ࡱ> Z\WXY @ yHbjbj00 LRR rrrrrrr4QQQhQT0.U"d(ddd f p s$^R rAuf fAuAu rrddӆӆӆAu rdrdӆAuӆӆ7rr_dU U$ QBOst00W`5F`_rrrr`r_AuAuӆAuAuAuAuAu IQ{XQTips for Math with the TI-89 Titanium  DATE \@ "yyyy MMMM dd" 2006 July 24, for the TI-89 Titanium OS 3.10 Dr. Wm J. Larson, International School of Geneva,  HYPERLINK "mailto:william.larson@ecolint.ch" william.larson@ecolint.ch Corrections welcome.  TOC \o "1-3" \h \z \u  HYPERLINK \l "_Toc136402056" Operating System  PAGEREF _Toc136402056 \h 2  HYPERLINK \l "_Toc136402057" Getting Started  PAGEREF _Toc136402057 \h 2  HYPERLINK \l "_Toc136402058" The Green, Yellow & Purple Symbols  PAGEREF _Toc136402058 \h 2  HYPERLINK \l "_Toc136402059" The Two Minus Keys  PAGEREF _Toc136402059 \h 2  HYPERLINK \l "_Toc136402060" The Two Equals Signs  PAGEREF _Toc136402060 \h 2  HYPERLINK \l "_Toc136402061" Exponents and Roots  PAGEREF _Toc136402061 \h 2  HYPERLINK \l "_Toc136402062" Scientific Notation  PAGEREF _Toc136402062 \h 2  HYPERLINK \l "_Toc136402063" Degrees & Radian Modes  PAGEREF _Toc136402063 \h 2  HYPERLINK \l "_Toc136402064" Exact and Approximate Modes  PAGEREF _Toc136402064 \h 2  HYPERLINK \l "_Toc136402065" More Symbols & Functions  PAGEREF _Toc136402065 \h 3  HYPERLINK \l "_Toc136402066" Parentheses  PAGEREF _Toc136402066 \h 3  HYPERLINK \l "_Toc136402067" Lists, Matrices & Vectors  PAGEREF _Toc136402067 \h 3  HYPERLINK \l "_Toc136402068" The Plus/Minus Sign  PAGEREF _Toc136402068 \h 3  HYPERLINK \l "_Toc136402069" The F1 to F8 Keys  PAGEREF _Toc136402069 \h 3  HYPERLINK \l "_Toc136402070" Clearing  PAGEREF _Toc136402070 \h 3  HYPERLINK \l "_Toc136402071" Shortcuts  PAGEREF _Toc136402071 \h 3  HYPERLINK \l "_Toc136402072" On Screen Syntax Help  PAGEREF _Toc136402072 \h 3  HYPERLINK \l "_Toc136402073" If the screen is too dark or light  PAGEREF _Toc136402073 \h 3  HYPERLINK \l "_Toc136402074" If the calculator is locked-up  PAGEREF _Toc136402074 \h 4  HYPERLINK \l "_Toc136402075" To quickly move to the end of an expression  PAGEREF _Toc136402075 \h 4  HYPERLINK \l "_Toc136402076" To reuse a previous entry  PAGEREF _Toc136402076 \h 4  HYPERLINK \l "_Toc136402077" To highlight text  PAGEREF _Toc136402077 \h 4  HYPERLINK \l "_Toc136402078" To copy text  PAGEREF _Toc136402078 \h 4  HYPERLINK \l "_Toc136402079" If you are lost  PAGEREF _Toc136402079 \h 4  HYPERLINK \l "_Toc136402080" To change the number of digits displayed  PAGEREF _Toc136402080 \h 4  HYPERLINK \l "_Toc136402081" The With key  PAGEREF _Toc136402081 \h 4  HYPERLINK \l "_Toc136402082" Insert Mode  PAGEREF _Toc136402082 \h 4  HYPERLINK \l "_Toc136402083" log x  PAGEREF _Toc136402083 \h 4  HYPERLINK \l "_Toc136402084" Pi  PAGEREF _Toc136402084 \h 4  HYPERLINK \l "_Toc136402085" order of operations  PAGEREF _Toc136402085 \h 4  HYPERLINK \l "_Toc136402086" Graphing  PAGEREF _Toc136402086 \h 5  HYPERLINK \l "_Toc136402087" Zoom  PAGEREF _Toc136402087 \h 5  HYPERLINK \l "_Toc136402088" Friendly Windows  PAGEREF _Toc136402088 \h 5  HYPERLINK \l "_Toc136402089" Trace  PAGEREF _Toc136402089 \h 6  HYPERLINK \l "_Toc136402090" To change the center of the graph  PAGEREF _Toc136402090 \h 6  HYPERLINK \l "_Toc136402091" To format a graph  PAGEREF _Toc136402091 \h 6  HYPERLINK \l "_Toc136402092" To cancel a graph  PAGEREF _Toc136402092 \h 6  HYPERLINK \l "_Toc136402093" Modes  PAGEREF _Toc136402093 \h 6  HYPERLINK \l "_Toc136402094" To store a window setting  PAGEREF _Toc136402094 \h 6  HYPERLINK \l "_Toc136402095" To depict an inequality  PAGEREF _Toc136402095 \h 6  HYPERLINK \l "_Toc136402096" Parametric Graphs  PAGEREF _Toc136402096 \h 6  HYPERLINK \l "_Toc136402097" Polar Graphs  PAGEREF _Toc136402097 \h 6  HYPERLINK \l "_Toc136402098" To find the Minima or Maxima of a function  PAGEREF _Toc136402098 \h 7  HYPERLINK \l "_Toc136402099" Solving a system of two equations I.e. Finding the Intersections of Two Graphs  PAGEREF _Toc136402099 \h 7  HYPERLINK \l "_Toc136402100" To find the x and y intercepts  PAGEREF _Toc136402100 \h 7  HYPERLINK \l "_Toc136402101" Tables  PAGEREF _Toc136402101 \h 7  HYPERLINK \l "_Toc136402102" Asymptotes  PAGEREF _Toc136402102 \h 8  HYPERLINK \l "_Toc136402103" Piecewise-Defined Graph  PAGEREF _Toc136402103 \h 8  HYPERLINK \l "_Toc136402104" Graphing a Real Function  PAGEREF _Toc136402104 \h 9  HYPERLINK \l "_Toc136402105" Complex Numbers  PAGEREF _Toc136402105 \h 9  HYPERLINK \l "_Toc136402106" Converting from Rectangular to Polar or Trigonometric Form  PAGEREF _Toc136402106 \h 9  HYPERLINK \l "_Toc136402107" Solving with Complex Numbers  PAGEREF _Toc136402107 \h 9  HYPERLINK \l "_Toc136402108" Graphing a Complex Function  PAGEREF _Toc136402108 \h 10  HYPERLINK \l "_Toc136402109" Converting from Rectangular to Polar Coordinates  PAGEREF _Toc136402109 \h 10  HYPERLINK \l "_Toc136402110" Other Commands  PAGEREF _Toc136402110 \h 10  HYPERLINK \l "_Toc136402111" The Solve & Zeros commands  PAGEREF _Toc136402111 \h 10  HYPERLINK \l "_Toc136402112" Solving inequalities  PAGEREF _Toc136402112 \h 11  HYPERLINK \l "_Toc136402113" Factoring and Expanding  PAGEREF _Toc136402113 \h 11  HYPERLINK \l "_Toc136402114" Permutations and Combinations  PAGEREF _Toc136402114 \h 11  HYPERLINK \l "_Toc136402115" Sequences and Series  PAGEREF _Toc136402115 \h 11  HYPERLINK \l "_Toc136402116" Matrices  PAGEREF _Toc136402116 \h 12  HYPERLINK \l "_Toc136402117" How to graph a Conic Equation  PAGEREF _Toc136402117 \h 12  HYPERLINK \l "_Toc136402118" How to Simplify Rational Functions  PAGEREF _Toc136402118 \h 13  HYPERLINK \l "_Toc136402119" To Solve a System of Equations with Solve(  PAGEREF _Toc136402119 \h 13  HYPERLINK \l "_Toc136402120" The Inverse  PAGEREF _Toc136402120 \h 13  HYPERLINK \l "_Toc136402121" Linear Interpolation  PAGEREF _Toc136402121 \h 14  HYPERLINK \l "_Toc136402122" Step functions  PAGEREF _Toc136402122 \h 14  HYPERLINK \l "_Toc136402123" To Simplify Expressions  PAGEREF _Toc136402123 \h 14  HYPERLINK \l "_Toc136402124" Binary, Hexadecimal & Decimal  PAGEREF _Toc136402124 \h 14  HYPERLINK \l "_Toc136402125" To delete functions, lists, tables, etc.  PAGEREF _Toc136402125 \h 14  HYPERLINK \l "_Toc136402126" Define  PAGEREF _Toc136402126 \h 15  HYPERLINK \l "_Toc136402127" Programming  PAGEREF _Toc136402127 \h 15  HYPERLINK \l "_Toc136402128" On Line Help  PAGEREF _Toc136402128 \h 15  HYPERLINK \l "_Toc136402129" Error Messages  PAGEREF _Toc136402129 \h 15  HYPERLINK \l "_Toc136402130" Error: Dimension  PAGEREF _Toc136402130 \h 15  HYPERLINK \l "_Toc136402131" Reset All Memory  PAGEREF _Toc136402131 \h 15  HYPERLINK \l "_Toc136402132" Calculus  PAGEREF _Toc136402132 \h 16  HYPERLINK \l "_Toc136402133" On The Home Screen  PAGEREF _Toc136402133 \h 16  HYPERLINK \l "_Toc136402134" Differentiation  PAGEREF _Toc136402134 \h 16  HYPERLINK \l "_Toc136402135" Numerical Differentiation  PAGEREF _Toc136402135 \h 16  HYPERLINK \l "_Toc136402136" Integration  PAGEREF _Toc136402136 \h 16  HYPERLINK \l "_Toc136402137" Limits  PAGEREF _Toc136402137 \h 16  HYPERLINK \l "_Toc136402138" Finding Epsilon in the Limit Definition  PAGEREF _Toc136402138 \h 16  HYPERLINK \l "_Toc136402139" Minima and Maxima  PAGEREF _Toc136402139 \h 16  HYPERLINK \l "_Toc136402140" On The Graph Screen  PAGEREF _Toc136402140 \h 17  HYPERLINK \l "_Toc136402141" Differentiation  PAGEREF _Toc136402141 \h 17  HYPERLINK \l "_Toc136402142" Integration  PAGEREF _Toc136402142 \h 17  HYPERLINK \l "_Toc136402143" Partial Fraction Decomposition  PAGEREF _Toc136402143 \h 17  HYPERLINK \l "_Toc136402144" Minima, Maxima, Inflection Points, Tangent Lines & Arc Length  PAGEREF _Toc136402144 \h 17  HYPERLINK \l "_Toc136402145" Convergence of a Sequence  PAGEREF _Toc136402145 \h 17  HYPERLINK \l "_Toc136402146" Taylor Series Approximations  PAGEREF _Toc136402146 \h 17  HYPERLINK \l "_Toc136402147" Riemann Sums  PAGEREF _Toc136402147 \h 18  HYPERLINK \l "_Toc136402148" Differential Equations  PAGEREF _Toc136402148 \h 18  HYPERLINK \l "_Toc136402149" Implicit differentiation  PAGEREF _Toc136402149 \h 18  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  HYPERLINK "http://education.ti.com" http://education.ti.com and navigate to  HYPERLINK "http://education.ti.com/educationportal/sites/US/productDetail/us_os_89titanium.html" http://education.ti.com/educationportal/sites/US/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 2nd key accesses the blue functions, e.g. 2nd p. 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 ( key, produces upper case letters, e.g. ( 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 2nd a-lock or just hold the ALPHA button down. To key several uppercase letters, key ( 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 7 is keyed as 7 ^ 3. nx = x1/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 [2nd 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 , i.e. correctly, in degree mode sin((p/6)r) evaluates as , i.e. correctly. The r symbol is not alpha R; it is keyed as 2nd MATH 2: Angle 2:r. Thus radian mode is recommended, because it can be overridden more easily than degree mode. ( is keyed as 2nd .) Exact and Approximate Modes In AUTO mode (recommended), fractions will be displayed as fractions (e.g.  2/3 , not 0.666667) and pi as  p . To change to Auto mode key MODE, 2nd %, 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 p 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. More Symbols & Functions Catalog contains all of the calculators functions (e.g. !, sinh-1, (, 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. 2nd CHAR 2: Math A: gives (, I: gives , N: gives . These are just symbols, not functions, e.g. trying to evaluate 3 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}p/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 x + y = 1, solve for y, i.e. y =  EMBED Equation.2 , and key y1 = {1, -1}(1-x^2). To solve the quadratic formula for 2x + 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 2nd 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)), you expect x-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 (. keying ( > gives (. keying ( = gives . keying ( ( accesses the Greek letters. For example ( ( alpha s gives (. 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. 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 2nd %, %, % 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 (2nd 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 ( 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) to have an inverse, limit its domain by keying y1 = (x - 2) | x e" 2 Insert Mode 2nd 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 ( logex is on the keyboard. log x ( log10x is not on the keyboard. To get it key CATALOG 5 %, %, %, %, ENTER. Or use logx = ln x/ln 10. E.g. ln 100000/ln 10 gives 5. Pi To enter p, use the p 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=. 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 equations 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 = p/24 = 7.5 and xscl = p/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  EMBED Equation.DSMT4  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 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) = x. 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 q (not t, x, y or z) as your independent variable. q is (q (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.) qmin, qmax & qstep, 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 & q. ( | Coordinates Rectangular will cause F3 Trace to display the coordinates x, y & q. 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 1st Curve? and then 2nd Curve? to choose the two functions whose intersections you want to find. If the cursor is on the correct 1st & 2nd 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 2nd 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 q, 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, Dtbl = 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 & Dtbl, 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, (, then key in another x value, etc. To change the cell width in TABLE key (| or F1 9: Format 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 -10100 for x < 2 and a horizontal line at 10100 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.  EMBED Equation.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.  EMBED Equation.2 , 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 ( 0 y2(x) = 3 - x + x | 0 < x and x ( 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 ( 1 would have drawn a 3 for all x, not just for 0 < x ( 1. The Boolean operator and is 2nd 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  EMBED Equation.2 can be written as, y10(x) = when (x > 1, x, x^2). To write  EMBED Equation.2 , you must use nested when(s, For example: y1(x) = when (x ( 1, when(0 < x, 3, -x), x) or y1(x) = when (x ( 0, -x, when(x( 1, 3, x)) Notice the difference in the inequalities used above and that the -x + x term is no longer needed. For example 4 - x, x < 1 f(x) = 1.5x + 1.5, 1 ( x ( 3 x + 3, x > 3 is entered as y1 = when(x<=3, when(x < 1, 4 - x^2, 1.5x + 1.5), x + 3) 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 ( 1, key y1 = when(x ( 1, x, undef). undef means undefined, so it won't draw anything if x > 1. Just type undef in, its 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 = x2/3 is considered to be defined only for x ( 0 and y = EMBED Equation.3  is considered to be defined only for x e" 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 = x2/3 is (incorrectly) graphed for all x, but y = EMBED Equation.3  is (correctly) graphed only for all x > 4. 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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 = x. 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 y term (e.g. y = x or x + y = 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 px + 3,x) gives x = {tan-1[(y-3)/2]}/p + @njp. (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]}/p + kp, k = an integer. To get the inverse function just drop the + @njp part. So the inverse function is f-1(x) = tan-1[(x-3)/2]/p. In this example DrawInv 2 tan px + 3 or graphing y1 = {tan-1[(x-3)/2]}/p + @njp, gives a series of curves, one on top of the other. On the other hand graphing y1 = {tan-1 [(x-3)/2]}/p 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 x1 and x2 bracket x.: y = y1 +  EMBED Equation.2  E.g. find f(5.8) = 25.2 + (25.8-25.2)(5.8-5)/(6-5) = 25.68 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 2nd 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(p) 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)) 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. If you want to use your newly-defined function, easier than keying in the letters is copying a variable from VAR-LINK. Key 2nd 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/TIStudentCenterHome.html or http://www-s.ti.com/cgi-bin/discuss/sdbmessage.cgi?databasetoopen=calculators&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 2nd d (expression, variable[, order]) (on the 8 key) does differentiation. E.g. d(x,x) ENTER gives 3x. 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 2nd derivative use 2. E.g. d(x,x,2) ENTER gives 6x. If order is negative, the result is an antiderivative. E.g. d(x,x,-2) ENTER gives x5/20, note the constant is dropped. Numerical Differentiation d( [the differentiation function above] takes the difference quotient with the lim Dx0. The function nDeriv(expression, var [, Dx] allows the user to explore the difference quotient if Dx is kept finite. The default value of Dx is 0.001. E.g. nDeriv(x4,x,.1) calculates [f(x+.1)-f(x-.1)]/.2, which gives 4 x (x + .01), whereas d(x4,x) gives 4x, i.e. the exact derivative. Integration 2nd (expression, variable[, lower][, upper]) (on the 7 key) [or equivalently F3 Calc 2: ( integrate] does integration. E.g. (x,x) ENTER gives x/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. (x,x,c) ENTER gives x/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 =  EMBED Equation.2 iff for any (, there is a ( such that if x is within ( of c, f(x) is within ( of L. So L ( = c (. To find (, use solve(. Key in solve(L +/- ( = f(c + (), (). You have to key in the + & then the - separately. You know c, L & (, so key them in as numbers. Since ( is the unknown, key it in as x. Whichever is the smaller x, is (. One of the xs may come out as a negative number, but ignore the minus sign. For example: For f(x) = x and c = 2, find delta such that epsilon = 0.1. L =  EMBED Equation.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, ( = 0.00829. Note that I rounded .0082989 down, (i.e. incorrectly), because rounding properly gives a slightly too large (, i.e. gives an ( 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) + 3, x) ENTER will find the value of x where y = (x-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-x,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 whats 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 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  EMBED Equation.3  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 1st Point and 2nd 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/dq. 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. 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Or Key ( TABLE to see the values. Or simply sum the entire sequence. E.g. try ((1/x, x, 1, () ENTER. Surprising? Taylor Series Approximations Taylor series approximations can be displayed. E.g. sinx = (((-1)^n * x^(2n+1) / (2n+1)!, n, 0, ()). Plot the first 6 terms of the Taylor series approximation, i.e. y1 = (((-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 x5/120 - x/6 + x. And taylor((ln(x), x, 2, 1) gives -(x-1)/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 -x - 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 Dx [ EMBED Equation.2 ], where Dx is the (uniform) width of the interval and f(zi) is the height of the ith interval. For a sum with n terms over the interval [a, b], Dx = (b-a)/n. The expression for zi depends on whether one is doing the left, middle or right Riemann sum (RL, RM or RR). 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. RL: (b-a)/n*((f(a+x*(b-a)/n),x,0,n-1) STO lreman(a,b,n) RM: (b-a)/n*((f(a+(x+)*(b-a)/n),x,0,n-1) STO mreman(a,b,n) RR: (b-a)/n*((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 p, with 100 intervals first key sin(x) STO f(x), then key mreman(0, p, 100). You can avoid having to typeH.I.m.n. //O0T00022:2<22222222f3h3l3n3p333333344B4F444555555557777777778 8>8@8B8F8T8V8ޟ jSh%"D h%"D5H* h%"DH*j h%"DEHU$j7< h%"DCJUVmHnHujh%"DU hjh%"Dh%"DOJQJ h%"DH* h%"D6 h%"D5h%"D jh%"D>V8X8l8n88888888888899&9F99999999::::,;.;;;;;<@.@2@@@fAhAAA^BtBuBBBBBBBBjh U hX:h OJQJ^JmH sH hX:h mH sH hX:h 5\mH sH  hjh h h%"DH*U h%"D5H* h%"D5h%"DOJQJh%"D jSh%"D7 mreman( in by keying 2nd VAR-LINK, scrolling down to mreman and keying ENTER. mreman(0, p, 100) gives 2. In exact mode the TI-89 gives the exact value, in this example, 4 cos (39p/200) + 4 cos (37p/200) + ... [less than 100 terms, because it goes through and simplifies them - very impressive, but useless] after about 90 seconds. So use auto mode. Differential Equations deSolve(Equation, IndependentVar, DependentVar ). deSolve( is F3 C. Example: deSolve(y'=x*y, x, y) gives y = @1  EMBED Equation.DSMT4 . @1 is the TI way of writing an undetermined constant. The ' in y' is 2nd =. Implicit differentiation impDif(Equation, IndependentVar, DependentVar [,order] ). impDif( is F3 D Example: impDif(x^2 y^3 = 1, x, y) gives (y' = ) -2y/3x     Tips f:^BuBBOChCCCCCCCCCCCC4H5HvHwHxHyH &dP$Tx^T`a$gdN$a$gd $Tx^T`a$gd BBCCC=C>CCCDCICKCOChCCCCCCCCCCCCCCH(H)H/H0H2H3H4H5HjHƼƼƴƭƟʔtlt_t[h>/h6rE0JCJmHnHuh>/0JCJjh>/0JCJUU h>/CJh<-jh<-UhX:h%"DmH sH hX:h 5\mH sH  hjh%"Dh h H*h OJQJ^Jh hX:h mH sH jh Uj?h h EHU)jK!H h CJUVmH nH sH tH #or Math with the TI-89 Calculator, page  PAGE 18 Tips for Precalculus with the TI-89 Calculator, page  PAGE 17 jHkHqHrHtHuHvHwHxHyHhX:h%"DmH sH h<-h>/ h>/CJh>/0JCJmHnHuh>/0JCJjh>/0JCJU @ 00&P :pH. A!S"S#S$S%n0 '? 0 00&P :pH. A!S"n#S$S%7 P ? 0 00&P :pH. 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