Tips for Beginners for the TI 89 Calculator

Tips for Math with the TI-89 Titanium© 2012 April 29, for the TI-89 Titanium OS 3.10 Dr. Wm J. Larson, International School of Geneva, [email protected] Corrections welcome. OPERATING SYSTEM ................................................... 2 GETTING STARTED ...................................................... 2 THE GREEN, YELLOW & PURPLE SYMBOLS .................... 2 THE TWO MINUS KEYS ................................................... 2 THE TWO EQUALS SIGNS ................................................ 2 EXPONENTS AND ROOTS ................................................. 2 SCIENTIFIC NOTATION..................................................... 2 DEGREES & RADIAN MODES ........................................... 2 EXACT AND APPROXIMATE MODES ................................. 2 MORE SYMBOLS & FUNCTIONS ....................................... 3 PARENTHESES ................................................................. 3 LISTS, MATRICES & VECTORS ......................................... 3 THE PLUS/MINUS SIGN.................................................... 3 THE F1 TO F8 KEYS......................................................... 3 CLEARING ....................................................................... 3 SHORTCUTS ..................................................................... 3 ON SCREEN SYNTAX HELP .............................................. 3 IF THE SCREEN IS TOO DARK OR LIGHT ............................. 3 IF THE CALCULATOR IS LOCKED-UP ................................. 4 TO QUICKLY MOVE TO THE END OF AN EXPRESSION ........ 4 TO REUSE A PREVIOUS ENTRY.......................................... 4 TO HIGHLIGHT TEXT ........................................................ 4 TO COPY TEXT ................................................................. 4 IF YOU ARE LOST ............................................................. 4 TO CHANGE THE NUMBER OF DIGITS DISPLAYED ............. 4 THE WITH KEY ................................................................ 4 INSERT MODE .................................................................. 4 LOG X .............................................................................. 4 PI ..................................................................................... 4 ORDER OF OPERATIONS .................................................... 4 GRAPHING ....................................................................... 4 ZOOM .............................................................................. 5 FRIENDLY WINDOWS ...................................................... 5 TRACE ............................................................................. 6 TO CHANGE THE CENTER OF THE GRAPH .......................... 6 TO FORMAT A GRAPH ...................................................... 6 TO CANCEL A GRAPH ....................................................... 6 MODES ............................................................................ 6 TO STORE A WINDOW SETTING ........................................ 6 TO DEPICT AN INEQUALITY .............................................. 6 PARAMETRIC GRAPHS ..................................................... 6 POLAR GRAPHS ............................................................... 6 TO FIND THE MINIMA OR MAXIMA OF A FUNCTION ......... 7 SOLVING A SYSTEM OF TWO EQUATIONS I.E. FINDING THE INTERSECTIONS OF TWO GRAPHS .................................... 7 TO FIND THE X AND Y INTERCEPTS .................................. 7 TABLES ........................................................................... 7 ASYMPTOTES .................................................................. 8 PIECEWISE-DEFINED GRAPH ........................................... 8 GRAPHING A REAL FUNCTION ......................................... 9 COMPLEX NUMBERS ................................................... 9 CONVERTING FROM RECTANGULAR TO POLAR OR TRIGONOMETRIC FORM ................................................... 9 SOLVING WITH COMPLEX NUMBERS ............................... 9 GRAPHING A COMPLEX FUNCTION ................................ 10 CONVERTING FROM RECTANGULAR TO POLAR COORDINATES ............................................................... 10 OTHER COMMANDS ................................................... 10 THE SOLVE & ZEROS COMMANDS ................................. 10 SOLVING INEQUALITIES ................................................. 11 FACTORING AND EXPANDING........................................ 11 PERMUTATIONS AND COMBINATIONS ........................... 11 SEQUENCES AND SERIES ............................................... 11 MATRICES ..................................................................... 12 HOW TO GRAPH A CONIC EQUATION ............................. 12 HOW TO SIMPLIFY RATIONAL FUNCTIONS .................... 13 TO SOLVE A SYSTEM OF EQUATIONS WITH SOLVE( ...... 13 THE INVERSE................................................................. 13 LINEAR INTERPOLATION ............................................... 13 STEP FUNCTIONS ........................................................... 14 TO SIMPLIFY EXPRESSIONS ........................................... 14 BINARY, HEXADECIMAL & DECIMAL ............................ 14 TO DELETE FUNCTIONS, LISTS, TABLES, ETC. ................. 14 DEFINE .......................................................................... 14 PROGRAMMING .......................................................... 15 ON LINE HELP .............................................................. 15 ERROR MESSAGES ..................................................... 15 ERROR: DIMENSION ...................................................... 15 RESET ALL MEMORY .................................................... 15 CALCULUS ..................................................................... 15 ON THE HOME SCREEN ................................................. 15 DIFFERENTIATION ......................................................... 15 NUMERICAL DIFFERENTIATION ..................................... 16 INTEGRATION ................................................................ 16 LIMITS ........................................................................... 16 FINDING EPSILON IN THE LIMIT DEFINITION ................. 16 MINIMA AND MAXIMA .................................................. 16 ON THE GRAPH SCREEN ............................................... 16 DIFFERENTIATION ......................................................... 16 INTEGRATION ................................................................ 16 PARTIAL FRACTION DECOMPOSITION ........................... 17 MINIMA, MAXIMA, INFLECTION POINTS, TANGENT LINES & ARC LENGTH ............................................................. 17 CONVERGENCE OF A SEQUENCE .................................... 17 TAYLOR SERIES APPROXIMATIONS ............................... 17 RIEMANN SUMS ............................................................ 17 DIFFERENTIAL EQUATIONS ........................................... 18 IMPLICIT DIFFERENTIATION ........................................... 18 Tips for Math with the TI-89 Calculator, page 2 Operating System Just as your PC can be upgraded to WinXP, your TI-89 can be upgraded to OS 3.10. In both cases overall you get a better experience. To find your OS select TOOLS on the HOMESCREEN, select option A: ABOUT To download OS 3.10 go to http://education.ti.com and navigate to http://education.ti.com/educationportal/sites/U S/productDetail/us_os_89titanium.html. Installing OS 3.10 will remove all data including preloaded Graphing Calculator Software Applications (Apps) for example Stats/List Editor. Getting Started The Blue, Green, Yellow & Grey Symbols The blue 2 nd key accesses the blue functions, e.g. 2 nd t. The green + key accesses the green functions, e.g. + Y=. The gray alpha key accesses the gray functions, producing lower case letters, e.g. alpha A gives “a”. The l key, produces upper case letters, e.g. l A produces “A”. However the TI-89 isn't case- sensitive, 'A' is treated just like 'a', even built- in commands that have capital letters in them can be typed in lowercase, e.g., 'cSolve' can be typed 'csolve'. It will change to 'cSolve' once you hit ENTER. Alpha Lock: To key several lowercase letters, key 2 nd a-lock or just hold the ALPHA button down. To key several uppercase letters, key l alpha. To exit alpha lock, key alpha. To type a space, key alpha (-). The Two Minus Keys Two different keys are needed to enter -3 - 4. Use the (-) key (left of ENTER) for -3 & the - key (above +) for - 4. The Two Equals Signs Use the ENTER key to evaluate 3 + 4, not the = key. The = key is used, e.g. with solve(x^2 = 4, x). Exponents and Roots 7³ is keyed as 7 ^ 3. n \x = x 1/n , so you can key in x^(1÷n). Example 9^(1÷2) gives 3. Or you can use root(9,2). root( is Math [2 nd 5] 1: Number D: root(. Scientific Notation 6 × 10 -8 is entered as 6 EE(-)8. It appears on the screen as 6.E-8, but on your homeworks and tests you must copy that in proper scientific notation, i.e. as 6 × 10 -8 . Key the EE button only once. Degrees & Radian Modes To change from degrees to radians or vice versa, key MODE, then Angle. The Angle mode is displayed at the bottom center of the home screen as RAD or DEG. However the Angle mode can be overridden with ° & r . E.g.. in radian mode sin(30°) evaluates as ½, i.e. correctly, in degree mode sin((t/6) r ) evaluates as ½, i.e. correctly. The r symbol is not alpha R; it is keyed as 2 nd MATH 2: Angle 2: r . Thus radian mode is recommended, because it can be overridden more easily than degree mode. (° is keyed as 2 nd °.) Exact and Approximate Modes In AUTO mode (recommended), fractions will be displayed as fractions (e.g. “2/3”, not 0.666667) and pi as “t”. To change to Auto mode key MODE, 2 nd ▼, Exact/Approx, 1: Auto. The current Exact/Approx mode setting is displayed at the bottom center of the home screen as AUTO or EXACT or APPROX. To convert a fraction (e.g. 2/3) or t or \5 to a decimal, key + ENTER, instead of just ENTER or key one of the numbers with a decimal point, e.g. 2.÷3 will display as .66667. In Approximate mode results are always displayed as a decimal. Tips for Math with the TI-89 Calculator, page 3 More Symbols & Functions CATALOG contains all of the calculator’s functions (e.g. !, sinh -1 , E, nCr, nDeriv, abs.) It’s very long. To get close to your desired command, key the first letter of the command and then ▼ down. It’s not necessary to key alpha and the first letter of the command. For example to put seq( on the entry line, key CATALOG, 3, ▼, ▼, ▼, ▼, ENTER. 2 nd CHAR 2: Math A: gives , I: gives ², N: gives ±. These are just symbols, not functions, e.g. trying to evaluate 3² gives an error. Parentheses Use ( & ) for parentheses, not [ & ] or { & }. Lists, Matrices & Vectors {} delimits a list. E.g. {1, 2, 3} + 4 gives {5 6 7} {2, 1, .5} & sin({1, 2, 3}t/6) gives {2 \3/2 .5}. [] delimits a matrix or vector. The Plus/Minus Sign The list {1,-1} is effectively a ± sign. E.g. to graph x² + y² = 1, solve for y, i.e. y = ± 1 2 ÷ x , and key y1 = {1, -1}\(1-x^2). To solve the quadratic formula for 2x² + 3x - 4 = 0, key (-3 +{-1, 1}\(9 - 4 × 2 × -4) ) ÷ 4. This then displays both solutions. The F1 to F8 Keys The meanings of the F1 to F5 and 2 nd F6 to F8 keys are given on the top of the screen and depend upon which window is currently displayed. Pull down the menus in F1 to F8 and choose the desired operation. Some options may be unreadable. This means that the option is unusable in the current situation. A menu item (e.g. 5:approx) can be chosen either by scrolling down to highlight its line and keying ENTER or by keying its number (e.g. 5) If the result of algebra is a number Accidentally storing a number to a variable (This is surprisingly easy to do.) will produce unexpected results. If you type in, for example, expand((x-2)²), you expect “x²-4x+4”. If instead you get “4”, “x” probably has the value x = 0. To reset it, key Delvar x. Also see the next subject. Clearing ÷ is the back space key for erasing a single mistaken key stroke. If the cursor is sitting at the end of a line, CLEAR will erase the entire line. If the cursor is sitting in the middle of a line, CLEAR will erase the part of the line to the right of the cursor. If you want to clear the entire home screen, key F1 8: Clear Home. F6 Clean Up 1: Clear a-z will delete the definitions of any 1-character variables, i.e. “x”, but not “xx”. F6 Clean Up 2: NewProb does Clear Home, Clear a-z, deselects all plots & graphs. Shortcuts keying + ÷ gives ! (factorial). keying + < gives s. keying + > gives >. keying + = gives =. keying + ( accesses the Greek letters. For example + ( alpha s gives o. keying + EE gives a keyboard map with all of the + shortcuts. On Screen Syntax Help There are many commands which require several parameters, e.g. seq(, requires an expression, the name of the variable, starting & ending values and optionally the step size. If you do not remember the syntax and your manual is not handy, the syntaxes are given in CATALOG. Key CATALOG, 3 (to go to “s”) and ▼ to seq(. The syntax: EXPR,VAR,LOW,HIGH[,STEP] appears on the bottom line of the screen. The square brackets around STEP mean that the parameter STEP is optional. If you do not type it, the default value will be used. If the screen is too dark or light Key + + to make the characters darker or + - to make the characters lighter. Tips for Math with the TI-89 Calculator, page 4 If the calculator is locked-up If the screen won't come on, try turning the contrast up by keying Diamond + continuously for a few seconds. In case the calculator was actually off, key the ON button once and again try turning up the contrast. If the screen comes on dark, try turning the contrast down by keying Diamond - continuously for a few seconds. If this does not work, replace all of the batteries with new ones. If that doesn't fix it, again check the contrast. If that doesn't fix it, reset the memory by removing one battery and holding the (-) & ) buttons down replacing the battery and then holding them for another 5 seconds. If that doesn't fix it, remove all of the batteries, including the lithium watch battery. If that doesn't fix it, again check the contrast. To reload the operating system hold down 2nd, leftarrow, rightarrow and ON (2nd , hand and ON for the 92+/V200) simultaneously. If the screen comes on with an error message and then turns off again, reload the OS. To quickly move to the end of an expression To get to the start or end of a long expression or list, key 2 nd ◄, ▲, ▼ or ► as needed. To reuse a previous entry To reuse a previous entry, repeatedly key ▲ until the entry is highlighted. Then key ENTER, this will place the previous entry in the entry line, where it can be edited as necessary. Or you can key ENTRY (2 nd ENTER) repeatedly until the desired entry is in the home line To highlight text To highlight text (e.g. for copying or deleting) hold down l and highlight left or right with ◄ or ►. To copy text Highlight it as explained above then key COPY. Move the cursor to the place where the text is desired then key PASTE (+ESC). If you are lost If you are lost in some unfamiliar screen, key ESC to back up one screen or HOME to return home. To change the number of digits displayed To change the number of digits displayed, key MODE Display Digits. I recommend E: FLOAT, because the calculator displays all available digits in case you need them. The With key | means “with”. E.g. 2 + 3 - x | x = 5 ENTER gives 17. This key is very useful for limiting the domain of a function. For example if you want y = (x - 2)² to have an inverse, limit its domain by keying y1 = (x - 2)² | x ≥ 2 Insert Mode 2 nd INS toggles back & forth between insert & overtype mode. In insert mode (recommended) the cursor is a thin line between characters. In overtype mode the cursor highlights a character. log x ln x ÷ log e x is on the keyboard. log x ÷ log 10 x is not on the keyboard. To get it key CATALOG 5 ▲, ▲, ▲, ▲, ENTER. Or use log x = ln x/ln 10. E.g. ln 100000/ln 10 gives 5. Pi To enter t, use the t key, not 3.14 or 22/7. order of operations Your calculator knows the order of operations. E.g. 4 + 3 × 2 will be evaluated as 10. If you meant (4 + 3) × 2, key in the parentheses. Graphing To enter an equation to be graphed, key + Y=. Tips for Math with the TI-89 Calculator, page 5 Type in the equation. Key F4 3 to select or deselect an equations. Only selected equations are graphed. Key F6 Style to pick the way the graph is displayed (dotted, thick, etc.). This is useful if more than one equation is selected. If the window is too big (i.e. the graph is a tiny unhelpful squiggle) or too little (i.e. the important features of the graph are off the screen), try ZoomStd. If that does not work, reset the window size with + WINDOW. Inside + WINDOW: xmin & xmax set the values of x on the left and right sides of the window, similarly for ymin & ymax. xscl (“x scale”) sets the distance between tick marks on the x-axis, similarly for yscl. xres sets pixel resolution; 1 = highest resolution; 10 = lowest resolution; 2 is default. The lower the resolution, the faster a graph is drawn. See the discussion of friendly windows below. From OS 3.10 on xres is grayed out and set to 1, if the Discontinuity Detection (F1 Tools 9: Format) is set to ON. The default is ON. Zoom To view the equation’s graph, key + GRAPH. Inside + GRAPH, key F2 Zoom to resize the window (i.e. to change the maximum and / or the minimum value of x and / or y that is displayed). You might want to zoom out so that you can see all of the main features of the graph - intercepts, asymptotes, min/max and behavior as x ÷ ±·. You might want to zoom in so as to precisely determine an x- intercept or to understand a puzzling behavior of the graph. Inside + GRAPH F2 Zoom 1: ZoomBox zooms in on a box you draw. 2: ZoomIn & 3: ZoomOut zoom in & out by the amount you set in C: SetFactors. The default is 4 4: ZoomDec & 8: ZoomInt set friendly windows. See the discussion of friendly windows below. 5: ZoomSqr scales x & y the same, so circles look round, squares look square and ± lines look ±. 6: ZoomStd sets x & y min = -10, x & y max = 10 and x & y scl = 1. If nothing appears on the screen, try this first. 7: ZoomTrig is useful for graphing trig functions. It sets the pixel size = t/24 = 7.5° and xscl = t/2 = 90°. 9: ZoomData, for use with scatterplots or histograms data, sets xmin & xmax to match the data. A: ZoomFit resizes y to fit the graph. F2 Zoom can also be accessed from inside + WINDOW. Friendly Windows A friendly window is a window where the x coordinates of the pixel elements are round numbers, e.g. 1, 2, 3, 4, ... or 0.1, 0.2, 0.3, 0.4, ... This is very helpful if you want to see a hole in a graph. For example ( 1)( 2) ( 1) x x y x ÷ + = ÷ has the same graph as y = x + 2 except that there is a hole in the line at x = 1. In + GRAPH, Zoom 4: ZoomDec (decimal) sets the pixel size to 0.1 and the window dimensions to -7.9 < x < 7.9, -3.8 < y < 3.8. Zoom 8: ZoomInt (integer) sets the pixel size to 1.0 and allows you to use the arrows to move to the center to the part of the graph you wish to investigate. (ZoomDec does not allow this option.) If you choose to center the graph at the origin, the window size is -79 < x < 79, -38 < y < 38. But xres determines how many pixels are actually traceable. (Skipping pixels speeds up the graphing process.) What you actually trace is 1, 2, 3, 4, ... or 0.1, 0.2, 0.3, 0.4, ... (for ZoomInt & ZoomDec and respectively) times xres. If you really want 1, 2, 3, 4, ... or 0.1, 0.2, 0.3, 0.4, ... (normally you do want that), set xres to 1. The default is 2. The scale set by ZoomInt is often the wrong size, i.e. you get steps of 1.0, but need steps of 0.0001 or maybe 1000, etc. If you need steps of 0.1, then use ZoomDec. But if the region you want to trace is off the screen, you are out of luck with ZoomDec. It is possible to develop a formula to calculate the window size to set to solve this problem, but that is tedious. Here is a trick. In +GRAPH set F2 Tips for Math with the TI-89 Calculator, page 6 Zoom C: Set factors, set xFact and yFact to 10 (or if you prefer a smaller zoom step size to \10 - the default zoom step size is 4). Select F2 Zoom 8: ZoomInt. Set the center where you need it. Now Select F2 Zoom 2: ZoomIn twice (or once if you set the zoom factor to 10). Now you have step sizes of 0.1. If you need step sizes of 0.01, repeat, etc. Trace Inside + GRAPH, F3 Trace puts a cursor on the graph & displays the coordinates of the cursor. The cursor can be moved along the curve with ◄ or ► or by typing an x value and ENTER. This can be used for finding intercepts or other solutions to the equation, for reading out data points in a scatter plot or histogram heights. Unfortunately since trace moves from pixel to pixel, it usually does not land exactly on the desired point. Therefore if the coordinates are needed accurate to 3 significant figures, do not use trace. To change the center of the graph To change the center of the graph, move the cursor to the desired center and key ENTER. To format a graph To format a graph (rectangular vs. polar, grid on/off, label axes on/off, etc.) key +( or in Y=, Window or Graph, key F1 9: Format. To cancel a graph To cancel a graph while it is being plotted, key ON. Modes The normal graphing mode is MODE Graph = 1: FUNCTION. Use this e.g. to graph y1(x) = x². The other modes are used for parametric, polar sequence, 3D, and differential equations graphs To store a window setting To store a window setting with F2 Zoom B: Memory 2: ZoomSto. You can recall your stored setting with B: Memory 3: ZoomRcl. To depict an inequality To depict an inequality (e.g. y > 2x + 3) on the x- y plane by graphing the inequality as an equality solved for y (e.g. y1(x) = 2x + 3) and then shading above or below the graphed line, depending on whether the inequality was actually > or <. (You want to shade above in our example.) To set the shading in Y=, key F6 Style 7: Above or 8: Below as needed. There are 4 shading patterns which are automatically cycled through. So 4 different inequalities can be displayed. Parametric Graphs To make a parametric graph key MODE Graph = 2: PARAMETRIC. A parametric graph is made on the x-y axes by defining x = f(t) & separately y = f(t). Thus in parametric mode, you must type in a pair of equations. E.g. in Y= xt1 = sin 2t yt1 = sin 3t To view the graph of the above set x & y min/max = ±1 & use radian mode. You will get a pretty Lissajous figure. You must use t (not x, y or z) as your independent variable. + Window now has (in addition to xmax, yscl, etc.) tmin, tmax & tstep, which you may need to set. Polar Graphs To graph in polar coordinates key MODE Graph = 3: POLAR. The Y= screen will now read r1=, etc. You must use u (not t, x, y or z) as your independent variable. u is +u (above the ^ key). Use ZoomSqr to set the correct proportions or do it by hand by setting xmin & xmax to twice ymin & ymax. + Window now has (in addition to xmax, yscl, etc.) umin, umax & ustep, which you may need to set. If some functions are selected, they might graph along with your polar graph. To turn them all off, in Y= key F5 ALL 1: All Off, 3: Functions Off or 5: Data Plots Off as needed. + | Coordinates ÷ Polar will cause F3 Trace to display the coordinates r & u. + | Coordinates ÷ Rectangular will cause F3 Trace to display the coordinates x, y & u. Tips for Math with the TI-89 Calculator, page 7 To find the Minima or Maxima of a function In + GRAPH to find the Minima or Maxima of a function use F5 Math 3: Minimum or 4: Maximum. You will be prompted to choose an x value on each side of the zero. Solving a system of two equations I.e. Finding the Intersections of Two Graphs Solving a system of two equations means finding the intersections(s) of their graphs. Solve the 2 equations for y. Graph them using the + Y= and + GRAPH. If the intersection(s) do not appear on the screen, zoom out (or better by using + Window reset x or y min or max until the intersection(s) can be seen on the screen). Key F5 Math 5: intersection. In case you have other functions displayed on the screen besides the ones you want to solve, you will be prompted with “1 st Curve?” and then “2 nd Curve?” to choose the two functions whose intersections you want to find. If the cursor is on the correct 1 st & 2 nd curves, just press ENTER to answer each question, otherwise use the ▲ or ▼ key to move to the correct curve(s). Or better still, use Y= and then F4 to uncheck the unneeded curves. Then you will be prompted “Lower Bound?” Use the ◄ key to move the cursor to the left of the intersection and key ENTER or if you know roughly the x-coordinate of the intersection, key in an x value less than the x- coordinate of the intersection. You will be prompted “Upper Bound?” Use the ► key to move the cursor to the right of the intersection and key ENTER or if you know roughly the x-coordinate of the intersection, key in an x value more than x-coordinate of the intersection. “Lower” is a bit confusing. Remember it means “left”, not “below”. In response to “Upper Bound?”, similarly choose an x-value to the right of the intersection. To move the cursor faster, use 2 nd left or right arrow. Then the (x, y) coordinate of the intersection will be displayed at the bottom of the screen. To find the x and y intercepts To find x and y intercepts, graph the equation and zoom in or out with F2 Zoom (or better by using + Window reset x & y min & max until the x and y intercepts can be seen on the screen). To use the following method the intercepts must appear on the screen. To find the y-intercept, key F3 Trace, then 0, then ENTER. This will move the cursor to x = 0, i.e. the y-intercept. The y-intercept will be displayed in the lower right of the screen. To find the x-intercept, key F5 Math, then 2: Zero. You will be asked for the “Lower Bound?”. Either place the cursor to the left of the x-intercept with the ◄ key and key ENTER or if you know roughly the value of the x-intercept, key in an x value less than the value of the x-intercept. Then you will be asked for the “Upper Bound?” Either place the cursor to the right of the x-intercept with the ► key and key ENTER or key in an x value greater than the value of the x-intercept. Tables To make a table, create a function in +Y=. +TABLE produces a table of the functions selected in +Y =. Depending on the MODE setting, this could be functions y1, y2, etc, or parametric functions y1t, x1t, etc, or polar angle u, etc. Below I assume you are in function mode. There is a choice of ways to choose which x values to display. In TABLE SETUP, which can be accessed with + TblSet or inside TABLE with F2 Setup AUTO table automatically generates a series of values for x or you can choose them yourself with ASK. For example key y1(x) = sin x. Deselect any previously selected functions. Set the table parameters with tblStart = -90, Atbl = 15 (assuming you are in degree mode & want a table of sin x for every 15° starting at -90°). Graph <-> Table = OFF (unless you want to use xmin & xres to set tblStart & Atbl, which is not recommended). Independent = AUTO. ENTER If you want to choose your own values of x, use Ask. E.g. in + TABLE key F2 Setup, Independent ÷ Ask, ENTER. Then key in your x value, ENTER, ¹, then key in another x value, etc. To change the cell width in TABLE key +| or F1 9: Format Tips for Math with the TI-89 Calculator, page 8 Asymptotes The TI-89 does not draw asymptotes, but because of the way it draws a curve, fake asymptotes sometimes appear. It evaluates y at the center (in x) of each pixel, draws a dot there and connects the dots. Thus where an asymptote should appear, a slightly crooked fake asymptote might be drawn. If you want to get rid of fake asymptotes, in Y= set the style to dot. Or set the scale so that a pixel element falls on the asymptote. E.g. for y = 1/(x- 1) there is a vertical asymptote at x = 1. F2 Zoom 4: ZoomDec and + WINDOW xres = 1 will put a pixel element at x = 1 and thus no fake asymptote will appear. Discontinuity Detection From OS 3.10 on fake asymptotes no longer appear, if the Discontinuity Detection (F1 Tools 9: Format) is set to ON. The default is ON. You can access the Graph Formats window from either the Graph screen itself, the Y= Editor, or the Window Editor. Unfortunately xres = 1 makes graphing slow and Discontinuity Detection ON makes it even slower. If you want to draw an asymptote. If you know where an asymptote is, you could key its equation in - perhaps in a different style and add it to your graph. Horizontal asymptotes can be easily keyed in. E.g. a horizontal asymptote at y = 2 would be graphed as y2(x) = 2. Vertical asymptotes can only be approximated, e.g. for a vertical asymptote at x = 2, use y2(x) = 10^100(x-2), which is a line which goes through the point (2, 0) and has such a steep slope that it will appear perfectly vertical. Alternatively type y2(x) = when(x<2,-10^100, 10^100). This draws a horizontal line at -10 100 for x < 2 and a horizontal line at 10 100 for x > 2. Both are way off screen. What you see is the vertical line connecting these two lines. Alternatively type LineVert 2 in the home screen, not in Y=. You cannot trace or find intersections, etc. of a line drawn with LineVert. Piecewise-Defined Graph To display a piecewise-defined graph, e.g. f x x x x x ( ) , , = > s ¦ ´ ¹ 1 1 2 , inside Y= key y1 = when(x > 1, x, else x^2) when( is in CATALOG. If the graph has more than 2 pieces, e.g. g x ( ) = ÷ s < s ¦ ´ ¦ ¹ ¦ x, x 0 3, 0 x 1 x, 1 < x , you can use nested when functions or a user- defined function. Unfortunately the logic of nested when functions is hard to follow, especially since you have to read it inside the small entry line. It is easier is to enter a piecewise defined equation as separate equations, then selecting each with F4 3. E.g.: y1(x) = -x | x s 0 y2(x) = 3 - x + x | 0 < x and x s 1 y3(x) = x | 1< x The unexpected “- x + x” term was used to get an x into the expression. Without an x in the expression “y2(x) = 3 |0 < x and x s 1” would have drawn a 3 for all x, not just for 0 < x s 1. The Boolean operator and is 2 nd MATH 8: Test 8: and or CATALOG and. Unfortunately if you want to then graph g(x - 3) or if you need a table of g(x) or if you want to find values e.g. g(-3), then you need to express the function in one entry. If so the g(x) above can then be defined using when(test, expression when test true, expression when test false). For example f x x x x x ( ) , , = > s ¦ ´ ¹ 1 1 2 can be written as, y10(x) = when (x > 1, x, x^2). To write g x ( ) = ÷ s < s ¦ ´ ¦ ¹ ¦ x, x 0 3, 0 x 1 x, 1 < x , you must use nested when(s, For example: y1(x) = when (x s 1, when(0 < x, 3, -x), x) or y1(x) = when (x s 0, -x, when(xs 1, 3, x)) Notice the difference in the inequalities used above and that the -x + x term is no longer needed. For example 4 - x², x < 1 f(x) = 1.5x + 1.5, 1 s x s 3 x + 3, x > 3 is entered as y1 = when(x<=3, when(x < 1, 4 - x^2, 1.5x + 1.5), x + 3) Tips for Math with the TI-89 Calculator, page 9 Once you have defined a function in Y =, you can use it in the entry line. E.g. with the above definition, y1(-25) evaluates as 25 To display f(x)={x, x s 1, key y1 = when(x s 1, x, undef). undef means undefined, so it won't draw anything if x > 1. Just type “undef “ in, it’s not in the catalog! Graphing a Real Function The domains of some functions are restricted, because they produce complex results for some x. For example y y = x 2/3 is considered to be defined only for x > 0 and y = 4 2 ÷ ÷ x x is considered to be defined only for x ≥ 4. The TI- 89 can be commanded to show just these parts by changing the complex format mode, but the required setting is a bit inconsistent. In complex REAL mode y = x 2/3 is (incorrectly) graphed for all x, but y = 4 2 ÷ ÷ x x is (correctly) graphed only for all x > 4. In complex RECTANGULAR mode y = x 2/3 is (correctly) graphed for x ≥ 0, but y = 4 2 ÷ ÷ x x is (incorrectly) graphed only for all x > 4 and also for x ≤ 2. In complex POLAR mode an attempt to graph either function crashes. Solving Inequalities Graphically Example y1 = x+2 y2 = -x²+4 Find the values of x such that y1<y2. In y3 type when(y1(x)<y2(x),1,0). Get when( from the catalogue. Turn off y1 & y2 and key GRAPH. The solution, a horizontal line at y = 1 for -2 < x < 1, is displayed. The disadvantage of this method is that it’s impossible to get the exact endpoints with this method. Complex Numbers Converting from Rectangular to Polar or Trigonometric Form To convert a complex number in standard (i.e. rectangular, i.e. a + bi) form to polar (i.e.. r e iu ) form, set the mode to complex polar with MODE Complex Format 3: POLAR, key in the complex number in standard form & press ENTER. The way the result is displayed depends on what modes you are in. You may need to reset the Exact/Approx mode or the Angle mode (i.e. Degree vs. Radian mode). For example: In exact, radian mode \2 + \2i ENTER gives 2 e it/4 . In approx, radian mode \2 + \2i ENTER gives 2 e 0.785 i . In exact, radian mode \5 + i ENTER gives e i arctan\5/5 \3 \2. In approx, polar, degree mode \2 + \2i ENTER gives 2.449Z24.095. Etc. Apparently the TI-89 cannot be commanded to display in trigonometric form (e.g. 2 cos 45° + 2i sin 45°). To convert a complex number in polar (i.e. re iu ) form to a complex number in standard (i.e. rectangular, i.e. a + bi) form, set the mode to complex rectangular with MODE Complex Format 2: RECTANGULAR, key in the complex number in standard form and press ENTER. Again the way the result is displayed depends on what modes you are in. For example In approx mode 2e it/4 ENTER gives 1.414 + 1.414 i. In exact mode 2e it/4 ENTER gives \2 + \2i. Solving with Complex Numbers cSolve() or cZeros() can be used to find the roots of complex numbers. For example ³\i = cSolve(x³ = i, x) = cZeros(x³ - i, x) = {-i, -\3/2 + ½ i, \3/2 + ½ i} DeMoivre’s Theorem can be used to display and trace (i.e. read out) the roots of complex numbers. For example DeMoivre’s Theorem says that the cube roots of 2i = 2 e it/2 are ³\2 e i(t/2+2tk)/3 , with k = 0, 1, 2. To graph these values set the Graph mode to Polar. In Y= key 2^(1/3). In WINDOW set umin to the angle of the first root (i.e. (the argument of 2i) / n = (t/2)/3 = t/6). Set ustep to 2t/n = 2t/3. Set the window size appropriately, i.e. a little bigger than ³\2 e.g. ±2. Key GRAPH. Key F2 Zoom 5: ZoomSqr (to make the plot “circular”). Key F3 Trace. The first root, 1.091 + .630 i will be displayed (as 1.091, .630). Key ► to display Tips for Math with the TI-89 Calculator, page 10 the second root -1.091 + .630 i. Key ► again to display the third (& last) root 0 - 1.260 i. There are at least five possible ways (!) that the results of cSolve() or cZeros() can be displayed. E.g. for x² = 4i, i.e. finding the square roots of 2i, the answer could be displayed in exact polar radian form e iu , e.g. 2e it/4 and 2e it/4 or approx polar radian form e iu , e.g. 2 e 0.785 i and 2e -2.366 i or polar degree form (r Z u), e.g. (2 Z 45) and 2 Z -135) or exact standard form a + b i, e.g. \2 + \2 i and -\2 - \2 i, or approximate standard form a + bi, e.g. 1.414 + 1.414i & -1.414 - 1.414i. Apparently the TI-89 cannot be commanded to display the results in trigonometric form (e.g. 2 cos 45° + 2i sin 45°). Which form is displayed is controlled by 3 (!) mode settings. Mode: Complex Format 3: Polar will always display the result in polar form. Mode: Complex Format 1: Real or 2: Rect will always display the result in standard form. Mode: Exact/Approx 2: Exact will always display the result in Exact form, i.e. e it/4 or \2/2 + \2/2 i, depending on the Complex Format mode. Mode: Exact/Approx 3: Approximate will always display the result in Approximate form, i.e. e 0.785 i or .707 + .707 i, depending on the Complex Format mode. In Angle 1: Radian mode the polar radian form, r e iu , is displayed, i.e. e it/4 . In Angle 2: Degree mode the polar degree form, (r Z u) i.e. (1 Z 45), is displayed . Graphing a Complex Function A complex function, z = x(t) + y(t)i, can be graphed on the complex axis in parametric mode as xt1 = x(t) and yt1 = y(t). Or if z is not explicitly written with real and imaginary parts, i.e. just as z = f(t), use xt1 = real(f(t)) and yt1 = imag(f(t)). Converting from Rectangular to Polar Coordinates To convert from Rectangular to Polar Coordinates key [x,y] 2 nd MATH 4: Matrix L: Vector ops 4: ¬ Polar For example [1, \(3)]: ¬ Polar gives [2 Z t/3] To convert from Polar to Rectangular Coordinates key [r,Zu] 2 nd MATH 4: Matrix L: Vector ops 5: ¬ Rect For example [2, Z t/3] ¬ Rect gives [1 \3] The “Z ” is 2 nd EE. The “,” does not display in the answer, but it is there. Other Commands The Solve & Zeros commands To solve an equation key F2 Algebra 1: solve( in the entry line. E.g. solve(2x + 3 = 7,x) gives x = 2. This single line will do about half of your homework or test problems! If I want you to do the problem yourself by hand, I will require you to show your work or I will prohibit calculators on part of the test. Some equations cannot be solved analytically (i.e. with algebra), e.g. x = cos x. With a grapher there are several ways to solve it. One way is graphically by plotting y1 = x, y2 = cos x and in + GRAPH find the intersection of two functions using F5 Math 5: intersection. You will be prompted to choose two functions and an x value on each side of the intersection. Or in the entry line use solve(x = cos(x), x), giving “x = .739085133”. Or set the equation equal to zero (e.g. x - cosx = 0) and in the entry line use zeros(x - cos(x), x), giving “x = .739085133”. Zeros is F2 4: zeros(. Or graph y1 = x - cosx and find the zeros of the graph with F5 Math 2: Zeros. You will be prompted to choose an x value on each side of the zero. For a more detailed explanation of how to use the zeros feature, see below. The above two Zeros functions are different. The one used in the entry line does not ask for a range to search and might not find all zeros. Of all of the above methods, I recommend using intersection in the GRAPH screen, because by looking at the graphs you can understand what is going on, because if there is more than one intersection you can choose the one you want and because setting the equation equal to zero or equal to y involves extra algebra and is therefore not as intuitive. Either Solve() or Zeros() can be used for most problems. For example Solve(x² - 4 = 0, x) or Zeros(x² - 4, x) will both give 2 & -2. Tips for Math with the TI-89 Calculator, page 11 There are three Solve( & two Zeros() commands: Solve(, Zeros(, cSolve(, cZeros( and nSolve(. If you want complex solutions, you must use cSolve() or cZeros(). Solve() or Zeros() or nSolve() will give only the real solutions. Solve() is 2 nd Math 9: Algebra 1: Solve(). Zeros() is 2 nd Math 9: Algebra 4: Zeros(). nSolve() is 2 nd Math 9: Algebra 8: nSolve(). cSolve() is 2 nd Math 9: Algebra A: Complex 1: cSolve(). cZeros() is 2 nd Math 9: Algebra A: Complex 3: cZeros(). nSolve looks for only one approximate real solution, but is faster than Solve() or Zeros(). Solve is very powerful. - It can solve symbolic equations, e.g. Solve(a-x^2 + b*x + c = 0, x) will give the quadratic formula. - It can find the intersection of two equations, e.g. Solve(x^2 + y^2 = 4 and y = x^2, {x, y}) will give (±1.250, 1.562) i.e. the intersections of the circle and the parabola. (“and” is 2 nd Math 8: Test 8: and.) - Solve(x 4 - 1 = 0, x) will give x = 1 or x = -1, but cSolve(x 4 - 1 = 0, x) will give x = i or x = -i or x = 1 or x = -1. - To speed up the process, you can also include a guess e.g. Solve(x^2 = 2, x = 1.4) will give 1.414, although the equation must be more complicated before the speed-up is noticeable. Solving inequalities OS 3.10 can solve inequalities. Example solve(abs(x - 1) < 3, x) gives -2 < x < 4 Factoring and Expanding To factor an expression key F2 Algebra 2: factor( in the entry line. E.g. factor(x^2 - 5x + 6, x) gives (x-3) (x-2). E.g. factor(x³ + 3x²y + 3xy² + y³) ENTER gives (x + y)³. E.g. cfactor(x 4 -1) gives (x-1) (x+1) (x-i) (x+i) To expand an expression key F2 Algebra 3: expand( in the entry line. E.g. expand(x + y)^3) ENTER gives x³ + 3x²y + 3xy² + y³. Permutations and Combinations Combinations are the number of ways of selecting r objects from n objects where order does not count = n C r = n! / r! (n-r)! is also written as n r | \ | . | . Permutations are the number of ways of selecting r objects from n objects where order does count = n P r = n! / (n-r)! n C r is 2 nd MATH 7: Probability 3: n C r ( n P r is 2 nd MATH 7: Probability 2: n P r ( Sequences and Series A sequence is an ordered list of numbers. A series is the summation of the terms of a sequence. They can be calculated using 2 nd MATH 3: List 1: seq(. Seq takes the parameters (expression, variable name, begin, end [, step]). Step is optional, the default is 1. E.g. to display the first 6 terms of the arithmetic sequence a n = 3n + 2, key seq(3x+2, x, 1, 6) ENTER which gives {5 8 11 14 17 20}. Note I used “x” rather than “n” as my variable to make the keying easier. seq(3x+2, x, 3, 5, ½) ENTER gives {11 25/2 14 31/2 17}. To find the sum of a sequence you can use sum(seq( or E(. The sum & E features are different. Sum just takes a list to sum, for example sum({1, 2, 3}) gives 6. E( takes parameters (expression, variable name, begin, end). E.g. to find the sum of the first 4 terms of the geometric sequence a n = 5-2 n , key sum(seq(5*2 x , x, 1, 4) ENTER or key E(5*2 x , x, 1, 4) ENTER, either of which give 150. Since E( is the same as sum(seq(, it’s easier to just use E(. But E( does not accept a step size, i.e. the step size is set to 1. If another step size is needed, e.g. for Riemann sums, then you must use sum(seq(. For example for f(x) = 2 x to find f(2) + f(2.2) + f(2.4 + ... +f(4), use sum(seq(2 x ,x,2,4,.2)), which gives 96.700. Or you can get clever with E(. To find the sum of the n terms of f(x) from x = a to b (i.e. with step size =Ax = (b-a)/n use E(f(a+Ax×x),x,0,n). For example for f(x) = 2 x to find the sum of the 10 terms f(2) + f(2.2) + f(2.4) + ... +f(4), Ax = (4- 2)/10 = .2, use E(2 2+.2 x , x, 0, n), which, of course, also gives 96.700. Tips for Math with the TI-89 Calculator, page 12 The result can contain symbols, e.g. E(x,x,1,n) ENTER gives n(n+1)/2. To find the partial sums of a sequence use cumSum(. E.g. key 2 nd MATH 3: List 7: cumSum(seq(1/x², x, 1, 4)) ENTER gives {1 5/4 49/36 205/144} A sequence graph is will display a sequence. Set MODE Graph SEQUENCE. Key + Y=. Enter the formula for the sequence in u1. If an initial value is needed, e.g. if a recursive formula for the sequence is entered in u1, enter the initial value in ui1, otherwise leave it blank. E.g. key in u1 = 3n+2. You must use “n” as your variable in sequence mode. Key + GRAPH F2 Zoom A: ZoomFit. The axes are n and u1. You can use F3 Trace to read out the values of u1 (they are called yc). Matrices To create a matrix key APPS Data/Matrix Editor, 3: New, Type 2: Matrix, Folder: Main, Variable: (Give it a name, e.g. a, which you get by typing =, because by default alpha is already turned on.). To type the row & column dimensions you must key alpha to turn alpha off. Determinant is in MATH (2 nd 5) 4: Matrix 2: det( Inverse is ^-1 Example Solve 3 4 5 6 2 1 3 2 5 x y z x y z x y ÷ + = ÷ + = ÷ ÷ = with 3 4 5 6 1 2 1 , 1 , 3 2 0 5 x a b X y z ÷ | | | | | | | | | = ÷ = ÷ = | | | | | | ÷ \ . \ . \ . this can be written a X = b. So the solution is X = a^-1 b giving 26 / 7 43/14 10 / 7 | | | | | \ . Matrices in the Home screen The TI-89 requires several keystrokes before you begin typing the matrix vales. The TI- 84 is needs fewer keystrokes. Therefore some prefer to type a matrix in the home screen Example Solve 3 4 5 6 2 1 3 2 5 x y z x y z x y ÷ + = ÷ + = ÷ ÷ = [[3,-4,5][1,-2,1][3,-2,0]]^-1 [[6][-1][5]] gives 26 / 7 43/14 10 / 7 | | | | | \ . . How to graph a Conic Equation Since a conic equation is not solved for y, it cannot be directly entered into Y=. But by treating x as a constant, a conic can be rearranged into a quadratic equation in y. Then the quadratic formula can be used to solve for y. The general form of the conic, Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, can be rearranged to Cy 2 + (Bx + E)y+(Ax 2 + Dx + F) = 0. Note that this now has the form of a quadratic in y. Applying the quadratic formula gives y Bx E Bx E C Ax Dx F C = ÷ + ± + ÷ + + ( ) ( ) ( ) 2 2 4 2 . There is one problem with keying in this formula, there is no “±” button. The solution is easy, the list {1, - 1} is effectively a “±” sign. If you have one conic to graph, key the above formula in for y1= with the letters (A, B, C, etc.) replaced with their values. For example for x² - 4xy + 4y² + 10x - 30 = 0 (i.e. A = 1, B = -4, C = 4, D = 10, E = 0, F = -30), key in y1 = (4x+{1,-1}\((-4x)^2-4*4(x^2+10x- 30)))÷(2*4). If you want to graph several conics, it is faster to create a function called conic(x) which contains the above formula, i.e. in the home screen key (-(b*x+e)+{1,-1}*\((b*x+e)^2- 4*c*(a*x^2+d*x+f)))÷ (2*c) ÷ conic(x), where ÷ is the STO÷ button. Note that you must explicitly type b*x, not bx, otherwise bx will be interpreted as a variable name, not b times x. Then for each conic just key in the values of A, B, C, etc. For the above example, key 1 ÷ A, -4 ÷ B, 4 ÷ C, 10 ÷ D, 0 ÷ E and -30 ÷ F. Now set y1(x) = conic(x), and graph it. Tips for Math with the TI-89 Calculator, page 13 How to Simplify Rational Functions To do long division of one polynomial by another, use expand(. For example expand((x^2+4x+6)/(x+2), x) gives 2/(x+2)+x+2. Expand is F2 Algebra 3: expand(. MODE Split Screen 3 LEFT-RIGHT or 2: TOP- BOTTOM is useful with graphs. Make one screen the graph and use the other screen to adjust the window and to set the style (dot, thick, etc.) of the graph. To Solve a System of Equations with Solve( To solve a system of three equations and three unknowns key solve(eqn1 and eqn2 and eqn3, {var1,var2,var3}). E.g. to solve x + 4y - z = 6, 3x + 2y + 3z = 16 and 2x - y + z = 3, key solve(x+4y-z=6 and 3x+2y+3z=16 and 2x-y+z=3,{x,y,z}), which gives x=1 and y=2 and z=3. The Inverse To find analytically the inverse of f(x), written f -1 (x), key solve(y = f(x), x). Then exchange x and y by hand. Solve( is F2 Algebra 1: solve(. For example solve(y = 2x + 1, x) gives x = (y-1)/2. So f -1 (x) = (x-1)/2. Solve(y = ln(x), x) gives x = e y . So f -1 (x) = e x . There are several ways to graph the inverse. 1) You can first find f -1 (x) analytically as above and then just key it in in Y =. For example y = x² Solve it for y, i.e. y = ±\x. This can be graphed as y ={1,-1}\x. The list {1,-1} is effectively a ± symbol. 2) Or use parametric mode: xt1 = t yt1 = f(t) will produce a graph of f(x). xt2 = yt1(t) yt2 = t will produce a graph of f -1 (x). For example, xt1 = t, yt1 = t^2, will produce a graph of y = x², xt2 = yt1(t), yt2 = t, will produce a graph of x = y², its inverse. You may need to adjust the values of tMin and tMax to allow for the range of y values you want. 3) Or type on the command line DrawInv x^2, which draws the inverse of y = x². All of the above is for the inverse relationship i.e. where you do not require that the inverse be a function. If you want the inverse function and the inverse relationship is not already a function, it does not work. To obtain the graph of the inverse function use Method 1. If the function did not have an inverse function because it contained a y² term (e.g. y = x² or x² + y² = 1), proceed as in 1 above, but remove the {1, -1}. If f(x) contains a transcendental function, (i.e. one which cannot be inverted algebraically, e.g. tan x), use the inverse transcendental, e.g. tan -1 x. For example solve(y = 2 tan tx + 3,x) gives x = {tan -1 [(y-3)/2]}/t + @njt. (The j stands for an integer. @nj is the name of an integer constant. If the solution has two constants, they will be named @n1 and @n2.), i.e., in more standard notation, x = {tan -1 [(y-3)/2]}/t + kt, k = an integer. To get the inverse function just drop the + @njt part. So the inverse function is f -1 (x) = tan -1 [(x-3)/2]/t. In this example DrawInv 2 tan tx + 3 or graphing y1 = {tan -1 [(x-3)/2]}/t + @njt, gives a series of curves, one on top of the other. On the other hand graphing y1 = {tan -1 [(x-3)/2]}/t gives just the one curve of the function, called the principal branch. Linear Interpolation If you have a table of data, e.g.: x y = f(x) 4 24.7 5 25.2 6 25.8 and wish to find a y-value corresponding to an x- value between those x-values in the table, assume the (unknown) function y = f(x) is locally linear and use linear interpolation. Use the following equation, where x 1 and x 2 bracket x.: y = y 1 + (y - y )(x - x ) x - x 2 1 2 1 1 E.g. find f(5.8) = 25.2 + (25.8-25.2)(5.8-5)/(6-5) = 25.68 Tips for Math with the TI-89 Calculator, page 14 Step functions [x] = the greatest integer less than or equal to x = int(. This function is also called floor(. For example, int(3.5) returns 3 and int(-3.5) returns -4. If you want the smallest integer greater than or equal to x, use ceiling(. For example, ceiling(1.5) returns 2. ceiling(-1.5) returns -1. If you want the integer closest to zero, i.e. simply removing any fractional part, use iPart(. For example, iPart(1.5) returns 1 and iPart(-1.5) returns -1. These can be found in the MATH 1: Number. To Simplify Expressions Usually expressions are simplified automatically, but sometimes this does not happen or the default form is unsatisfactory. Both expand( and factor( can be used (sometimes back to back) to force a simplification of an expression or equation. Also propFrac( is useful in some cases. If trig is involved, tExpand( and tCollect( can be used in a similar fashion. All of these are in F2 Algebra. Binary, Hexadecimal & Decimal Decimal means base 10, binary means base 2 & hexadecimal means base 16. How do you tell the calculator you are entering a binary or hexadecimal number? You type 0b before a binary number & 0h before a hexadecimal number. For example to enter 7 in binary, key 0b111. To enter 15 in hexadecimal, key 0hF. How do you set the mode of the answer of a calculation? To display your answer in binary, key MODE, Base, 3: BIN. To display your answer in hexadecimal key MODE, Base, 2: HEX. To display your answer in decimal (i.e. the normal way) key MODE, Base, 1: DEC. You can enter the number in any mode, the answer will be displayed in the chosen mode. For example in decimal mode 0b111 + 0b1 gives 8. In binary mode 0b111 + 0b1 gives 0b1000. In hexadecimal mode 0b111 + 0b1 gives 0h8. You must always use the prefixes. For example in hex mode 11 + 3 gives 0hE, i.e. 11 is interpreted as decimal. How can you convert a number from one base to another? There are two ways. Firstly you can set the mode to the desired base and key in the number with the correct prefix. For example in decimal mode 0b111 ENTER gives 7. In hex mode 15 ENTER gives 0hF. Alternatively you can use the convert base commands in 2nd MATH D: Base, which are 1:(Hex, 2:(Bin & 3:(Dec. For example, 0b11010(Hex gives 0h1A and 0b11010 (Dec gives 26. These commands only work in EXACT or AUTO, not APPROX mode. To delete functions, lists, tables, etc. To delete tables, etc. key 2 nd VAR-LINK. All of your (user defined) variables: tables, functions, lists, text entries are listed. Highlight the table, etc. you want to delete. Key ÷ Enter. If you want to delete several variables, select all to be deleted with key F4 3, then key ÷ Enter as above. If you want to delete all but a few variables, key F5 All 1: Select All. Then use F4 3 to deselect those tables you do not want to delete. By default all user defined variables are placed in the MAIN folder unless you specified otherwise. You cannot delete the MAIN folder. Define F4 Other 1: Define can be used to define a function. E.g. Define xxt(x) = 3x^2. Then xxt(5) ENTER gives 75. Notice that I used letters available on the keyboard to make typing the name faster. You would have gotten an error message if you had tried to define xt1(x), because xt1(x) is a system variable. There is another way to define a function. 1÷cos(x) STO¬ sec(x) defines a new function, sec(x) which can, for example, be graphed by entering y1 = sec(x) or evaluated by typing sec(t) which gives -1. An erroneous set of keystrokes can result in defining x & y. This is surprisingly easy to do by accident. E.g. if x has been defined as 2, expand((x+2)²) will give 16. If something like this happens, F6 Clean Up 1: Clear a-z ENTER ENTER will clear (i.e. delete) the definitions of any 1-character variables, i.e. “x”, but not “xx”. Tips for Math with the TI-89 Calculator, page 15 If you want to use your newly-defined function, easier than keying in the letters is copying a variable from VAR-LINK. Key 2 nd VAR- LINK. Highlight the variable (e.g. the sec(x) function created above) ENTER, sec(x) now appears in the entry line. Programming To type in a program, key APPS 7: Program Editor 7: Program Editor 3: New Type: Program. Give it a name, e.g. Variable: example. The lines :example() :Prgm : :EndPrgm then appear automatically. Note the first line is the name you assigned to the program. Type in your program between Prgm and EndPrgm. Key ENTER after you finish each line. When you are done typing the program, key HOME. To save on typing key F3 I/O to insert “Disp”, “Input”, “Output”, etc. into your program. To run the program, type its name in the entry line on the home screen, e.g. example(). Note that the () were not used above when naming the program. Lbl creates a label which the program can later jump to with a Goto statement. Input [prompt], variable prints the prompt on the screen. After you type in a value for the variable and key ENTER, that value is assigned to the variable. Disp displays the current contents of the Program I/O screen. To leave the Program I/O screen, i.e. stop a running program, key F5 PrgmIO or key ON ENTER HOME. On Line Help TI provides dozens of forums on its TI Calculators, for example there are forums entitled “TI-89”, “Precalculus”; “Calculus”, etc. Look at links below http://education.ti.com/index.html or http://education.ti.com/student/TIStudentCenterH ome.html or http://www-s.ti.com/cgi- bin/discuss/sdbmessage.cgi?databasetoopen=calc ulators&topicarea=TI-89/92+Plus&do_2=1 Error Messages Error: Dimension Error: Dimension means that the variable you are using, e.g. x, is not the correct dimension for the function you are using, e.g. Factor(x^2 - 9). This is probably because by accident you stored something in x. Either use F4 Other 4:DelVar x or F6 Clean Up 1: Clear a-z. Reset All Memory To reset all memory key: 2nd, 6, F1, 3, Enter Calculus The two main operations of calculus are differentiation and integration, of course. Both of these can be done easily either numerically or symbolically. They can be done on the home screen with the keyboard or on the graph screen. On The Home Screen Differentiation 2 nd d (expression, variable[, order]) (on the 8 key) does differentiation. E.g. d(x³,x) ENTER gives 3x². E.g. d(f(x)÷g(x),x) ENTER gives the quotient rule (although in a nonstandard form). To find f’(a) [where a is a number] use “|” E.g. d((x-1)÷(x+1)),x) |x = 3 ENTER gives 1/8. Order is optional, the default is 1. For a 2 nd derivative use 2. E.g. d(x³,x,2) ENTER gives 6x. If order is negative, the result is an antiderivative. E.g. d(x³,x,-2) ENTER gives x 5 /20, note the constant is dropped. Tips for Math with the TI-89 Calculator, page 16 Numerical Differentiation d( [the differentiation function above] takes the difference quotient with the lim Ax÷0. The function nDeriv(expression, var [, Ax] allows the user to explore the difference quotient if Ax is kept finite. The default value of Ax is 0.001. E.g. nDeriv(x 4 ,x,.1) calculates [f(x+.1)-f(x-.1)]/.2, which gives 4 x (x² + .01), whereas d(x 4 ,x) gives 4x³, i.e. the exact derivative. Integration 2 nd }(expression, variable[, lower][, upper]) (on the 7 key) [or equivalently F3 Calc 2: }( integrate] does integration. E.g. }(x²,x) ENTER gives x³/3 E.g. }(x cosx,x) ENTER gives cosx + x sinx E.g. }(1/x,x,1,2) ENTER gives ln2 lower is added as a constant of integration if upper is omitted. E.g. }(x²,x,c) ENTER gives x³/3 + c F3 Calc has other useful operations. Limits F3 Calc 3: limit( will find a limit. Syntax: limit(expression,var,point[,direction]). The optional direction parameter indicates from which side a one-sided limit is taken. If direction is a positive number, the limit is from the right, if direction is a negative number, the limit is from the left. E.g. limit(3x/(1-2x), x, ·) ENTER gives -2/3. e.g. limit(1/x, x, 0) = undef E.g. limit(1/x, x, 0, 1) = · (from the right side) E.g. limit(1/x,x,0,-1) = -· (from the left side) Appendix A has further examples. Finding Epsilon in the Limit Definition L = lim ( ) x c f x ÷ iff for any c, there is a o such that if x is within o of c, f(x) is within c of L. So L ± c = c ± o. To find c, use solve(. Key in solve(L +/- c = f(c + o), o). You have to key in the + & then the - separately. You know c, L & c, so key them in as numbers. Since o is the unknown, key it in as “x”. Whichever is the smaller x, is o. One of the x’s may come out as a negative number, but ignore the minus sign. For example: For f(x) = x³ and c = 2, find delta such that epsilon = 0.1. L = lim ( ) x f x ÷2 = 8. So key in solve(8+/-.1=(2+x)^3,x). Key in the + & then the - separately. The - gives x = .0083683. The + gives x = .0082989. Since .0082989 is smaller, o = 0.00829. Note that I rounded .0082989 down, (i.e. “incorrectly”), because rounding properly gives a slightly too large o, i.e. gives an c larger than 0.1. Minima and Maxima F3 Calc 6: fMin( and 7: fMax( will find a minimum and maximum respectively between Lower and Upper Bounds. E.g. F3 Calc 6: fMin((x-2)² + 3, x) ENTER will find the value of x where y = (x-2)² + 3 has a minimum i.e. x = 2. Arc Length F3 Calc 8: arcLen( will find the arc length. E.g. F3 Calc 8: arcLen( \(1-x²,x,-1,1) ENTER gives 3.14159. On The Graph Screen Plot the graph as usual. I recommend finding minima and maxima and numerical differentiation and integration, etc. be done on the graph screen, because then you can see what’s going on, instead of plugging away blindly. Differentiation To find the derivative key F5 Math 6: Derivatives 1: dy/dx. Then you will be prompted “dy/dx at?” Either key in the desired x value and key ENTER or move the cursor to the desired x value and key ENTER. E.g. graphing y = (x-1)÷(x+1) and in response to the prompt keying in 3 ENTER gives .125 Integration To find the integral key F5 Math 7: }f(x)dx. You will be prompted for the “Lower Limit?” Either key in the desired x value and key ENTER or move the cursor to the desired x Tips for Math with the TI-89 Calculator, page 17 value and key ENTER. You will be prompted similarly for the “Upper Limit?” E.g. graphing y = 1/x and in response to the prompts keying in 1 ENTER and then 2 ENTER gives .69315. Partial Fraction Decomposition Use expand( for partial fraction decomposition. expand( is F2 Algebra 3:expand( E.g. expand(1÷(x^2-5x+6)) gives 2 1 3 1 ÷ ÷ ÷ x x Minima, Maxima, Inflection Points, Tangent Lines & Arc Length F5 Math has several other useful operations. 3: Minimum and 4: Maximum will find a minimum and maximum respectively between Lower and Upper Bounds. 8: Inflection will find an inflection point between Lower and Upper Bounds. A: Tangent will draw a tangent line at the entered point and display its equation. B: Arc will find the arc length between “1 st Point” and “2 nd Point”. Other options are available in other modes. For example, in polar mode 6: Derivatives offers the options of 1: dy/dx or 4: dr/du. In parametric mode 6: Derivatives offers the options of 1: dy/dx or 2: dy/dt or 3: dx/dt. Convergence of a Sequence Convergence of a sequence can be displayed. E.g. Key in u1(n) = -.8u1(n-1) + 3.6, ui1 = -4 and + GRAPH. The sequence can be seen to converge to 2. Or Key + TABLE to see the values. Or simply sum the entire sequence. E.g. try E(1/x², x, 1, ·) ENTER. Surprising? Taylor Series Approximations Taylor series approximations can be displayed. E.g. sinx = E((-1)^n * x^(2n+1) / (2n+1)!, n, 0, ·)). Plot the first 6 terms of the Taylor series approximation, i.e. y1 = E((-1)^n * x^(2n+1)/((2n+1)!), n, 0, 5)) and also y2 = sin x on the same screen. The value of x where the approximation diverges from sin x is very striking. Or to find the numerical value of the differences define y3(x) = y1(x) - y2(x), deselect y1 & y2, key + TABLE. Use MODE Display digits 9 & +| to set the cell width to 12 (for the maximum number of decimal places) and use F2 Setup to select where the table starts and the step size of the table. taylor() performs a Taylor series expansion. The parameters are taylor(expression, variable, order[, point]), where expression is the expression to be expanded. Point is the point about which the series is expanded. Its default value is zero. The Taylor polynomial includes non-zero terms of integer degrees from zero through order expanded in powers of (variable - point). For example taylor((sin(x), x, 5) gives x 5 /120 - x³/6 + x. And taylor((ln(x), x, 2, 1) gives -(x- 1)²/2 + x - 1. If the taylor series would include non-integer powers, the TI-89 must be forced to give the expansion with a trick. For example taylor(1/(x*(x-1)), x, 3) gives taylor(1/(x*(x-1)), x, 3, 0), but expand(taylor(1/(x-1)/x), x, 3), x) gives -x² - x - 1/x. And taylor(e^(\(x)), x, 2) gives taylor(e^(\(x)), x, 2, 0), but taylor(e^(t), t, 2)| t = \(x) gives x/2 +\x + 1. Riemann Sums Riemann Sums are approximations to the definite integral. One sums Ax | f z i i n ( ) = ¿ 1 ], where Ax is the (uniform) width of the interval and f(z i ) is the height of the i th interval. For a sum with n terms over the interval [a, b], Ax = (b-a)/n. The expression for z i depends on whether one is doing the left, middle or right Riemann sum (R L , R M or R R ). Fortunately http://tifaq.calc.org/p2.htm#11.6 shows how to create functions to do these 3 sums and also how to create functions to do the Trapezoid and Simpson’s rules. (Check out this awesome site!) Here’s how: Type the following on the home screen, one at a time. R L : (b-a)/n-E(f(a+x-(b-a)/n),x,0,n-1) STO÷ lreman(a,b,n) R M : Tips for Math with the TI-89 Calculator, page 18 (b-a)/n-E(f(a+(x+½)-(b-a)/n),x,0,n-1) STO÷ mreman(a,b,n) R R : (b-a)/n-E(f(a+x-(b-a)/n),x,1,n) STO÷ rreman(a,b,n) Trapezoid rule: (lreman(a,b,n)+rreman(a,b,n))/2 STO÷ trapez(a,b,n) Simpson’s rule: (2*rreman(a,b,n)+trapez(a,b,n))/3 STO÷ simpson(a,b,n) * Now store your function in f(x) with STO÷ f(x). For example to evaluate the midpoint Riemann sum for sin x from 0 to t, with 100 intervals first key sin(x) STO÷ f(x), then key mreman(0, t, 100). You can avoid having to type mreman( in by keying 2 nd VAR-LINK, scrolling down to mreman and keying ENTER. mreman(0, t, 100) gives 2. In exact mode the TI-89 gives the exact value, in this example, 4 cos (39t/200) + 4 cos (37t/200) + ... [less than 100 terms, because it goes through and simplifies them - very impressive, but useless] after about 90 seconds. So use auto mode. Differential Equations deSolve(Equation, IndependentVar, DependentVar ). deSolve( is F3 C. Example: deSolve(y'=x*y, x, y) gives y = @1 2 2 x e . @1 is the TI way of writing an undetermined constant. The ' in y' is 2 nd =. Implicit differentiation impDif(Equation, IndependentVar, DependentVar [,order] ). impDif( is F3 D Example: impDif(x^2 y^3 = 1, x, y) gives (y' = ) -2y/3x key  ENTER. correctly. The Two Equals Signs Use the ENTER key to evaluate 3 + 4.g. The = key is used. 2nd  The green  key accesses the green functions.10 will remove all data including preloaded Graphing Calculator Software Applications (Apps) for example Stats/List Editor.10 go to http://education. 1: Auto. It appears on the screen as 6. key alpha (-). 'A' is treated just like 'a'. select option A: ABOUT To download OS 3. “2/3”.g. To change to Auto mode key MODE.g.66667. In Approximate mode results are always displayed as a decimal. page 2 Operating System Just as your PC can be upgraded to WinXP. i. Key the EE button only once. e. in degree mode sin((r) evaluates as ½.g. Installing OS 3. Exponents and Roots 7³ is keyed as 7 ^ 3.e. (° is keyed as 2nd °. as 6 10-8.e. not 0.g.3 will display as . Scientific Notation 6 10-8 is entered as 6 EE(-)8. e. Alpha Lock: To key several lowercase letters.2). 2nd ▼.  Y=. However the TI-89 isn't casesensitive.e.. To type a space. e. In both cases overall you get a better experience. e. root( is Math [2nd 5] 1: Number D: root(. fractions will be displayed as fractions (e.g. key alpha. . e. However the Angle mode can be overridden with ° & r. 'cSolve' can be typed 'csolve'.) Exact and Approximate Modes In AUTO mode (recommended). with solve(x^2 = 4. because it can be overridden more easily than degree mode. n x = x1/n. alpha A gives “a”. not the = key.com and navigate to http://education.key (above +) for .10. x). The Two Minus Keys Two different keys are needed to enter -3 . but on your homeworks and tests you must copy that in proper scientific notation.g. Use the (-) key (left of ENTER) for -3 & the .g. The current Exact/Approx mode setting is displayed at the bottom center of the home screen as AUTO or EXACT or APPROX. e. i. Exact/Approx. producing lower case letters.ti.4. 2.html. 2/3) or  or 5 to a decimal. key  alpha. Example 9^(12) gives 3. i. key MODE. your TI-89 can be upgraded to OS 3. Green.E-8. The r symbol is not alpha R.com/educationportal/sites/U S/productDetail/us_os_89titanium. it is keyed as 2nd MATH 2: Angle 2:r. E..  A produces “A”. key 2nd a-lock or just hold the ALPHA button down.g.4. Getting Started The Blue. Yellow & Grey Symbols The blue 2nd key accesses the blue functions. The gray alpha key accesses the gray functions. To key several uppercase letters. so you can key in x^(1n).666667) and pi as “”. To convert a fraction (e.ti.g. To find your OS select TOOLS on the HOMESCREEN. produces upper case letters. It will change to 'cSolve' once you hit ENTER. Thus radian mode is recommended. then Angle. The Angle mode is displayed at the bottom center of the home screen as RAD or DEG. correctly. instead of just ENTER or key one of the numbers with a decimal point. e. even builtin commands that have capital letters in them can be typed in lowercase. Or you can use root(9. in radian mode sin(30°) evaluates as ½. The  key. To exit alpha lock. Degrees & Radian Modes To change from degrees to radians or vice versa.Tips for Math with the TI-89 Calculator. The syntax: EXPR. solve for y. the name of the variable. If you do not remember the syntax and your manual is not handy. The Plus/Minus Sign The list {1. {1. ▼. The F1 to F8 Keys The meanings of the F1 to F5 and 2nd F6 to F8 keys are given on the top of the screen and depend upon which window is currently displayed. for example. e. If the cursor is sitting at the end of a line. Parentheses Use ( & ) for parentheses. keying  EE gives a keyboard map with all of the shortcuts. ▼. key CATALOG. trying to evaluate 3² gives an error. e. starting & ending values and optionally the step size. requires an expression. These are just symbols. deselects all plots & graphs. y = ± 1  x2 . keying  = gives .) will produce unexpected results. To get close to your desired command. Clearing  is the back space key for erasing a single mistaken key stroke. N: gives ±. sinh-1. Matrices & Vectors {} delimits a list. but not “xx”.5} & sin({1. i. -1}(1-x^2). If the screen is too dark or light Key  + to make the characters darker or  .5}. Key CATALOG. To reset it. expand((x-2)²). not [ & ] or { & }. nDeriv. To solve the quadratic formula for 2x² + 3x . Also see the next subject. and key y1 = {1. 2nd CHAR 2: Math A: gives .STEP] appears on the bottom line of the screen. If instead you get “4”. [] delimits a matrix or vector. the syntaxes are given in CATALOG. For example  ( alpha s gives . 5:approx) can be chosen either by scrolling down to highlight its line and keying ENTER or by keying its number (e. . Clear a-z. F6 Clean Up 2: NewProb does Clear Home. . Some options may be unreadable. 3} + 4 gives {5 6 7} {2.g. ▼.to make the characters lighter.HIGH[.) It’s very long. the default value will be used. If the cursor is sitting in the middle of a line. key (-3 +{-1. This then displays both solutions.g. 3 (to go to “s”) and ▼ to seq(. 3. Shortcuts keying   gives ! (factorial).VAR. If you type in. F6 Clean Up 1: Clear a-z will delete the definitions of any 1-character variables. abs.g. key Delvar x. to graph x² + y² = 1. page 3 More Symbols & Functions CATALOG contains all of the calculator’s functions (e. If you do not type it. keying  > gives .e.g.Tips for Math with the TI-89 Calculator. “x” probably has the value x = 0. seq(. Lists. ENTER.g. “x”. CLEAR will erase the part of the line to the right of the cursor. 3}/6) gives {2 3/2 .e. . E. If you want to clear the entire home screen. If the result of algebra is a number Accidentally storing a number to a variable (This is surprisingly easy to do. This means that the option is unusable in the current situation. nCr. Pull down the menus in F1 to F8 and choose the desired operation. 1}(9 . A menu item (e. 2. key F1 8: Clear Home. ▼.4  2 -4) )  4.g. It’s not necessary to key alpha and the first letter of the command. you expect “x²-4x+4”.-1} is effectively a ± sign. 5) On Screen Syntax Help There are many commands which require several parameters.4 = 0. 1.LOW. E. CLEAR will erase the entire line. key the first letter of the command and then ▼ down. I: gives ². keying  ( accesses the Greek letters. not functions. i. keying  < gives . For example to put seq( on the entry line.g. The square brackets around STEP mean that the parameter STEP is optional. 2. !. ▼ or ► as needed. including the lithium watch battery. remove all of the batteries. page 4 If the calculator is locked-up If the screen won't come on.2)² | x ≥ 2 Insert Mode 2nd INS toggles back & forth between insert & overtype mode. rightarrow and ON (2nd . In insert mode (recommended) the cursor is a thin line between characters. Graphing To enter an equation to be graphed. ln 100000/ln 10 gives 5. again check the contrast. To quickly move to the end of an expression To get to the start or end of a long expression or list.14 or 22/7. In overtype mode the cursor highlights a character. ▲. E. this will place the previous entry in the entry line. limit its domain by keying y1 = (x . try turning the contrast up by keying Diamond + continuously for a few seconds. ▲. This key is very useful for limiting the domain of a function. ▲. for copying or deleting) hold down  and highlight left or right with ◄ or ►. If that doesn't fix it. order of operations Your calculator knows the order of operations. The With key | means “with”. For example if you want y = (x . If you are lost If you are lost in some unfamiliar screen. try turning the contrast down by keying Diamond continuously for a few seconds. key  Y=. I recommend E: FLOAT. key 2nd ◄. use the key. If that doesn't fix it. Or use log x = ln x/ln 10. because the calculator displays all available digits in case you need them.g. If the screen comes on dark. To reload the operating system hold down 2nd. 2 + 3  x | x = 5 ENTER gives 17. key ESC to back up one screen or HOME to return home.Tips for Math with the TI-89 Calculator. Or you can key ENTRY (2nd ENTER) repeatedly until the desired entry is in the home line log x ln x  logex is on the keyboard. hand and ON for the 92+/V200) simultaneously. ▲. ENTER. key the ON button once and again try turning up the contrast. To change the number of digits displayed To change the number of digits displayed. To copy text Highlight it as explained above then key COPY. If this does not work.g. log x  log10x is not on the keyboard. To highlight text To highlight text (e. key MODE Display Digits. To reuse a previous entry To reuse a previous entry. replace all of the batteries with new ones. key in the parentheses. If that doesn't fix it. E. again check the contrast. leftarrow. In case the calculator was actually off. Then key ENTER.g. Pi To enter . To get it key CATALOG 5 ▲. If that doesn't fix it.2)² to have an inverse. reload the OS. If the screen comes on with an error message and then turns off again.g. E. repeatedly key ▲ until the entry is highlighted. not 3. Move the cursor to the place where the text is desired then key PASTE (ESC). reset the memory by removing one battery and holding the (-) & ) buttons down replacing the battery and then holding them for another 5 seconds. . 4 + 3 2 will be evaluated as 10. If you meant (4 + 3) 2. where it can be edited as necessary. e. 2 is default. or 0. Key F6 Style to pick the way the graph is displayed (dotted.).10 on xres is grayed out and set to 1.1 and the window dimensions to -7. the window size is -79 < x < 79. sets xmin & xmax to match the data.. 1 = highest resolution. asymptotes. It is possible to develop a formula to calculate the window size to set to solve this problem.8 < y < 3.2.3. (ZoomDec does not allow this option. Friendly Windows A friendly window is a window where the x coordinates of the pixel elements are round numbers.. A: ZoomFit resizes y to fit the graph. xres sets pixel resolution.3. etc. .4. x & y max = 10 and x & y scl = 1. Here is a trick. See the discussion of friendly windows below. 4.1. ( x  1)( x  2) For example y  has the same ( x  1) graph as y = x + 2 except that there is a hole in the line at x = 1. Key F4  to select or deselect an equations. (Skipping pixels speeds up the graphing process. The lower the resolution. or 0. key  GRAPH. 0. 2.5° and xscl = /2 = 90°. 7: ZoomTrig is useful for graphing trig functions. From OS 3. (normally you do want that). F2 Zoom can also be accessed from inside  WINDOW. 4. If you really want 1. See the discussion of friendly windows below. if the Discontinuity Detection (F1 Tools 9: Format) is set to ON. or 0. -38 < y < 38. In GRAPH set F2 Zoom To view the equation’s graph.0001 or maybe 1000.) What you actually trace is 1. .9.8. squares look square and lines look. It sets the pixel size = /24 = 7.4. 0.. 10 = lowest resolution. 0. .2. 0. 2. 1. If that does not work.) If you choose to center the graph at the origin. 0. so circles look round. try ZoomStd. Zoom 8: ZoomInt (integer) sets the pixel size to 1. The scale set by ZoomInt is often the wrong size.. key F2 Zoom to resize the window (i.e.g. then use ZoomDec.Tips for Math with the TI-89 Calculator.1. . to change the maximum and / or the minimum value of x and / or y that is displayed).e. .intercepts. i. the important features of the graph are off the screen)... -3. 0.9 < x < 7. 6: ZoomStd sets x & y min = -10. xscl (“x scale”) sets the distance between tick marks on the x-axis. . similarly for ymin & ymax. 2. but that is tedious.0 and allows you to use the arrows to move to the center to the part of the graph you wish to investigate. The default is ON. But if the region you want to trace is off the screen. min/max and behavior as x ± You might want to zoom in so as to precisely determine an xintercept or to understand a puzzling behavior of the graph. set xres to 1. reset the window size with  WINDOW.0. for use with scatterplots or histograms data. 4. 0. you get steps of 1. If you need steps of 0. similarly for yscl. The default is 4 4: ZoomDec & 8: ZoomInt set friendly windows.1.3. thick. But xres determines how many pixels are actually traceable. but need steps of 0. The default is 2. try this first. Inside  WINDOW: xmin & xmax set the values of x on the left and right sides of the window. you are out of luck with ZoomDec. If nothing appears on the screen. e. page 5 Type in the equation... This is very helpful if you want to see a hole in a graph. 9: ZoomData. If the window is too big (i. 3. . the faster a graph is drawn. 3. 5: ZoomSqr scales x & y the same.. 0. You might want to zoom out so that you can see all of the main features of the graph . Only selected equations are graphed..4. 3. Inside  GRAPH F2 Zoom 1: ZoomBox zooms in on a box you draw. 0. 2: ZoomIn & 3: ZoomOut zoom in & out by the amount you set in C: SetFactors. In  GRAPH. This is useful if more than one equation is selected. the graph is a tiny unhelpful squiggle) or too little (i. (for ZoomInt & ZoomDec and respectively) times xres.2. etc.1.. Inside  GRAPH.e. Zoom 4: ZoomDec (decimal) sets the pixel size to 0.. yscl.  is  (above the ^ key). which you may need to set. repeat. y &  To cancel a graph To cancel a graph while it is being plotted. Use this e. You will get a pretty Lissajous figure. 3: Functions Off or 5: Data Plots Off as needed.  | Coordinates  Rectangular will cause F3 Trace to display the coordinates x. x. You can recall your stored setting with B: Memory 3: ZoomRcl. To format a graph To format a graph (rectangular vs. The cursor can be moved along the curve with ◄ or ► or by typing an x value and ENTER. etc. you must type in a pair of equations. for reading out data points in a scatter plot or histogram heights. To turn them all off. To depict an inequality To depict an inequality (e. This can be used for finding intercepts or other solutions to the equation. they might graph along with your polar graph. y1(x) = 2x + 3) and then shading above or below the graphed line. in Y= key F5 ALL 1: All Off. F3 Trace puts a cursor on the graph & displays the coordinates of the cursor. polar. it usually does not land exactly on the desired point. depending on whether the inequality was actually > or <.) tmin. in Y= xt1 = sin 2t yt1 = sin 3t To view the graph of the above set x & y min/max = ±1 & use radian mode. Select F2 Zoom 8: ZoomInt. and differential equations graphs To store a window setting To store a window setting with F2 Zoom B: Memory 2: ZoomSto. Window or Graph. page 6 Zoom C: Set factors. A parametric graph is made on the x-y axes by defining x = f(t) & separately y = f(t). To change the center of the graph To change the center of the graph.g. There are 4 shading patterns which are automatically cycled through. Modes The normal graphing mode is MODE Graph = 1: FUNCTION. Polar Graphs To graph in polar coordinates key MODE Graph = 3: POLAR.  | Coordinates  Polar will cause F3 Trace to display the coordinates r & . max & step.g. 3D. (You want to shade above in our example. etc. y or z) as your independent variable. to graph y1(x) = x². Thus in parametric mode.) key  or in Y=.Tips for Math with the TI-89 Calculator. yscl. which you may need to set.1. If you need step sizes of 0. etc. The other modes are used for parametric. etc. Trace Inside  GRAPH. E. key F1 9: Format.  Window now has (in addition to xmax. You must use  (not t. You must use t (not x.g. y > 2x + 3) on the xy plane by graphing the inequality as an . Now you have step sizes of 0. Set the center where you need it. y or z) as your independent variable. key ON. label axes on/off. polar sequence.g. So 4 different inequalities can be displayed. etc. Use ZoomSqr to set the correct proportions or do it by hand by setting xmin & xmax to twice ymin & ymax. Unfortunately since trace moves from pixel to pixel.01. grid on/off. key F6 Style 7: Above or 8: Below as needed. do not use trace.) To set the shading in Y=. equality solved for y (e. Therefore if the coordinates are needed accurate to 3 significant figures.the default zoom step size is 4). tmax & tstep. set xFact and yFact to 10 (or if you prefer a smaller zoom step size to 10 .  Window now has (in addition to xmax.) min. Now Select F2 Zoom 2: ZoomIn twice (or once if you set the zoom factor to 10). Parametric Graphs To make a parametric graph key MODE Graph = 2: PARAMETRIC. move the cursor to the desired center and key ENTER. If some functions are selected. The Y= screen will now read r1=. Below I assume you are in function mode. To find the y-intercept. Key F5 Math 5: intersection. Solving a system of two equations I. Solve the 2 equations for y. then 2: Zero.e. Then you will be prompted “Lower Bound?” Use the ◄ key to move the cursor to the left of the intersection and key ENTER or if you know roughly the x-coordinate of the intersection. If the cursor is on the correct 1st & 2nd curves. which can be accessed with  TblSet or inside TABLE with F2 Setup AUTO table automatically generates a series of values for x or you can choose them yourself with ASK. not “below”. y2. In response to “Upper Bound?”. To find the x-intercept. just press ENTER to answer each question. or polar angle . key F5 Math. This will move the cursor to x = 0.Tips for Math with the TI-89 Calculator. Depending on the MODE setting. You will be prompted “Upper Bound?” Use the ► key to move the cursor to the right of the intersection and key ENTER or if you know roughly the x-coordinate of the intersection. E. Graph <-> Table = OFF (unless you want to use xmin & xres to set tblStart & tbl. ENTER If you want to choose your own values of x. use Ask. ENTER. To move the cursor faster. “Lower” is a bit confusing.g. In TABLE SETUP. Tables To make a table. Deselect any previously selected functions. Or better still. Set the table parameters with tblStart = -90. To change the cell width in TABLE key | or F1 9: Format To find the x and y intercepts To find x and y intercepts. Independent  Ask. the x and y intercepts can be seen on the screen). Then key in your x value. then ENTER. Graph them using the  Y= and  GRAPH. key F3 Trace. Remember it means “left”. create a function in Y=. page 7 To find the Minima or Maxima of a function In  GRAPH to find the Minima or Maxima of a function use F5 Math 3: Minimum or 4: Maximum. graph the equation and zoom in or out with F2 Zoom (or better by using  Window reset x & y min & max until . ENTER. y) coordinate of the intersection will be displayed at the bottom of the screen. . key in an x value less than the xcoordinate of the intersection. you will be prompted with “1st Curve?” and then “2nd Curve?” to choose the two functions whose intersections you want to find. key in an x value more than x-coordinate of the intersection. otherwise use the ▲ or ▼ key to move to the correct curve(s). key in an x value less than the value of the x-intercept. this could be functions y1. In case you have other functions displayed on the screen besides the ones you want to solve. Independent = AUTO. which is not recommended). For example key y1(x) = sin x. To use the following method the intercepts must appear on the screen. or parametric functions y1t. etc. in  TABLE key F2 Setup. Either place the cursor to the left of the x-intercept with the ◄ key and key ENTER or if you know roughly the value of the x-intercept. the y-intercept. x1t. use Y= and then F4 to uncheck the unneeded curves. etc. etc. Finding the Intersections of Two Graphs Solving a system of two equations means finding the intersections(s) of their graphs. You will be asked for the “Lower Bound?”. There is a choice of ways to choose which x values to display. If the intersection(s) do not appear on the screen. etc. tbl = 15 (assuming you are in degree mode & want a table of sin x for every 15° starting at -90°). similarly choose an x-value to the right of the intersection. Then you will be asked for the “Upper Bound?” Either place the cursor to the right of the x-intercept with the ► key and key ENTER or key in an x value greater than the value of the x-intercept. The y-intercept will be displayed in the lower right of the screen.e. i. You will be prompted to choose an x value on each side of the zero. then key in another x value. TABLE produces a table of the functions selected in Y =. then 0. Then the (x. use 2nd left or right arrow. zoom out (or better by using  Window reset x or y min or max until the intersection(s) can be seen on the screen).  x. g(-3).5x + 1. You cannot trace or find intersections. 1< x  you can use nested when functions or a userdefined function. Thus where an asymptote should appear.3) or if you need a table of g(x) or if you want to find values e.  x. etc. E. 0  x  1 . If you want to draw an asymptote.x^2. x  1 f (x)   2 can x . E. For example  x. when(0 < x. . x. Horizontal asymptotes can be easily keyed in. x) or y1(x) = when (x  0.x + x | 0 < x and x  1 y3(x) = x | 1< x The unexpected “. -x). You can access the Graph Formats window from either the Graph screen itself. x. Or set the scale so that a pixel element falls on the asymptote. expression when test false). draws a dot there and connects the dots. or the Window Editor. but because of the way it draws a curve. else x^2) when( is in CATALOG. x^2).-10^100. Unfortunately if you want to then graph g(x . not just for 0 < x  1. x  0  g( x )  3. F2 Zoom 4: ZoomDec and  WINDOW xres = 1 will put a pixel element at x = 1 and thus no fake asymptote will appear. x)) Notice the difference in the inequalities used above and that the -x + x term is no longer needed. Without an x in the expression “y2(x) = 3 |0 < x and x  1” would have drawn a 3 for all x. If the graph has more than 2 pieces.x².perhaps in a different style and add it to your graph. not in Y=. The Boolean operator and is 2nd MATH 8: Test 8: and or CATALOG and. 0) and has such a steep slope that it will appear perfectly vertical. if the Discontinuity Detection (F1 Tools 9: Format) is set to ON. a slightly crooked fake asymptote might be drawn. you could key its equation in . 1  x  3 Piecewise-Defined Graph To display a piecewise-defined graph. e.x + x” term was used to get an x into the expression. E.g. fake asymptotes sometimes appear. when(x 1.g. a horizontal asymptote at y = 2 would be graphed as y2(x) = 2. then selecting each with F4 .5). of a line drawn with LineVert.5x + 1. for y = 1/(x1) there is a vertical asymptote at x = 1. 1< x  you must use nested when(s. For example 4 . This draws a horizontal line at -10100 for x < 2 and a horizontal line at 10100 for x  2. x  1 inside Y= key y1 = when(x > 1. 1. Vertical asymptotes can only be approximated. expression when test true. x  1 f (x)   2 .Tips for Math with the TI-89 Calculator. What you see is the vertical line connecting these two lines.g. Alternatively type y2(x) = when(x<2. Alternatively type LineVert 2 in the home screen. 0  x  1 . To write  x.10 on fake asymptotes no longer appear. especially since you have to read it inside the small entry line. e. 10^100).g. The default is ON. x + 3) f(x) = 1. Unfortunately the logic of nested when functions is hard to follow. page 8 Asymptotes The TI-89 does not draw asymptotes. then you need to express the function in one entry. x  1 be written as. Both are way off screen. the Y= Editor. x .g.  x. Unfortunately xres = 1 makes graphing slow and Discontinuity Detection ON makes it even slower. for a vertical asymptote at x = 2.g. in Y= set the style to dot.: y1(x) = -x | x  0 y2(x) = 3 .g. -x. It is easier is to enter a piecewise defined equation as separate equations.5. 3. x < 1 x + 3. If you want to get rid of fake asymptotes. when(x < 1. 3. x  0  g( x )  3. 4 . Discontinuity Detection From OS 3. which is a line which goes through the point (2. e. For example: y1(x) = when (x  1. x > 3 is entered as y1 = when(x<=3.  x. use y2(x) = 10^100(x-2). If so the g(x) above can then be defined using when(test. If you know where an asymptote is. y10(x) = when (x > 1. It evaluates y at the center (in x) of each pixel. a horizontal line at y = 1 for -2 < x < 1. The way the result is displayed depends on what modes you are in. In y3 type when(y1(x)<y2(x). radian mode 5 + i ENTER gives ei arctan5/5 3 2. Solving with Complex Numbers cSolve() or cZeros() can be used to find the roots of complex numbers. In WINDOW set min to the angle of the first root (i.e.e. In complex REAL mode y = x2/3 is (incorrectly) graphed for x2 all x. i. degree mode + 2i ENTER gives 2. with the above definition. For example y y = x2/3 is considered to be x2 defined only for x  0 and y = is x4 considered to be defined only for x ≥ 4. i. key y1 = when(x  1. You may need to reset the Exact/Approx mode or the Angle mode (i.e. The first root. In approx. r ei) form. (the argument of 2i) / n = (/2)/3 = /6).095. a + bi) form. rectangular. Set step to 2/n = 2/3.1.630 i will be displayed (as 1. Apparently the TI-89 cannot be commanded to display in trigonometric form (e. rectangular. Radian mode). with k = 0. Again the way the result is displayed depends on what modes you are in. set the mode to complex polar with MODE Complex Format 3: POLAR. For example: In exact. 1. -3/2 + ½ i.g.Tips for Math with the TI-89 Calculator. For example ³i = cSolve(x³ = i. Graphing a Real Function The domains of some functions are restricted.g. rei) form to a complex number in standard (i.e. x2 but y = is (incorrectly) graphed only for x4 all x > 4 and also for x ≤ 2. In complex RECTANGULAR mode y = x2/3 is (correctly) graphed for x ≥ 0. .e. because they produce complex results for some x. In exact. The disadvantage of this method is that it’s impossible to get the exact endpoints with this method.0). i. but y = is (correctly) graphed only x4 for all x > 4. E.e. In approx.630).44924. To convert a complex number in polar (i. x.i.e.e. In exact mode 2ei/4ENTER gives + 2i. To graph these values set the Graph mode to Polar.785 i. In complex POLAR mode an attempt to graph either function crashes. For example DeMoivre’s Theorem says that the cube roots of 2i = 2 ei/2 are ³2 ei(/2+2k)/3. 2. page 9 Once you have defined a function in Y =. polar. x) = {-i. For example In approx mode 2ei/4ENTERgives + i. In Y= key 2^(1/3). ±2.e. you can use it in the entry line. undef means undefined.. a + bi) form to polar (i. but the required setting is a bit inconsistent. x) = cZeros(x³ . Degree vs. Key ► to display Complex Numbers Converting from Rectangular Polar or Trigonometric Form to To convert a complex number in standard (i. The TI89 can be commanded to show just these parts by changing the complex format mode. so it won't draw anything if x > 1. x  1. radian mode + 2i ENTER gives 2 ei/4. y1(-25) evaluates as 25 To display f(x)={x. 2 cos 45° + 2i sin 45°). Key F2 Zoom 5: ZoomSqr (to make the plot “circular”). undef). Just type “undef “ in. Get when( from the catalogue. Solving Inequalities Graphically Example y1 = x+2 y2 = -x²+4 Find the values of x such that y1<y2. it’s not in the catalog! ENTER. set the mode to complex rectangular with MODE Complex Format 2: RECTANGULAR. Etc. read out) the roots of complex numbers. 1. key in the complex number in standard form & press . Set the window size appropriately.091. key in the complex number in standard form and press ENTER. is displayed. The solution.091 + . Key F3 Trace. radian mode + 2i ENTER gives 2 e0. Key GRAPH. 3/2 + ½ i} DeMoivre’s Theorem can be used to display and trace (i.g. Turn off y1 & y2 and key GRAPH. a little bigger than ³2 e.e. with algebra). Key ► again to display the third (& last) root 0 . For a more detailed explanation of how to use the zeros feature.4.260 i. 1. Which form is displayed is controlled by 3 (!) mode settings.e. e. Zeros is F2 4: zeros(. (1  45). because if there is more than one intersection you can choose the one you want and because setting the equation equal to zero or equal to y involves extra algebra and is therefore not as intuitive.2 i.g. z = x(t) + y(t)i. You will be prompted to choose an x value on each side of the zero. (r  ) i. To convert from Polar to Rectangular Coordinates key [r. Some equations cannot be solved analytically (i.091 + .y] 2nd MATH 4: Matrix L: Vector ops 4:  Polar For example [1.414 . is displayed. 2 e 0. Mode: Complex Format 3: Polar will always display the result in polar form. or approximate standard form a + bi. In Angle 1: Radian mode the polar radian form.e. Or if z is not explicitly written with real and imaginary parts.785 i or . giving “x = . i. depending on the Complex Format mode. see below. 3)]:  Polar gives [2  /3] . e. x . I will require you to show your work or I will prohibit calculators on part of the test. This single line will do about half of your homework or test problems! If I want you to do the problem yourself by hand. There are at least five possible ways (!) that the results of cSolve() or cZeros() can be displayed. can be graphed on the complex axis in parametric mode as xt1 = x(t) and yt1 = y(t).g. y2 = cos x and in  GRAPH find the intersection of two functions using F5 Math 5: intersection. E. i. Apparently the TI-89 cannot be commanded to display the results in trigonometric form (e.707 i. i.e. Or set the equation equal to zero (e.e.g. Other Commands The Solve & Zeros commands To solve an equation key F2 Algebra 1: solve( in the entry line. (2  45) and 2  -135) or exact standard form a + b i. 2ei/4 and 2ei/4 or approx polar radian form ei. E.cosx = 0) and in the entry line use zeros(x . I recommend using intersection in the GRAPH screen.] 2nd MATH 4: Matrix L: Vector ops 5:  Rect For example [2.1. use xt1 = real(f(t)) and yt1 = imag(f(t)). finding the square roots of 2i. One way is graphically by plotting y1 = x. Mode: Complex Format 1: Real or 2: Rect will always display the result in standard form. Mode: Exact/Approx 2: Exact will always display the result in Exact form. giving “x = . In Angle 2: Degree mode the polar degree form. Graphing a Complex Function A complex function. 2 cos 45° + 2i sin 45°).g. e 0. the answer could be displayed in exact polar radian form ei. x) will both give 2 & -2. For example Solve(x² . Or in the entry line use solve(x = cos(x). With a grapher there are several ways to solve it.” does not display in the answer. because by looking at the graphs you can understand what is going on. for x² = 4i.cos(x). The one used in the entry line does not ask for a range to search and might not find all zeros.e. Of all of the above methods. just as z = f(t).x) gives x = 2. You will be prompted to choose two functions and an x value on each side of the intersection.785 i and 2e -2. i.g. Converting from Rectangular Polar Coordinates to To convert from Rectangular to Polar Coordinates key [x.Tips for Math with the TI-89 Calculator. x) or Zeros(x² .e. Mode: Exact/Approx 3: Approximate will always display the result in Approximate form.cosx and find the zeros of the graph with F5 Math 2: Zeros. i. depending on the Complex Format mode.1.739085133”. The “. solve(2x + 3 = 7. e.g. e.739085133”. ei/4 .4 = 0.g.414 + 1.  + 2 i and -2 . e. but it is there.e.414i. ei/4 or 2/2 + 2/2 i. e. x). The above two Zeros functions are different.g. page 10 the second root -1. x). r ei.707 + .g. is displayed .366 i or polar degree form (r  ). x = cos x.  /3]  Rect gives [1 3] The “ ” is 2nd EE. Or graph y1 = x .414i & -1.630 i. Either Solve() or Zeros() can be used for most problems.g. To find the sum of the n terms of f(x) from x = a to b (i. To find the sum of a sequence you can use sum(seq( or (. +f(4). x = (42)/10 = . Or you can get clever with (. factor(x³ + 3x²y + 3xy² + y³) ENTER gives (x + y)³. cZeros() is 2 Math 9: Algebra A: Complex 3: cZeros(). x.700. e. to find the sum of the first 4 terms of the geometric sequence an = 52n. variable name. y}) will give (±1.1 = 0. E.1) < 3. n). 0. If you want complex solutions.2. 1.g. For example for f(x) = 2x to find f(2) + f(2.2 x. Solve(x^2 = 2. x. Solve() or Zeros() or nSolve() will give only the real solutions.1 = 0.2) + f(2. x) will give x = i or x = -i or x = 1 or x = -1.g.e. e. cSolve(. cSolve() is 2nd Math 9: Algebra A: Complex 1: cSolve(). Note I used “x” rather than “n” as my variable to make the keying easier. 1. for Riemann sums.  It can find the intersection of two equations. x. the step size is set to 1. you must use cSolve() or cZeros().n).. also gives 96. 2. which gives 96.. key sum(seq(5*2x. begin.g. e.g. key seq(3x+2. end).. 4) ENTER. then you must use sum(seq(. page 11 There are three Solve( & two Zeros() commands: Solve(. Step is optional. Example solve(abs(x . of course. Solve(x^2 + y^2 = 4 and y = x^2. 5. Zeros(. +f(4). x) gives -2 < x < 4 Factoring and Expanding To factor an expression key F2 Algebra 2: factor( in the entry line.700.  To speed up the process.0. Since ( is the same as sum(seq(. step]). x) will give x = 1 or x = -1. E. but is faster than Solve() or Zeros(). for example sum({1. 1.Tips for Math with the TI-89 Calculator. .10 can solve inequalities. the default is 1.g. you can also include a guess e. begin.)  Solve(x . the intersections of the circle and the parabola. x) will give the quadratic formula. nSolve looks for only one approximate real solution. either of which give 150. expand(x + y)^3) ENTER gives x³ + 3x²y + 3xy² + y³. x. 4 nd Permutations and Combinations Combinations are the number of ways of selecting r objects from n objects where order does not count = nCr = n! / r! (n-r)! is also written as  n    r . variable name.562) i. factor(x^2 .g.x. x.4 + . Permutations are the number of ways of selecting r objects from n objects where order does count = nPr = n! / (n-r)! n Cr nPr is 2nd MATH 7: Probability 3: nCr( is 2nd MATH 7: Probability 2: nPr( Sequences and Series A sequence is an ordered list of numbers. Solve(ax^2 + b*x + c = 0.g.250.5x + 6.. end [.x.g. which. ( takes parameters (expression. E. use sum(seq(2x. but cSolve(x4.4. They can be calculated using 2nd MATH 3: List 1: seq(.2)). 1. E. ½) ENTER gives {11 25/2 14 31/2 17}. 3.e. For example for f(x) = 2x to find the sum of the 10 terms f(2) + f(2. If another step size is needed. Seq takes the parameters (expression. 3}) gives 6.4) will give 1.414. x) gives (x-3) (x-2). to display the first 6 terms of the arithmetic sequence an = 3n + 2. with step size =x = (b-a)/n use (f(a+xx).  It can solve symbolic equations. {x.2. The sum &  features are different. nSolve() is 2nd Math 9: Algebra 8: nSolve(). E. x = 1.4) + . although the equation must be more complicated before the speed-up is noticeable. i. But ( does not accept a step size. cZeros( and nSolve(. Solving inequalities OS 3. Solve is very powerful.g.g. 4) ENTER or key (5*2x.e. E. (“and” is 2nd Math 8: Test 8: and. cfactor(x4-1) gives (x-1) (x+1) (x-i) (x+i) To expand an expression key F2 Algebra 3: expand( in the entry line. 6) ENTER which gives {5 8 11 14 17 20}. it’s easier to just use (..2) + f(2. A series is the summation of the terms of a sequence. seq(3x+2. use (22+. Zeros() is 2nd Math 9: Algebra 4: Zeros(). Sum just takes a list to sum. Solve() is 2nd Math 9: Algebra 1: Solve(). (x.  10 / 7    How to graph a Conic Equation Since a conic equation is not solved for y.-1}*((b*x+e)^24*c*(a*x^2+d*x+f))) (2*c)  conic(x). it cannot be directly entered into Y=. Key  GRAPH F2 Zoom A: ZoomFit. Set MODE Graph SEQUENCE. Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.1][3. Enter the formula for the sequence in u1. because by default alpha is already turned on. can be rearranged to Cy2 + (Bx + E)y+(Ax2 + Dx + F) = 0. Then for each conic just key in the values of A. Now set y1(x) = conic(x). C = 4. 0  E and -30  F. e. You can use F3 Trace to read out the values of u1 (they are called yc). If you want to graph several conics. To find the partial sums of a sequence use cumSum(. Note that you must explicitly type b*x. otherwise leave it blank. b   1 . The solution is easy. D = 10. a. Determinant is in MATH (2nd 5) 4: Matrix 2: det( Inverse is ^-1 Example Solve 3x  4 y  5 z  6 . Type 2: Matrix. There is one problem with keying in this formula. where  x  2 y  z  1 3x  2 y 5  3 4 5  6  x       with a   1 2 1  . . The general form of the conic. and graph it. i. Then the quadratic formula can be used to solve for y. 3: New.5][1. X   y   3 2 0  5 z       this can be written a X = b. Note that this now has the form of a quadratic in y. Therefore some prefer to type a matrix in the home screen Example Solve 3x  4 y  5 z  6 x  2 y  z  1 3x  2 y 5 [[3. Folder: Main. etc. The TI- is the STO button. in the home screen key (-(b*x+e)+{1. Key  Y=.0]]^-1 [[6][-1][5]] gives  26 / 7     43 /14  .e. key in u1 = 3n+2.) replaced with their values. B.n) ENTER gives n(n+1)/2.30 = 0 (i. If you have one conic to graph. E.g. -4  B.-2. To type the row & column dimensions you must key alpha to turn alpha off. E.). So the solution is X = a^-1 b  26 / 7    giving  43 /14   10 / 7    Matrices in the Home screen The TI-89 requires several keystrokes before you begin typing the matrix vales. The axes are n and u1. page 12 The result can contain symbols.-4. You must use “n” as your variable in sequence mode. Applying the quadratic formula gives y  ( Bx  E)  ( Bx  E)2  4C( Ax2  Dx  F) 2C Matrices To create a matrix key APPS Data/Matrix Editor. 4  C. there is no “±” button. enter the initial value in ui1. key 1  A. But by treating x as a constant. key in y1 = (4x+{1. For the above example. 10  D. key the above formula in for y1= with the letters (A. 4)) ENTER gives {1 5/4 49/36 205/144} A sequence graph is will display a sequence.g.g. etc.g. Variable: (Give it a name. 1. not bx. it is faster to create a function called conic(x) which contains the above formula.g. For example for x² .-2. a conic can be rearranged into a quadratic equation in y.1.4xy + 4y² + 10x . A = 1. B. e. E = 0. F = -30). which you get by typing =. 1} is effectively a “±” sign.Tips for Math with the TI-89 Calculator. if a recursive formula for the sequence is entered in u1. 84 is needs fewer keystrokes. e. If an initial value is needed. C.e.-1}((-4x)^2-4*4(x^2+10x30)))(2*4). B = -4. C. not b times x. key 2nd MATH 3: List 7: cumSum(seq(1/x².x. otherwise bx will be interpreted as a variable name. the list {1. x. e. use expand(.e. one which cannot be inverted algebraically. i. There are several ways to graph the inverse. If f(x) contains a transcendental function. This can be graphed as y ={1. e. which gives x=1 and y=2 and z=3. On the other hand graphing y1 = {tan-1 [(x-3)/2]}/ gives just the one curve of the function. Linear Interpolation If you have a table of data.2)(5.y.g.g. to solve x + 4y . For example y = x² Solve it for y. x) gives x = (y-1)/2. e. E. yt1 = t^2. x) gives 2/(x+2)+x+2. Solve( is F2 Algebra 1: solve(.e. but remove the {1.8 and wish to find a y-value corresponding to an xvalue between those x-values in the table. To Solve a System of Equations with Solve( To solve a system of three equations and three unknowns key solve(eqn1 and eqn2 and eqn3. x) gives x = ey. So f -1(x) = (x-1)/2. y = ±x. For example. {var1. where x1 and x2 bracket x.{x.z}). yt2 = t. Expand is F2 Algebra 3: expand(.8) = 25. xt1 = t. written f-1(x).g.x) gives x = {tan-1[(y-3)/2]}/ + @nj. gives a series of curves.y + z = 3.g.g. Then exchange x and y by hand.x1 ) x 2 . For example solve(y = 2 tan x + 3. All of the above is for the inverse relationship i. The list {1.var3}).) of the graph. k = an integer. where you do not require that the inverse be a function. If the solution has two constants.Tips for Math with the TI-89 Calculator.7 25. Use the following equation. 1) You can first find f-1(x) analytically as above and then just key it in in Y =. -1}. x = {tan-1[(y-3)/2]}/ + k. will produce a graph of y = x². called the principal branch.-1} is effectively a ± symbol. 2) Or use parametric mode: xt1 = t yt1 = f(t) will produce a graph of f(x).. 3x + 2y + 3z = 16 and 2x .x1 E.: x 4 5 6 y = f(x) 24.8-5)/(6-5) = 25.68 . Solve(y = ln(x). If you want the inverse function and the inverse relationship is not already a function. key solve(y = f(x).e.2 25. in more standard notation. For example expand((x^2+4x+6)/(x+2).: y = y1 + (y 2 . proceed as in 1 above. will produce a graph of x = y². MODE Split Screen 3 LEFT-RIGHT or 2: TOPBOTTOM is useful with graphs. 3) Or type on the command line DrawInv x^2. So the inverse function is f-1(x) = tan-1[(x-3)/2]/ In this example DrawInv 2 tan x + 3 or graphing y1 = {tan-1[(x-3)/2]}/ + @nj. y = x² or x² + y² = 1). page 13 How to Simplify Rational Functions To do long division of one polynomial by another.-1}x. Make one screen the graph and use the other screen to adjust the window and to set the style (dot. find f(5. (The j stands for an integer. use the inverse transcendental. To get the inverse function just drop the + @nj part.z = 6.). key solve(x+4y-z=6 and 3x+2y+3z=16 and 2x-y+z=3. xt2 = yt1(t) yt2 = t will produce a graph of f-1(x). So f -1(x) = ex. thick. tan-1 x. i. You may need to adjust the values of tMin and tMax to allow for the range of y values you want. it does not work. To obtain the graph of the inverse function use Method 1. @nj is the name of an integer constant. assume the (unknown) function y = f(x) is locally linear and use linear interpolation. If the function did not have an inverse function because it contained a y² term (e.2 + (25. x). its inverse. e. which draws the inverse of y = x².8-25. they will be named @n1 and @n2.g.var2. The Inverse To find analytically the inverse of f(x). one on top of the other. etc.y 1 )(x . tan x). For example solve(y = 2x + 1. (i. xt2 = yt1(t). e. Binary. but sometimes this does not happen or the default form is unsatisfactory. This function is also called floor(. F6 Clean Up 1: Clear a-z ENTER ENTER will clear (i. 0b11010Hex gives 0h1A and 0b11010 Dec gives 26.e. Both expand( and factor( can be used (sometimes back to back) to force a simplification of an expression or equation. 2:Bin & 3:Dec. If you want to delete all but a few variables.Tips for Math with the TI-89 Calculator. because xt1(x) is a system variable.5) returns 3 and int(-3. then key  Enter as above. In binary mode 0b111 + 0b1 gives 0b1000. Define xxt(x) = 3x^2. ceiling(-1. iPart(1.5) returns -1. An erroneous set of keystrokes can result in defining x & y. Then xxt(5) ENTER gives 75. For example. For example. tables. For example in decimal mode 0b111 + 0b1 gives 8. How do you set the mode of the answer of a calculation? To display your answer in binary. Base. which are 1:Hex. E. Notice that I used letters available on the keyboard to make typing the name faster. ceiling(1. 1cos(x) STO sec(x) defines a new function. use iPart(. page 14 Step functions [x] = the greatest integer less than or equal to x = int(. “x”. i. Then use F4  to deselect those tables you do not want to delete. 3: BIN.g. text entries are listed. delete) the definitions of any 1-character variables. For example to enter 7 in binary. key 2nd VAR-LINK. There is another way to define a function. 2: HEX. These can be found in the MATH 1: Number. You must always use the prefixes. for example. These commands only work in EXACT or AUTO. .5) returns -1. etc. lists. Hexadecimal & Decimal Decimal means base 10. To delete functions. You can enter the number in any mode. All of your (user defined) variables: tables. Key  Enter.5) returns -4. To Simplify Expressions Usually expressions are simplified automatically. Base. To display your answer in hexadecimal key MODE. expand((x+2)²) will give 16. tExpand( and tCollect( can be used in a similar fashion. 1: DEC.g. For example in decimal mode 0b111 ENTER gives 7.e. If trig is involved. key F5 All 1: Select All. Also propFrac( is useful in some cases. Highlight the table. For example. etc. To display your answer in decimal (i. if x has been defined as 2. i. E.e. not APPROX mode. etc. binary means base 2 & hexadecimal means base 16. but not “xx”.5) returns 1 and iPart(-1.e. To delete tables. sec(x) which can. In hexadecimal mode 0b111 + 0b1 gives 0h8. the normal way) key MODE. You would have gotten an error message if you had tried to define xt1(x). select all to be deleted with key F4 . be graphed by entering y1 = sec(x) or evaluated by typing sec() which gives -1. How can you convert a number from one base to another? There are two ways. key MODE. lists. To enter 15 in hexadecimal. you want to delete. For example in hex mode 11 + 3 gives 0hE. In hex mode 15 ENTER gives 0hF. 11 is interpreted as decimal. By default all user defined variables are placed in the MAIN folder unless you specified otherwise. For example. int(3. Define F4 Other 1: Define can be used to define a function. If you want the integer closest to zero. key 0b111. All of these are in F2 Algebra. If something like this happens. How do you tell the calculator you are entering a binary or hexadecimal number? You type 0b before a binary number & 0h before a hexadecimal number. If you want the smallest integer greater than or equal to x. If you want to delete several variables. use ceiling(. i.5) returns 2. functions. Alternatively you can use the convert base commands in 2nd MATH D: Base. Firstly you can set the mode to the desired base and key in the number with the correct prefix. You cannot delete the MAIN folder. simply removing any fractional part. This is surprisingly easy to do by accident. key 0hF. Base. the answer will be displayed in the chosen mode. Key ENTER after you finish each line.g.x. To leave the Program I/O screen. variable[.ti. Input [prompt]. Factor(x^2 .g. Enter Calculus The two main operations of calculus are differentiation and integration.-2) ENTER gives x5/20. To run the program.x) |x = 3 ENTER gives 1/8.g. “Output”. On The Home Screen Differentiation 2nd d (expression. d(x³. d(x³. They can be done on the home screen with the keyboard or on the graph screen.Tips for Math with the TI-89 Calculator. Disp displays the current contents of the Program I/O screen. example().g. key HOME. into your program. E. of course.html or http://www-s. Note the first line is the name you assigned to the program. e. To find f’(a) [where a is a number] use “|” E. Order is optional. d(f(x)g(x). E. variable prints the prompt on the screen. F1.ti.ti. 3. This is probably because by accident you stored something in x. the result is an antiderivative. d((x-1)(x+1)). Variable: example. order]) (on the 8 key) does differentiation. d(x³. After you type in a value for the variable and key ENTER. stop a running program. E. Error Messages Error: Dimension Error: Dimension means that the variable you are using. that value is assigned to the variable. Both of these can be done easily either numerically or symbolically.g.g.g. key APPS 7: Program Editor 7: Program Editor 3: New Type: Program. Look at links below http://education.com/student/TIStudentCenterH ome. or http://education.cgi?databasetoopen=calc ulators&topicarea=TI-89/92+Plus&do_2=1 Programming To type in a program. i. the sec(x) function created above) ENTER. E. Reset All Memory To reset all memory key: 2nd. the default is 1. Note that the () were not used above when naming the program. x.com/index.9). The lines :example() :Prgm : :EndPrgm then appear automatically. “Calculus”. e.x. note the constant is dropped.2) ENTER gives 6x. e. sec(x) now appears in the entry line.x) ENTER gives the quotient rule (although in a nonstandard form).e. On Line Help TI provides dozens of forums on its TI Calculators. To save on typing key F3 I/O to insert “Disp”. type its name in the entry line on the home screen.g.html . e. is not the correct dimension for the function you are using. 6. page 15 If you want to use your newly-defined function. If order is negative.x) ENTER gives 3x².g. etc.com/cgibin/discuss/sdbmessage. easier than keying in the letters is copying a variable from VAR-LINK. etc. key F5 PrgmIO or key ON ENTER HOME. When you are done typing the program. for example there are forums entitled “TI-89”. “Precalculus”. Type in your program between Prgm and EndPrgm. Lbl creates a label which the program can later jump to with a Goto statement.g. Key 2nd VARLINK. For a 2nd derivative use 2. Either use F4 Other 4:DelVar x or F6 Clean Up 1: Clear a-z. Give it a name. Highlight the variable (e. “Input”. I recommend finding minima and maxima and numerical differentiation and integration. Minima and Maxima F3 Calc 6: fMin( and 7: fMax( will find a minimum and maximum respectively between Lower and Upper Bounds. the limit is from the left. because rounding properly gives a slightly too large . x) ENTER will find the value of x where y = (x-2)² + 3 has a minimum i.x). If direction is a positive number.gives x = . = f(c + ).g. Finding Epsilon Definition L= lim xc f ( x ) iff in the Limit for any . if direction is a negative number. ) ENTER gives -2/3. Since . Limits F3 Calc 3: limit( will find a limit.e. instead of plugging away blindly.c) ENTER gives x³/3 + c F3 Calc has other useful operations.x.g. E.1)]/.  = 0. The default value of x is 0. E. Key in the + & then the .e.-1) = - (from the left side) Appendix A has further examples. F3 Calc 8: arcLen( (1-x². To find . x.0083683. graphing y = (x-1)(x+1) and in response to the prompt keying in 3 ENTER gives . the limit is from the right. limit(1/x. limit(3x/(1-2x). Arc Length F3 Calc 8: arcLen( will find the arc length. etc.0082989 is smaller.x. Integration 2nd (expression. L = xlim 2 f ( x ) = 8. E.1)-f(x-.x. Then you will be prompted “dy/dx at?” Either key in the desired x value and key ENTER or move the cursor to the desired x value and key ENTER.g. limit(1/x. x = 2.. 1/x. The + gives x = . so key them in Integration To find the integral key F5 Math 7: f(x)dx.g.1) ENTER gives 3. You have to key in the + & then the separately.point[.1. Key in solve(L +/. You know c. x². use solve(.2. but ignore the minus sign.var. The . lower][. limit(1/x. (i. 0. E. x². The function nDeriv(expression. x cosx. the exact derivative. 0) = undef E.0082989.-1.g.Tips for Math with the TI-89 Calculator. 1) =  (from the right side) On The Graph Screen Plot the graph as usual. One of the x’s may come out as a negative number. key it in as “x”.2) ENTER gives ln2 lower is added as a constant of integration if upper is omitted. find delta such that epsilon = 0.g.x. i. Since  is the unknown. e. Syntax: limit(expression. Whichever is the smaller x. variable[.1) calculates [f(x+. be done on the graph screen. ).x) gives 4x³. whereas d(x4. which gives 4 x (x² + .125 E. nDeriv(x4. f(x) is within  of L.0082989 down.1. For example: For f(x) = x³ and c = 2.1=(2+x)^3. x] allows the user to explore the difference quotient if x is kept finite.e. You will be prompted for the “Lower Limit?” Either key in the desired x value and key ENTER or move the cursor to the desired x .e.g. page 16 Numerical Differentiation d( [the differentiation function above] takes the difference quotient with the lim x0.g.0. var [. Note that I rounded . as numbers. E. L & .g.x. there is a  such that if x is within  of c.00829.001.direction]).1. E. because then you can see what’s going on. F3 Calc 6: fMin((x-2)² + 3. E. The optional direction parameter indicates from which side a one-sided limit is taken.g.g. “incorrectly”). upper]) (on the 7 key) [or equivalently F3 Calc 2: ( integrate] does integration. gives an  larger than 0. x. x. Differentiation To find the derivative key F5 Math 6: Derivatives 1: dy/dx.separately. is . i.x) ENTER gives cosx + x sinx E.01). So  key in solve(8+/-.g. So L ±  = c ± .14159.x) ENTER gives x³/3 E. variable. Or simply sum the entire sequence. Plot the first 6 terms of the Taylor series approximation. One sums x   f ( z i ) ]. 3.n) RM: Taylor Series Approximations Taylor series approximations can be displayed. 3: Minimum and 4: Maximum will find a minimum and maximum respectively between Lower and Upper Bounds. ui1 = -4 and  GRAPH. order[. Inflection Points. y1 = ((-1)^n * x^(2n+1)/((2n+1)!). x = (b-a)/n. 2)| t = (x) gives x/2 +x + 1.69315. The parameters are taylor(expression. key  TABLE.g.org/p2. i. b]. 5)) and also y2 = sin x on the same screen. And taylor((ln(x).x . x. RM or RR). Display digits 9 & | to set the cell width to 12 (for the maximum number of decimal places) and use F2 Setup to select where the table starts and the step size of the table. Other options are available in other modes. Use MODE . x. where expression is the expression to be expanded.0. expand( is F2 Algebra 3:expand( E. 0).Tips for Math with the TI-89 Calculator.g. E. 8: Inflection will find an inflection point between Lower and Upper Bounds. 0. sinx = ((-1)^n * x^(2n+1) / (2n+1)!. n. x.g. The value of x where the approximation diverges from sin x is very striking. 2.n-1) STO lreman(a.1/x. i 1 n Convergence of a Sequence Convergence of a sequence can be displayed. Key in u1(n) = -. You will be prompted similarly for the “Upper Limit?” E. 5) gives x5/120 .1. point]). Point is the point about which the series is expanded. A: Tangent will draw a tangent line at the entered point and display its equation. Surprising? where x is the (uniform) width of the interval and f(zi) is the height of the ith interval. For example taylor((sin(x). x.g. x. For example taylor(1/(x*(x-1)).8u1(n-1) + 3. The Taylor polynomial includes non-zero terms of integer degrees from zero through order expanded in powers of (variable . expand(1(x^2-5x+6)) gives 1 1  x3 x2 Minima. 3) gives taylor(1/(x*(x-1)). n. t. graphing y = 1/x and in response to the prompts keying in 1 ENTER and then 2 ENTER gives . try (1/x².6. middle or right Riemann sum (RL. E. The expression for zi depends on whether one is doing the left. 1) gives -(x1)²/2 + x . Riemann Sums Riemann Sums are approximations to the definite integral. (Check out this awesome site!) Here’s how: Type the following on the home screen.x. For example. ) ENTER.y2(x).calc. RL: (b-a)/n(f(a+x(b-a)/n). 0. E. Partial Fraction Decomposition Use expand( for partial fraction decomposition. B: Arc will find the arc length between “1st Point” and “2nd Point”. taylor() performs a Taylor series expansion. Fortunately http://tifaq.b. 3). If the taylor series would include non-integer powers. in polar mode 6: Derivatives offers the options of 1: dy/dx or 4: dr/d.x³/6 + x. page 17 value and key ENTER. x. 0). )). 2) gives taylor(e^((x)). x) gives -x² . Maxima.point). one at a time. deselect y1 & y2. And taylor(e^(x)). the TI-89 must be forced to give the expansion with a trick. x.6 shows how to create functions to do these 3 sums and also how to create functions to do the Trapezoid and Simpson’s rules. but expand(taylor(1/(x-1)/x). The sequence can be seen to converge to 2. but taylor(e^(t). Or to find the numerical value of the differences define y3(x) = y1(x) . Or Key  TABLE to see the values. x. Tangent Lines & Arc Length F5 Math has several other useful operations. In parametric mode 6: Derivatives offers the options of 1: dy/dx or 2: dy/dt or 3: dx/dt.g.e.htm#11. For a sum with n terms over the interval [a. Its default value is zero. 1. 2. b.x.n))/3 STO simpson(a. x. y) gives y = @1 e . . but useless] after about 90 seconds.Tips for Math with the TI-89 Calculator.b. scrolling down to mreman and keying ENTER. Example: deSolve(y'=x*y.0. because it goes through and simplifies them . In exact mode the TI-89 gives the exact value. You can avoid having to type mreman( in by keying 2nd VAR-LINK.b. x2 2 Implicit differentiation impDif(Equation.n)+trapez(a.n))/2 STO trapez(a. @1 is the TI way of writing an undetermined constant.b.n-1) STO mreman(a. IndependentVar.n) STO rreman(a.n)+rreman(a.b. 100) gives 2. .order] ).. y) gives (y' = ) -2y/3x . DependentVar [. impDif( is F3 D Example: impDif(x^2 y^3 = 1.n) Simpson’s rule: 2*rreman(a. in this example.b.x. with 100 intervals first key sin(x) STO f(x).very impressive. So use auto mode. IndependentVar.b.n) RR: (b-a)/n(f(a+x(b-a)/n). 100).n) Trapezoid rule: (lreman(a. [less than 100 terms.n) * Now store your function in f(x) with STO f(x). The ' in y' is 2nd =. x.1. 4 cos (39/200) + 4 cos (37/200) + . mreman(0.b.. For example to evaluate the midpoint Riemann sum for sin x from 0 to . deSolve( is F3 C. page 18 (b-a)/n(f(a+(x+½)(b-a)/n). DependentVar ). then key mreman(0. Differential Equations deSolve(Equation. 8.5439 0/85.8 
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