan (expression)
Returns the tangent of angle or expression, which can be real or complex. An angle is interpreted as degrees or radians according to the current angle mode. In any angle mode, you can designate an angle as degrees or radians by using the or r designator, respectively, from the MATH ANGLE menu.

tan list

In Degree angle mode: tan {0,45,60} b {1.73205080757}
Returns a list in which each element is the tangent of the corresponding element in list.
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tan L1

tanL1 number or tanL1 (expression)
In Radian angle mode: tanL1.5 b In Degree angle mode: tanLb 45 In Radian angle mode: tanL1 {0,.2,.5} b {0.19739555985.463 tanh 1.2 b.833654607012.463647609001
Returns the arctangent of number or expression, which can be real or complex.

tanL1 list

Returns a list in which each element is the arctangent of the corresponding element in list.
tanh number or tanh (expression)
Returns the hyperbolic tangent of number or expression, which can be real or complex.

tanh list

Returns a list in which each element is the hyperbolic tangent of the corresponding element in list.
tanh {0,1.2} b {0.833654607012}

tanh L1

tanh L1 number or tanh L1(expression)

tanhLb

Sets the window variable values to produce square pixels where @x=@y, and then updates the graph screen. The center of the current graph (not necessarily the axes intersection) becomes the center of the new graph. In other types of zooms, squares may look like rectangles and circles may look like ovals. Use ZSqr for a more accurate shape.
y1=(8 2Nx 2):y2=Ly1 b ZStd b

ZSqr b

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Sets the window variables to the standard default values, and then updates the graph screen.
Func graphing mode: xMin=L10 xMax=10 xScl=1 yMin=L10 yMax=10 yScl=1
Pol graphing mode: qMin=0 xMin=L10 yMin=L10 qMax=6.28318530718 (2p) xMax=10 yMax=10 qStep=.130899693899 (p/24) xScl=1 yScl=1 Param graphing mode: tMin=0 xMin=L10 yMin=L10 tMax=6.28318530718 (2p) xMax=10 yMax=10 tStep=.130899693899 (p/24) xScl=1 yScl=1 DifEq graphing mode: tMin=0 xMin=L10 yMin=L10 tMax=6.28318530718 (2p) xMax=10 yMax=10 tStep=.130899693899 (p/24) xScl=1 yScl=1 tPlot=0 difTol=.001
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Sets the window variables to preset values appropriate for plotting trig functions in Radian angle mode (@x=p/24), and then updates the graph screen.
xMin=L8.24668071567 xMax=8.24668071567 xScl=1.5707963267949 (p/2) yMin=L4 yMax=4 yScl=1

y1=sin x b ZStd b

ZTrig b
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TI-86 Menu Map.. 380 Handling a Difficulty... 392 Error Conditions... 393 Equation Operating System (EOS).. 397 TOL (The Tolerance Editor) - ). 398 Computational Accuracy.. 399 Support and Service Information.. 400 Warranty Information.. 402
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TI-86 Menu Map

This section presents the TI-86 menus as they appear on the TI-86 keyboard, starting at the top. If a menu has items that display other menus, the other menus follow directly below the main menu. In the program editor, the appearance of some menus changes slightly. The menu map omits user-created-name menus, such as the LIST NAMES and CONS USER menus.

BASE - (Hexadecimal) menu, 67 BASE BIT menu, 69 BASE BOOL (Boolean) menu, 68 BASE CONV (Conversion) menu, 68 BASE menu, 66 BASE TYPE menu, 67 base type symbol, 67 batteries, 2, 16-18 battery compartment, 16 BCKUP (memory backup), 237 Bin (binary), 35, 272 4Bin (to binary), 68, 272 binary integer, 271 binary number base, 35, 66 Boolean operators, 68, 268, 325, 328, 370 bound={L1E99,1E99}, 204 bounds, 204 BOX (GRAPH ZOOM menu), 14, 92, 93 Box (stat plot), 272 BOX (ZOOM menu), 208 break (program), 222 BREAK menu, 26
CALC (Calculus) menu, 54 calculating derivatives, 7 calculation interrupting, 26 calculus functions, 54 CATALOG, 25, 38 Quick-Find Locator, 262 CATLG (CATALOG), 43 CATLG-VARS (CATALOG Variables) menu, 43 changing TI-86 settings, 39 CHAR (Character) menu, 45 CHAR GREEK menu, 46 CHAR INTL (International) menu, 46 CHAR MISC (Miscellaneous) menu, 46 characters, 19 alpha, 22 blue, 21, 22 case, 22
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contrast adjusting, 2, 18 CONV (Conversions) menu, 62 CONV AREA menu, 63 CONV ENRGY (Energy) menu, 64 CONV FORCE menu, 64 CONV LNGTH (Length) menu, 63 CONV MASS menu, 64 CONV POWER menu, 64 CONV PRESS (Pressure) menu, 64 CONV SPEED menu, 64 CONV TEMP (Temperature) menu, 8, 63 CONV TIME menu, 63 CONV VOL (Volume) menu, 63 conversions 4Bin, 272 4Dec, 279 4DMS, 51, 285 4Frac, 52, 298 4Hex, 303 4Oct, 327 4Pol, 336 4REAL, 156 conversions (continued) 4Rec, 343 4Sph, 360 Eq4St, 227 li4vc, 160 St4Eq(, 227, 361 vc4li, 160 converting a value expressed as a rate, 65 converting Fahrenheit to Celsius, 8 converting units of measure, 61 CoordOff, 84, 275 CoordOn, 84, 275 copying variable value, 41 corr (correlation coefficient), 193 cos (cosine), 48, 186, 276 cos L1 (arccosine), 48, 276 cosh (hyperbolic cosine), 51, 277 cosh L1 (inverse hyperbolic cosine), 51, 277 CPLX (complex number variables), 43, 71 cross(, 173, 277 cSum( (cumulative sum), 160, 278 current entry, 19 clearing, 23 current item, 38 cursor, 17, 22 ALPHA, 22 alpha, 22 changing, 23 direction keys, 23 entry, 22 free-moving, 128, 144, 205 full, 22 insert, 22 location, 19, 20, 21, 25 moving, 23 second, 22 selection, 38 trace, 90 curves drawing, 107 CUSTOM menu, 44 clearing items, 45 copying items, 44 Customer Support, 392 4Cyl (to cylindrical), 174, 278
CylV (cylindrical vector coordinate mode), 36, 278
(decimal), 278 data type selection screen, 42 Dec (decimal number base mode), 278 Dec (decimal), 35, 65 4Dec (to decimal), 279 decimal, 20 decimal mode, 34, 35, 65 fixed (012345678901), 35 floating, 35 decimal number, 278 decimal number base, 35 decimal point, 35 degree angle mode, 35, 75, 279 degree complex-number mode, 70 degree entry (), 279 degrees, 51 degrees/minutes/seconds form, 51 DELc (delete column), 179 DELET, 60

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differential equations (continued) graphing, 132, 137, 139, 141, 142 initial conditions editor, 136 mode, 144 Q'n equation variables, 135 setting axes, 137 setting graph format, 133 setting graphing mode, 132 solving, 139 tracing, 144 using EVAL, 150 window variables, 135 differentiation modes, 36 difTol (tolerance), 136 dim (dimension), 173, 184, 281 dimL (dimension of list), 159, 282 DirFld (direction field), 134, 282 Disp (display), 216, 283 DispG (display graph), 283 display, 17 display contrast adjusting, 17, 18 displaying a menu, 31 DispT (display table), 284 DIST (distance), 96, 98 division (/), 284 division symbol, 3 4DMS (to degrees/ minutes/seconds), 51, 285 dot(, 173, 285 dr/dq, 122 DRAW, 75, 88 DrawDot, 84, 285 DrawF (draw function), 103, 107, 286 drawing circles, 106 differential equation graphs, 145 freehand points, lines, curves, 107 function, tangent line, inverse function, 107 line segments, 105 lines, 105, 106 parametric graphs, 130 points, 108 polar graphs, 122 drawing tools, 101 drawings clearing, 103 drawings (continued) recalling, 102 saving, 102 DrawLine, 84, 286 DrEqu( (draw equation), 145, 287 DrInv (draw inverse), 103, 107, 287 DS<( (decrement and skip), 219, 288 DUPLICATE NAME menu, 241 dx/dt, 130 dxDer1 (exact differentiation), 36, 75, 288 dxNDer (numeric differentiation), 36, 75, 288 dy/dt, 130 dy/dx, 96, 99, 130
DELf (delete function), 77 DELi (delete element), 170 DELr (delete row), 179 Deltalst( (delta list), 160, 279 DelVar( (delete variable), 219, 280 der1( (first derivative), 54, 280 der2( (second derivative), 54, 280 derivatives calculating, 7 det (determinant), 183, 281 DFLTS (defaults), 232 DifEq (differential equation mode), 35, 74, 239, 281 differential equation editor, 134 differential equation graphs, 74 displaying, 138 drawing, 145 mode, 35 differential equations changing to first order, 142 defining graph, 132 drawing solutions, 148 DrEqu(, 287 editor, 134 EXPLR, 148
E (exponent), 48, 292 e^ (e raised to power), 288 editing equations, 205 editor menu, 33 eigVc (eigenvector), 183, 289 eigVl (eigenvalue), 183, 289
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P2Reg (quadratic regression), 190, 330 P3Reg (cubic regression), 190, 331 P4Reg (quartic regression), 190, 332 panning, 90 Par, 74 Param (parametric mode), 35, 239, 333 parametric equation deleting, 127 graphing, 126 selecting and deselecting, 127 parametric graphs, 74 default graph style, 126 defining, 125 displaying, 128 drawing, 130 equation editor, 126 free-moving cursor, 128 graph format, 128 graph tools, 128
99INDEX.DOC TI-86, Index, US English Bob Fedorisko Revised: 02/13/01 2:51 PM Printed: 02/27/01 1:29 PM Page 414 of 15
reusing, 28 PRGM (program names), 43 PRGM CTL menu, 218 PRGM I/O (Input/Output) menu, 215 PRGM menu, 214 prod (product), 52, 160, 338 program editor, 214 menus and screens, 215, 220 program flow, 56 programming assembly language, 225 calling a program, 224 copying a program, 225 creating programs, 214 defined, 214 deleting a program, 223 downloading assembly programs, 225 editing a program, 223 entering a command line, 220 getting started, 214 interrupting program, 222 running program, 221 using variables, 225 Prompt (PRGM I/O menu), 216, 338 prompts, 22 Eval x=, 76 Name=, 22, 39, 76 Rcl, 42 Sto, 212 PTCHG, 105 PtChg(, 338 PTOFF, 105, 108 PtOff(, 338 PTON, 105, 108 PtOn(, 338 PwrR (power regression), 190, 339 PxChg(, 103, 340 PxOff(, 103, 340 PxOn(, 103, 340 PxTest(, 103, 340 Quick-Find Locator (A to Z Reference), 262
(radian entry), 341 rAdd, 184 rAdd(, 340 Radian (angle mode), 35 radian angle mode, 75, 341 radian complex-number mode, 70 radian entry (r), 341 rand (random), 50, 341 randBin( (random binomial), 50, 341 randInt( (random integer), 50, 342 randM( (random matrix), 184, 342 randNorm( (random normal), 50, 342 random number, 50 RCGDB (recall graph database), 76, 88, 343 RcPic (recall picture), 76, 102, 343
Q'n equation variables, 135 Qrtl1, 193 Qrtl3, 193 Quick Zoom, 91 in parametric graphing, 129 in polar graphing, 120
RCPIC menu, 76 REAL, 43, 175, 185, 343 4REAL (to real number), 156, 170, 179 real number variables, 43 real numbers, 29 real portion of complex number, 71 4Rec (to rectangular), 71, 174, 343 recalling variable values, 18, 42 receiving transmitted data, 241 rectangular complex mode, 35 rectangular complex numbers, 70 rectangular complex-number form, 20 rectangular graph, 84 rectangular vector coordinates, 36 RectC (rectangular complex), 35, 344 RectGC (rectangular graph coordinates), 84, 344 RectV (rectangular vector coordinate mode), 36, 344 RECV (LINK menu), 236

SphereV (spherical vector coordinate mode), 36, 360 square ( 2), 360 square root (), 7, 360 St4Eq( (string to equation), 227, 361 STAT (statistical result variables), 43 STAT CALC (Calculations) menu, 189 STAT menu, 188 Stat Plot changing on/off status, 81 setting up, 195 turning on and off, 195 STAT PLOT menu, 195 STAT PLOT status screen, 194 STAT VARS (Statistical Variables) menu, 192 statistical analysis, 188 results, 192 statistical data entering, 189 plotting, 194, 195 STGDB (store graph database), 76, 88, 361 STOa, 210 STOb, 210 Stop, 219, 362 Store, 18 store symbol, 22 store to variable (), 362 storing a graph display, 102 storing data, 39 storing equation coefficients, 210 storing equation results, 210 STOx, 210 STPIC (store picture), 76, 88, 362 STPIC menu, 76 StReg (store regression equation), 190, 362 string, 29 concatenating, 226 creating, 226 defined, 226 storing, 226, 227 string entry, 363 STRNG (string variables), 43 STRNG (String) menu, 227 STYLE, 77 sub( (subset of string), 227, 363 submatrix displaying, 181 subroutines, 224 subtraction (N), 363 sum, 52, 160, 364 sum of elements of list, 52 Sx (statistical result variable), 193 syntax error, 27 syntax of function, 25 syntax of instruction, 25
T (transpose), 367 table, 110 clearing, 114 displaying, 110 navigating, 111 setting up, 113 setup editor, 113 TABLE menu, 110 Table menus, 112 table setup editor, 113 tan (tangent), 48, 364 tan L1 (arctangent), 48, 365 tangent line drawing, 107
tanh (hyperbolic tangent), 51, 365 tanh L1 (inverse hyperbolic tangent), 51, 365 TANLN (tangent line), 96, 99 TanLn(, 103, 107, 366 TBLST (table setup editor), 112, 113 TEST menu, 55 TEXT, 105 Text(, 366 Then, 218, 305, 306 TI-GRAPH LINK, 235 tMax, 127, 136 tMin, 127, 136 TOL (Tolerance Editor), 398 tPlot, 136 TRACE, 88 TRACE (cursor), 75 Trace (Graph menu), 367 TRACE (Solver menu), 207 trace cursor, 75, 90, 144, 205 in parametric graphing, 128 in polar graphing, 120 moving, 90, 121, 129 panning, 90 Quick Zoom, 91
99INDEX.DOC TI-86, Index, US English Bob Fedorisko Revised: 02/13/01 2:51 PM Printed: 02/27/01 1:29 PM Page 417 of 15
selecting an item, 33 user-created constants, 43, 58, 60 user-created zoom variables, 239 VARS EQU menu, 203 vc4li (vector to list), 160, 174, 369 vector, 29 brackets [ ], 369 complex, 171, 180 creating, 170 defined, 168 deleting from memory, 170 displaying, 171 editing dimension and elements, 172 forms, 168 operations, 173 using in an expression, 172 with math functions, 176 vector coordinate modes, 36 vector editor, 168 Vector Editor menu, 170 vector entry [ ], 369 VECTR (vector names), 43 VECTR CPLX (Complex) menu, 175 VECTR MATH menu, 173 VECTR menu, 169 VECTR NAMES menu, 169 VECTR OPS (Operations) menu, 173 VERT (vertical line), 104, 106, 369

Part B

Guide for Texas Instruments TI-86 Graphing Calculator
This Guide is designed to offer step-by-step instruction for using your TI-86 graphing calculator with the third edition of 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

Before you begin, check the TI-86 setup and be 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 Figures 1, 2, and 3.
Press 2nd MORE (MODE) and choose the settings shown in Figure 1 for the basic setup. Check the window format by pressing GRAPH MORE F3 (FORMT) and choose the settings shown in Figure 2. ! If you do not have the darkened choices shown in Figure 1 and Figure 2, use the arrow keys to move the blinking cursor over the setting you want to choose and press ENTER. Return to the home screen with EXIT or 2nd EXIT (QUIT). Note that EXIT
EXIT clears the menus from the bottom of the screen.
Specify the statistical setup as shown in Figure 3 by pressing 2nd (LIST) F5 [OPS] MORE
MORE 7 (L) 4 , MORE F3 [SetLE] ALPHA 7 (L) 2 , ALPHA 7 (L) 2 , ALPHA 7 (L) 3 , ALPHA ALPHA 7 (L) 5 , ENTER. You need this setup for some of the programs referred to
in this Guide to execute properly.

TI-86 Basic Setup

Figure 1

TI-86 Window Setup

Figure 2

TI-86 Statistical Setup

Figure 3
Copyright Houghton Mifflin Company. All rights reserved.

Chapter 1

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 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

* 15 +

. 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. Evaluate
( 3) 8.037. Use () for the negative symbol and 8 + 1.456
for the subtraction sign. To clear the home screen, press CLEAR. NOTE: The numerator and denominator must be enclosed in parentheses and 34 (3)4. Now, evaluate e3*0.027 and e3*0.027. Type e^ with 2nd LN (ex). The TI-86 will assume you mean e3*0.027 unless you use parentheses around the two values in the exponent to indicate e3*0.027. 2. USING THE ANS MEMORY Instead of again typing an expression that was just evaluated, use the answer memory by pressing 2nd () (ANS).
F 1 Calculate G 4 G 15 + H*
I1 J using this nice shortcut. J K
Type Ans-1 by pressing 2nd () (ANS) 2nd EE (x1). 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. The to a fraction key is obtained by pressing 2nd ! (MATH)
F5 [MISC] MORE F1 [!Frac].
The calculators symbol for times 1012 is E12. Thus, 7.945E12 means 7,945,000,000,000. The result 1.4675E6 means 1.4675*106, which is the scientific notation expression for 0.0000014675.

TI-86 Guide

4. STORING VALUES It may be beneficial to store numbers or expressions for later recall. To store a number, type it, press STO! (note that the cursor changes to the alphabetic cursor A ), press the key corresponding to the capital letter(s) naming the storage location, and then press ENTER. To join several short commands together, use 2nd. ( : ) between the statements. Note that when you join statements with a colon, only the value of the last statement is shown. WARNING: The STO! key locks the upper-case alphabetic cursor and ALPHA unlocks it. Always look at the screen when you are typing to be certain that you are not entering numbers when you intend to type letters and vice-versa. Store 5 in A and 3 in B, and then calculate 4A 2B. (Press ALPHA to return to the regular cursor.) To recall a value, press ALPHA , type the letter in which the value is stored, and then press ENTER.

Using x as the input variable, enter y1 = 3.622(1.093^ x). Return to the home screen by pressing 2nd EXIT (QUIT). Substitute 15 into the function with 2nd ALPHA (alpha) 0 (Y) 1 ( 15 ) Find the value by pressing ENTER. NOTE: You do not have to have the closing parenthesis on the right if nothing else follows it. To choose another graphing list location, say y2, just type the number corresponding to that functions location, 2, following the lower-case y. WARNING: You must use a lower case, not upper case, y in order for the TI-86 to recognize the function in the graphing list. You must also use a lower-case x for the x-variable. To now evaluate the function at other inputs, first recall the previous entry with 2nd ENTER (ENTRY). Then edit the expression to the new value. For instance, press 2nd ENTER (ENTRY), change 15 to 20 by pressing and typing 20, and then press ENTER to evaluate the function at x = 20. Evaluate y1 at x = 0 by recalling the previous entry with 2nd ENTER (ENTRY), change 20 to 0 with DEL , and then press ENTER. 1.1.1f EVALUATING OUTPUTS USING THE TABLE Function outputs can be determined by evaluating on the graphics screen, as discussed in Section 1.1.1c, or by evaluating on the home screen as discussed in Section 1.1.1e of this Guide. You can also evaluate functions using the TI-86 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 a value called TblStart and differ by a value called Tbl. Lets use the TABLE to evaluate the function v(t) = 3.622(1.093t) at t = 15. Even though you can use any of the function locations, we again choose to use y1. Press GRAPH F1 [y(x)=], clear the function locations, and enter 3.622(1.093^ x) in location y1 of the y(x)= list. After entering the function v in y1, choose the TABLE SETUP menu by pressing TABLE F2 [TBLST]. To generate a list of values beginning with 13 such that the table values differ by 1, enter 13 in the TblStart location and 1 in the Tbl location. Then choose AUTO in the Indpnt: location by having the cursor on that word and pressing ENTER. Press F1 [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 more decimal places at the bottom of the screen.

Return to the table set-up screen with F1 [TBLST]. To compute specific outputs rather than a list of values, choose ASK in the Indpnt: location. Press ENTER. (Note that when using ASK, the settings for TblStart and Tbl do not matter.) Press F1 [TABLE], type in the x-value(s) at which the function is to be evaluated, and press ENTER. Unwanted entries or values from a previous problem can be cleared with DEL. NOTE: 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 AUTO. 1.1.1g FINDING INPUT VALUES USING THE SOLVER Your calculator solves for the input values of any equation that is in the form "expression = constant". This means that all terms involving the variable must be on one side of the equation and constant terms must be on the other side before you enter the equation into the calculator. The expression can, but does not have to, use x as the input variable. The TI-86 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(x)= list or you can enter the equation in the solver. Return to the home screen with 2nd EXIT (QUIT). Access the solver by pressing 2nd GRAPH (SOLVER). 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 until only the eqn: line is on the screen and CLEAR to delete the old equation. You should then have the screen that is shown in the previous step. Lets now use the solver to answer the question in part e of Example 3 in Section 1.1: When did the land value reach $20,000? Because the land value is given by v(t) = 3.622(1.093t) thousand dollars where t is the number of years after the end of 1985, we are asked to solve the equation 3.622(1.093t) = 20. That is, we are asked to find the input value t that makes this equation a true statement. If you already have y1 = 3.622(1.093^ x) in the graphing list, you can refer to the function as y1 in the SOLVER. (Note that y1 can be entered by pressing the F-key under its location in the menu at the bottom of the screen.) If not, enter 3.622(1.093^ x) instead of y1 in the eqn: location. Press ENTER. Enter 20 in the exp: location under y1 to tell the TI-86 the rest of the equation.

An intercept is the where the graph crosses or touches an axis. Also remember that the xintercept of the function y = f(x) has the same value as the root or solution of the equation f(x) = 0. Thus, finding the x-intercept of the graph of f(x) c = 0 is the same as solving the equation f(x) = c.
We illustrate this method with a problem similar to the one in Activity 36 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 x such that 3x 0.8x2 + 4 = 2.3. Because this function is not given in a context, we have no indication of an interval of input values to use when drawing the graph. So, we 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 function (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 GRAPH F1 [y(x)=] and clear all locations with CLEAR. Enter the function 3x 0.8x2 + 4 2.3 in y1. You can enter x2 with x-VAR x2 or enter it with x-VAR ^ 2. Remember to use , not () , for the subtraction signs. NOTE: Whenever there are two menus at the bottom of the display screen, press EXIT to delete the bottom menu or press 2nd before pressing the F-key you want to access a certain command in the top menu. We give instructions assuming there is only one menu. Draw the graph with F3 [ZOOM] MORE F4 [ZDECM] or
F4 [ZSTD]. If you use the former, press F2 [WIND] and reset yMax to 5 to get a better view of the graph. (If you reset the
window, press F5 [GRAPH] to draw the graph.) To graphically find an x-intercept, i.e., a value of x at which the graph crosses the horizontal axis, press MORE F1 [MATH]
F1 [ROOT]. 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 input of the leftmost x-intercept is displayed as x = 0.5. Note that if this process does not return the correct value for the intercept 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.)

(alpha) 0 (y) 3 ( 5 ) ENTER to see the result. We find that
milk sales were T(5) $40.93. 1.2.2a CHECKING YOUR ANSWER FOR A COMPOSITE FUNCTION We illustrate this technique with the functions that are given on page 23 of Section 1.2 of Calculus Concepts: altitude = F(t) = 222.22t3 + 1755.95t2 + 1680.56t + 4416.67 feet above sea level where t is the time into flight in minutes and air temperature = A(F) = 277.897(0.99984F) 66 degrees Fahrenheit where F is the number of feet above sea level. Remember that when you enter functions in the y(x)= list, you must use x as the input variable. Clear the functions from the y(x)= list. Enter F in y1 by pressing

() 222. 22 x-VAR

^ 3 + 1755. 95 x-VAR

+ 1680

. 56 x-VAR
+ 4416. 67 ENTER. Enter A in y2 by pressing

. 99984 ^

277. 897 (

) 66 ENTER.

Enter the composite function (AoF)(x) = A(F(x)) = y2(y1) in y3 with 2nd ALPHA (alpha) 0 (y) 2 ( 2nd ALPHA (alpha) 0

(y ) 1 ) ENTER.

Next, enter your algebraic answer for the composite function in y4. Be certain that you enclose the exponent in y4 (the function 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 calculators 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. 1.2.2b 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. Press GRAPH F1 [y(x)=] and place the cursor in the line containing y1. Press F5 [SELCT] and use to move the cursor to the y2 line. Press F5 [SELCT]; 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 function dark. The SELCT key toggles between the function being off and being on. When you draw a graph, the TI-86 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 TABLE F2 [TBLST]. Move the cursor to ASK in the Indpnt: location and press ENTER. Press F1 (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. The TI-86 calls this an OVERFLOW error. 1.2.3 GRAPHING A PIECEWISE CONTINUOUS FUNCTION Piecewise continuous functions are used throughout the text. You will need to use your calculator to graph and evaluate outputs of piecewise continuous functions. Several methods can be used to draw the graph of a piecewise function. One of these is presented below using the function that appears in Example 2 of Section 1.2 in Calculus Concepts:

The population of West Virginia from 1985 through 1999 can be modeled by 23.373t + 3892.220 thousand people when 85 t < 90 P( t ) = 1.013t 2 + 193164t 7387.836 thousand people when 90 t 99. where t is the number of years since 1900.

R | S | T

Clear any functions that are in the y(x)= list. Using x as the input variable, enter each piece of the function in a separate location. We use locations y1 and y2. Next, we form the formula for the piecewise function in y3. Parentheses must be used around the function portions and the inequality statements that tell the calculator which side of the break point to graph each part of the piecewise function. Have the cursor in y2 and press to place the cursor in y3. Press ( 2nd ALPHA (alpha) 0 (Y) 1 ) ( x-VAR 2nd 2

(TEST) F2 [<] 90 )

ALPHA (alpha) 0 (Y) 2 ) ENTER.

2nd 2 (TEST) F5 [] 90 )

Your calculator draws graphs by connecting function outputs wherever the function is defined. However, this function breaks at x = 90. The TI-83 will connect the two pieces of P unless you tell it not to do so. Whenever you draw graphs of piecewise functions, set your calculator to Dot mode as described below so that it will not connect the function pieces at the break point. Turn off y1 and y2 and place the cursor on y3. Press MORE
F3 [STYLE]. Press F3 five more times and the slanted line2 to
the left of y3 should be a dotted line (as shown to the right). The function y3 is now in Dot mode. NOTE: The method described above places individual functions in Dot mode. The functions return to standard (Connected) mode when the function locations are cleared. If you want to put all functions in Dot mode at the same time, press GRAPH F2 [WIND] MORE F3
[FORMT], choose DrawDot in the third line, and press ENTER. However, if you choose to
set DOT mode in this manner, you must return to the window format screen, select DrawLine instead of DrawDot, and press ENTER to take the TI-86 out of Dot mode. Now, set the window. The function P is defined only when the input is between 85 and 99. So, we evaluate P(85), P(90), and P(99) to help when setting the vertical view. Note that if you attempt to set the window using ZFIT as described in Section 1.1. 1b of this Guide, the picture is not very good and you will probably want to manually reset the height of the window as described below.

2 The different graph styles you can draw from this location are described in more detail on page 79 in your TI-86 Graphing Calculator Guidebook.
We set the lower and upper endpoints of the input interval as xMin and xMax, respectively. Press GRAPH F2 [WIND], set xMin = 85, xMax = 99, yMin 1780, and yMax 1910. Press F5 [GRAPH] and use CLEAR to remove the menu. Reset the window if you want a closer look at the function around the break point. The graph to the right was drawn using xMin = 89, xMax = 91, yMin = 1780, and yMax = 1810. You can find function values by evaluating outputs on the home screen or using the table. Either evaluate y3 or carefully look at the inequalities in the function P to determine whether y1 or y2 should be evaluated to obtain each particular output.
1.3 Limits: Functions, Limits, and Continuity
The TI-86 table is an essential tool when you estimate end behavior numerically. Even though rounded values are shown in the table due to space limitations, the TI-86 displays at the bottom of the screen many more decimal places for a particular output when you highlight that output. 1.3.1a NUMERICALLY ESTIMATING END BEHAVIOR Whenever you use the TI-86 to estimate end behavior, set the TABLE to ASK mode. We illustrate using the function u that appears in Example 1 of Section 1.3 in Calculus Concepts: Press GRAPH F1 [y(x)=] and use F4 [DELf] to delete all previously-entered functions. Enter u(x) =
3x 2 + x. Be 10 x 2 + 3x + 2
certain to enclose both numerator and denominator of the fraction in parentheses.
Press TABLE F2 [TBLST]. Choose Ask in the Indpnt: location by placing the cursor over Ask and pressing ENTER. Press F1

[TABLE].

Delete any values that appear by placing the cursor over the first x value and repeatedly pressing DEL. To numerically estimate lim u(x), enter increasingly large values of x.
NOTE: The values you enter do not have to be those shown in the text or these shown in the above table provided the values you input increase without bound. CAUTION: Your instructor will very likely have you write the table you construct on paper. Be certain that if necessary, you highlight the rounded values in the output column of the table and look on the bottom of the screen to see what these values actually are. ROUNDING OFF: Recall that rounded off (also called rounded in this Guide) means that if one digit past the digit of interest if less than 5, other digits past the digit of interest are dropped. If one digit past the one of interest is 5 or over, the digit of interest is increased by 1 and the remaining digits are dropped.

The sequence of observed output values confirm our numerical estimate.
1.3.2a NUMERICALLY ESTIMATING THE LIMIT AT A POINT Whenever you numerically estimate the limit at a point, you should again set the TABLE to ASK mode. We illustrate using the function u that appears in Example 2 of Section 1.3 in Calculus Concepts: Have u(x) =
3x in some location of the y(x)= list, say y1. 9x + 2
Have TBLST set to Ask, and press TABLE F1 [TABLE] to return to the table. Delete the values currently in the table with DEL. To numerically estimate Because the output values and closer to, appear to become larger and larger, we estimate that the limit does not exist and write lim u(x) .
x 2/9 x 2/9 2/9 = 0.222222.
u(x), enter values to the left of, and closer
Delete the values currently in the table. To numerically estimate lim u(x), enter values to the right of, and closer and closer +
to, 2/9. Because the output values appear to become larger and larger, we estimate that lim + u(x) .
1.3.2b GRAPHICALLY ESTIMATING THE LIMIT AT A POINT A graph can be used to estimate a limit at a point or to confirm a limit that you estimate numerically. The procedure usually involves zooming in on a graph to confirm that the limit at a point exists or zooming out to validate that a limit does not exist. We again illustrate using the function u that appears in Example 2 of Section 1.3 in Calculus Concepts. Have the function u(x) =

y(x)= list,

3x entered in some location of the 9x + 2 F3 [ZOOM] MORE F4
say y1. A graph drawn with GRAPH F3 [ZOOM]
F4 [ZSTD] or with GRAPH [ZDECM] is not very helpful.

To confirm that

u(x) and

x 2/9+

u(x) do not exist, we are interested in values of x
that are near to and on either side of 2/9. Choose input values close to 0.222222 for the x-view and experiment with different y values until you find an appropriate vertical view. Use these values to manually set a window such as that shown to the right. Draw the graph with F5 [GRAPH]. The vertical line appears because the TI-86 is set to Connected mode. Place the TI-86 in DrawDot mode or place the function y1 in the y(x)= list in Dot mode (see page B16) and redraw the graph.

1 ENTER. Repeat the process to enter the output data, but store
these data in a list named L2. WARNING: If you do not enter and store the input data into a list named L1 and the output data into a list named L2, many of the programs in this Guide will not properly execute. 1.4.1b EDITING DATA

, , and/or

No matter how it is entered, the easiest way to edit data is using the sta-
tistical lists. If you incorrectly type a data value, use the cursor keys (i.e., the arrow keys ,
) 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.
1.4.1c 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 2nd + (STAT) F2 [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 and
to move the cursor over the name L2. Press CLEAR ENTER. Repeat
this procedure to clear the other lists. 1.4.1d 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 is given in the TI-86 Program Appendix at the Calculus Concepts web site. Consult the Programs category in Trouble Shooting the TI-86 in this Guide if you have questions about obtaining the programs. Have the data given in Table 1.19 in Section 1.4 of Calculus Concepts entered in your calculator. (See Section 1.4.1a of this Guide.) Exit the list menu with 2nd EXIT (QUIT). To run the program, press PRGM [NAMES] followed by the Fkey that is under the DIFF program location, and press ENTER. The message on the right appears on your screen. If you have not entered the data in L1 and L2, press F2 [Quit] and do so. Otherwise, press F1 [Yes] to continue. Press F1 [1st] to compute the first differences. Choose F4 to quit.
We use the other choices in the next chapter. You can also view the first differences in list L3.
The first differences are constant, so a linear function gives a perfect fit to these tax data.

use it! This is not a problem because the calculator pastes the entire equation it finds into the graphing list at the same time the function is found if you follow the instructions given above. NOTE: The TI-86 will use lists called xStat and yStat, which probably contain different data, if you do not specify lists L1 and L2 in the instruction to find the best-fit equation. It is possible to use lists other than L1 and L2 for the input and output data. However, if you do so, you must set one of the STAT PLOT locations to draw the scatter plot for those other lists (as described in Section 1.4.2b). To find the best-fit function, replace L1 and L2 by the other lists in the fit instruction. To paste the function into a location other than y1, just change the number 1 following y in the fit instructions to the number corresponding to the graphing location that you want. CAUTION: The r that is shown on the screen that first gives the linear equation is called the correlation coefficient. This and a quantity called r2, the coefficient of determination, are numbers that you will learn about in a statistics course. It is not appropriate3 to make use of these values in a calculus course. Graphing the Line of Best Fit: After finding a best-fit equation, you should always draw the graph of the function on a scatter plot to verify that the function provides a good fit to the data. Press EXIT F5 [GRAPH] to overdraw the function you pasted in the graphing list on the scatter plot of the data. (As we suspected from looking at the scatter plot and the first differences, this function provides a very good fit to the data.) 1.4.2d COPYING A GRAPH TO PAPER Your instructor may ask you to copy what is on your graphics screen to paper. If so, use the following ideas to more accurately perform this task. After using a ruler to place and label a scale (i.e., tick marks) on your paper, use the trace values (as shown below) to draw a scatter plot and graph of the line on your paper. Press GRAPH to return the modified tax data graph found in Section 1.4.2c to the screen. Press F4 [TRACE] and . The symbol P1 in the upper right-hand corner of the screen indicates that you are tracing the scatter plot of the data in Plot 1. Press to move the trace cursor to the linear function graph. The number in the top right of the screen tells you the location of the function that you are tracing (in this case, y1). Use and/or to locate values that are as nice as possible and mark those points on your paper. Use a ruler to connect the points and draw the line.

If you are copying the graph of a continuous curve rather than a straight line, you need to trace as many points as necessary to see the shape of the curve while marking the points on your paper. Connect the points with a smooth curve.
3 Unfortunately, there is no single number that can be used to tell whether one function better fits data than another. The correlation coefficient only compares linear fits and should not be used to compare the fits of different types of functions. For the statistical reasoning behind this statement, read the references in footnote 6 on page B-30.
1.4.3a ALIGNING DATA We return to the modified tax data entered in Section 1.4.2b. If you want L1 to contain the number of years after a certain year instead of the actual year, you need to align the input data. In this illustration, we shift all of the data points to 3 different positions to the left of where the original values are located. Press 2nd + (STAT) F2 [EDIT] to access the data lists. To copy the contents of one list to another list; for example, to copy the contents of L1 to L3, use and to move the cursor so that L3 is highlighted. Press ALPHA 7 (L) 1 ENTER. NOTE: This first step shown above is not necessary, but it will save you the time it takes to re-enter the input data if you make a mistake. Also, it is not necessary to first clear L3. However, if you want to do so, have the symbols L3 highlighted and press CLEAR ENTER. To align the input data as the number of years past 1999, first press the arrow keys ( and ) so that L1 is highlighted. Tell the TI-86 to subtract 1999 from each number in L1 with

ALPHA 7 (L) 1 1999.

Press ENTER. Instead of an actual year, the input now represents the number of years since 1999. Return to the home screen with 2nd EXIT (QUIT). Find the linear function by pressing 2nd ENTER (ENTRY) as many times as needed until you see the linear fit instruction. To enter this function in a different location, say y2, press and 2. Press ENTER and then press GRAPH F1 [y(x)=] to see the function pasted in the y2 location. Note: If you want the aligned function to be in y1, do not replace y1 with y2 before pressing ENTER to find the equation. To graph this equation on a scatter plot of the aligned data, first turn off the function in y1 (see page B-15 of this Guide). Press

5 Program LSLINE is for illustration purposes only. Actually finding the line of best fit for a set of data should be done
according to the instructions in Section 1.5.7 of this Guide. 6 Two articles that further explain best-fit are H. Skala, Will the Real Best Fit Curve Please Stand Up? Classroom Computer Capsule, The College Mathematics Journal, vol. 27, no. 3, May 1996 and Bradley Efron, Computer-Intensive Methods in Statistical Regression, SIAM Review, vol. 30, no. 3, September 1988.