ti 89 programming instructions

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Instructions for using a TI-89with Fundamentals of PhysicsJearl Walker

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Procedures useful starting in Chapters 1 and 2Turning On/Off and Changing ContrastMode SettingSubtraction and Negation KeysMultiplication and DivisionPowers and Scientific NotationReevaluating an ExpressionUsing an Answer in a Next CalculationEscaping, and Getting HomeConversion of UnitsEntering Alphabetic SymbolsStored Constants

Storing and Recalling a Value

Putting Greek Letters on the ScreenSolving Quadratic Equations

Calculator Sample Problem

More advanced procedures, useful starting in Chapter 2Making and Using a List

Calculator Sample ProblemGraphing a Function


Procedure useful in lab workLinear Regression


Basic procedures, useful starting in Chapter 3Trigonometric FunctionsVector NotationsVectors in Ma gnitude-Angle NotationVectors in Unit-vector NotationSwitching between Magnitude-angle Notation and Unit-vector NotationAngles for Three-dimensional VectorsAdding VectorsStoring a VectorOver riding the Mode Setting for Angles

Calculator Sample ProblemVector Multiplication

Product of a Scalar and a Vector

Dot (or Scalar) Product of Two VectorsCross (or Vector) Product of Two Vectors

Advanced procedures, useful starting in Chapter 4Numeric Solver


Introduction to Programming

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Custom Toolbar Menu

Procedure useful starting in Chapter 5Adding Force Vectors

Procedure useful starting in Chapter 7

Functional and Graphical Integration


Procedure for Chapter 14Multiple Gravitational Forces

Procedure for Chapter 16Graphing SHM


Procedure for Chapter 17Adding Waves


Procedure for Chapter 18Using log, Inverse log, and User-defined Functions

Procedure for Chapter 19Converting Temperatures

Procedure for Chapter 22Multiple Electrostatic Forces

Procedure for Chapter 28Simultaneous Linear Equations

Procedure for Chapter 38

User-defined Function

Turning On/Off and Changing ContrastTo turn on the calculator, press the ON key in the lower left corner.To turn it off, press the 2nd key near the upper left corner, releaseit, and then press the key with OFF written just above it (that is, theON key, where you are now using the 2nd purpose of the key).

The 2nd key is said to be an on-off switch. You activate it with onepress but you can change your mind and deactivate by pressing it again.When it is activated, 2nd appears in the lowest line on the screen(the status line).

You will need to change the contrast on the screen to adjust forthe room light and for the aging of the battery. To change thecontrast, press and hold down the green-diamond key (located just belowthe 2nd key) and then press and release the plus or minus keys once ortwice. Note the color coding: the green-diamond key activates theoperation noted in green on some other key. Here it is the contrast, assymbolized by the half-filled green circle located between the plus andminus keys.
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Mode SettingThe mode setting of the calculator determines how calculated valuesappear on the screen. To access the mode menu, press MODE. For mostproblems inFundamentals of Physics, you will probably want calculatedvalues to appear in scientific notation, such as 3.143E3, which means

3.143 103, or 3143 innormal notation. The current option for the

calculator is written on the line that begins Exponential Format. Tosee the options, move the cursor down to that line by using the downcursor key. (The oval cursor keys are near the upper right corner ofthe keyboard.) Then press the rightward cursor key to open up thesubmenu. (The presence of the submenu is indicated by the arrow at theend of the line.) In the submenu, move the cursor onto the Scientificoption (if it is not already there) and then press ENTER to make thatthe choice for notation.

The line for Display Digits in the mode menu is where you choosethe number of decimal places that calculated values will have. The lastdecimal place displayed in a calculated value is a rounded-off value,but the full number, with all its significant figures, is actually keptin the calculator. The menu sets only how the number is displayed.

You will probably want two or three decimal places. Use a cursor

key to move to the Display Digits line and then press the rightwardcursor key to open up the submenu. For now, choose the Fix 3 option bymoving the cursor down to it. Then press ENTER. Your results will thenappear in the style of 3.143E3 or 3143.000, depending on whether youhave chosen scientific notation or normal notation.

If the Pretty Print line does not show ON, move to it, press therightward cursor key for the submenu, and then press 2 for the ONoption. The option allows the calculator to stack ratios and otherwiseclean up expressions that may be difficult to read (and proof) whenthey are initially entered. You can spot an error more easily with sucha cleaned-up expression.

Next, move to the last line, which starts Exact/Approx (it is notinitially on the screen). If APPROXIMATE is not listed, open up the

submenu and press 3 for the APPROXIMATE option. We do this to avoidanswers appearing with terms such as 2, even though that might make ananswer exact.

The next two lines should already show Base and DEC, and UnitSystem and SI.

Finally, to save all these choices and to return to the previousscreen, press ENTER. (If you just press ESC, you escape to the previousscreen but none of your changes in the menu are saved.)

If you do a calculation and find that the displayed result is notin the style you wish, press MODE to access the mode menu, change themode to what you want (pressing ENTER for each change), press ENTER toreturn to the previous screen, and then press ENTER to reevaluate yourcalculation.

Entering a Calculation; Seeing the ResultTo enter an expression for calculation, the cursor must be on the entryline, which is just above the status line and bordered by thinhorizontal lines. If the cursor is not already there, press ESC to movethere. To clear the entry line of any old expression, press CLEAR onceor twice. (The first press of CLEAR removes everything to the right ofthe cursor, which is enough if the cursor is at the left side of theline.)


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When you then enter an expression, it appears on the entry line.You can change what you enter in several ways:

1. To delete all items to the right of the cursor, press CLEAR.2. To delete items to the left of the cursor, press the left-arrow

key one or more times.

3. To insert or overwrite at the point of the cursor, first examine thewidth of the cursor. A thin cursor means the calculator will insertnew items as you key them in. A wider cursor means that thecalculator will replace items with any new items as you key them in.To toggle between insert and overwrite, press the 2nd key, releaseit, and then press the key with INS (for insert) printed above it.

To move the cursor slowly, use the leftward and rightward cursor keys.To make it jump to the left or right end, press 2 nd and then theleftward or rightward cursor key.

Once you have pressed in an expression, evaluate it by pressingENTER. The expression is then duplicated in the lower part ofthe history area. The result of the evaluation appears either off tothe right or on the next lower line. Long expressions or evaluations

extend rightward off the screen, as indicated by the marker . You cansee the rest of the expression by moving the cursor up to that line(use the up cursor key) and then using the rightward cursor key toscroll the line leftward across the screen.

Subtraction and Negation KeysBecause the calculator has different keys for subtraction and negation,you must be alert about which key to press. If you want to enter avalue of -3, you use the negation key, which is marked as (-) on thebottom row of keys. If you want to subtract 3 from 5, you press

5, then the subtraction key (the key above the + key), then 3,

and finally ENTER. What do you get if you erroneously press

5, then the negation key (-), then 3,

and finally ENTER?

Multiplication and Division

The multiply key is marked as but it places an asterisk (or star) inthe entry line. The division key is marked as but it places a slash inthe entry line.

In a textbook or in handwritten notes, the expression A/3B meansthat Ais divided by the term 3B. Generally, whatever numbers or

symbols are shown in the first term to the right of a division sign(the slash) are all taken as being part of the divisor. However, on acalculator, the divisor is usually taken to be only the first number orsymbol after the division sign. For example, if you substitute 3for Aand 4 for B, a calculators evaluation ofA/3B as written yields4, not 0.25 because the calculator computes (3/3)4. Thus, you mustdivide by each symbol in the divisor as in 3/3/4 or use parentheses asin 3/(3*4). Obviously such an error would be very difficult to spotduring the pressure of an exam; your best defense against it is to


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practice using your calculator on the course homework to the point atwhich entering data on the calculator is mentally automatic.

Powers and Scientific NotationRaising a number or a symbol to a power is done with the ^ key. Thus, 3raised to the second power is 3^2. Pressing this into the entry line

and then pressing ENTER gives 9.000E0 when the mode is set to giveanswers in scientific notation with three decimal places. Similarly, 3raised to the power is entered as 3^(1/2) or 3^0.5 or 3^.5, whichgives 1.732E0.

For taking a square root, you can also use the square root , whichyou put in by pressing 2nd(that is, press and then release the 2nd keyand then press and release the multiply key, which has the symbolprinted above it as the 2nd operation of that key).

If you want to raise more than one term to a power, you must groupthe terms with parentheses. Here are some examples:

(12 + 46)3is entered as (12 + 4*6)^3

(3 + 56.7)is entered as (3 +5*6.7)(7 + 3B)(1/3)

is entered as (7 + 3B)^(1/3)

(Entering alphabetic symbols is explained below.) Caution: Whenyou raise a term to a power that is a fraction, such as 1/3 you mustuse parentheses to enclose the fraction or the calculator will use onlythe first number in the fraction as the power.

To enter numbers in scientific notation, which is also calledpowers-of-ten notation, we use the EE key. For example, to enter

3.05 105, press in 3.05, then press the EE key, and then press in 5.However, the style in which that number will be displayed in the

history area depends on the mode settings. Here are some examples:

3.050E5 for scientific notation with three decimal places 3.1E5 for scientific notation with one decimal place305000 for normal notation with no decimal places305000.00 for normal notation with two decimal places

Reevaluating an ExpressionWhen you press ENTER to evaluate an expression, the expression remainsin the entry line. You can change it and then press ENTER again toevaluate the new expression. To change it, press the rightward cursorkey to start at the right end of the expression, or press the leftwardcursor key to start at the left end of the expression.

You can move any expression or evaluation from the history areadown into the entry line. First, position the cursor in the entry lineat the point where you want the material from the history area. Thenuse the up cursor key to snake your way up through the evaluations andexpressions until you reach the one you want (you can go higher thanwhat initially shows on the screen). Then press ENTER to paste thatmaterial into the entry line, at the point where the cursor had been.You can then edit the material before you press ENTER again for a newevaluation. If you snake your way into the history area and decide not


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to paste anything from it, you can snake your way back down with thedown cursor key or, faster, simply press ESC to jump directly down.

Using an Answer in a Next CalculationIf you have just calculated an answer, you can use it in a newcalculation without pressing it in again. (Nicely, the answer is used

complete with all its significant figures, not just the rounded-offversion that appears on the screen, as set with the mode menu.) If theanswer is the first item in that new calculation, press CLEAR and thenjust start pressing in the other steps of the calculation. For example,if you want to square the answer, first press the ^ key and then 2 sothat

ans(1)^2

appears in the entry line. (The notation (1) means that this is thefirst answer listed in the history area, numbering upward through thearea.) Now press ENTER to evaluate the expression.

If the answer is not the first item in the new calculation, pressCLEAR and then start entering the new calculation but when you reach

where the previous answer is needed, press the 2nd key, release it,press the key with ANS printed above it, and then continue pressing inany remaining steps in the new calculation. Finally, press ENTER toperform the new calculation. For example,

(45^2 + ans(1)^2)^.5

takes the square root of the sum of 45 squared and the previous answersquared.

Escaping, and Getting HomeIn many of the instructions that follow, you go to screens other thanthe home screen. As you shall see, when you perform some operation in

one of those other screen, you save the results of the operation bypressing ENTER, which can take you back to the home screen. However, ifyou change your mind or just mess up, you get usually get back to thehome screen without saving anything by pressing ESC to escape or HOME.If those do not work, press 2nd and then QUIT.

Conversion of UnitsConversion of units on a TI-89 may not be worth the trouble because theprocess is slowed by finding menus and submenus. To reach the mainmenu, press the 2ndkey, release it, and then press the key with UNITSprinted above it. Use the down cursor key to move down the list (itextends down and off the screen) and note the various quantities, whichare listed alphabetically. Stop on Time and then open its submenu bypressing the rightward cursor key. The submenu extends down and off thescreen. You can use the down cursor key to move to any of the listedmeasures of time. If you stop on one and press ENTER, you would pastethat measure into the entry line back in the home screen, to which youthen jump. (Although temperature is listed in the UNITS menu,converting temperatures requires a special technique that is discussedmuch later in these notes, under Chapter 19.)

As an example, lets convert 10 mi/h to meters per second, with themode set for answers in scientific notation with two decimal places.Clear the entry line, press in 10, and then press 2nd UNITS. Except for


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the top line, the menu consists of quantities (such as acceleration),all in an alphabetic listing. We want length. To reach that unit eitherhold down the down cursor key to scroll down the list slowly or press2nd and then press the down cursor key once to jump down to the nextlower part of the menu. Then move the cursor to the Length line andthen press the rightward cursor key to open up the secondary menu forlength. Then move down to mi (for mile), press ENTER to choose it. Youare then back in the home screen with

10_mi

in the entry line. (Units automatically come with the underscore symbol

_ out front.) Press the divide key to put a slash in the entry line.Next, press 2ndUNITS to return the conversion menu, move down to theTime line and open up its submenu. Move down to hr (whichrepresents hour on a TI calculator) and press ENTER so that

10_mi/_hr

shows in the entry line. Now press 2ndand then the transfer operator ,

which is the second purpose of the MODE key. Next return to the unitsmenu and put in m for meters, use the divide key, and then return tothe units menu and put in s for seconds. Finally, when

10_mi/_hr_m/_s

shows in the entry line, press ENTER to evaluate the expression. Forthe mode set for Sci and two decimal places, we find that 10 mi/h isequivalent to 4.47E0 m/s, which means 4.47 m/s.

You can also convert units that are raised to a power. For example,to convert 9.8 m/s2 to feet per second-squared, put

9.8 _m/_s^2_ft/_s^2

in the entry line and then press ENTER. The result is about 32 ft/s2.You can avoid going to the units menu by pressing in the underscore

symbol and the alphabetic symbols yourself. The underscore symbol isthe green-diamond option of the MODE key. To get it, press and thenrelease the green-diamond key, and then press and release the MODE key.To get a letter, press the alpha key and then the key with the letterprinted above it. (Putting in the alphabetic symbols is explained morefully below, but this explanation will suffice for now.)

Whether you enter an acceleration unit in this way or via the unitsmenu, putting them on the screen is time consuming (not something youwant to do during an exam). A faster technique is to store anacceleration unit under a name and then use that name where you needthe unit. For example, clear the entry line, press in _m/_s^2 and then

press STOso that

_m/_s^2

is in the entry line. Now press in _ms2 as the name under which theactual unit is to be stored. Here are the steps: To put in theunderscore symbol, press the green-diamond key and then the MODE key.To put in ms2, press alpha, then the key with M printed above it, thenalpha again, then the key with S printed above it, and then 2. (You


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could pick a shorter name, but you need the underscore out front tosignal the calculator that this is a unit.) You then have

_m/_s^2_ms2

in the entry line. Next press ENTER to store the unit under the name.Later, whenever you need the unit m/s2, press in _ms2 instead.

Caution: Dont confuse a unit with a division or ratio ofdata. For example, the ratio of 10 m to 2 s is

10_m/(2_s)

on the screen, which gives 5_m/_s and thus 5 m/s as the answer. If youleave out the parentheses, you get 5_m_s and thus 5 ms, which means 5milliseconds.

Entering Alphabetic SymbolsTo make the letters x, y, z, and t appear on the screen, press theirkeys. To make any other letter appear, use the alpha key or the up-arrow key , to turn on the lowercase or uppercase alphabetic modes,

respectively. Then when you press any key with a letter printed aboveit, that letter appears on the screen. You can also get the equal

symbol = (above STO) and a blank space (above the negation key).Here are some example of how to enter letters:

h Press alpha (note the lowercase a on the status line, just belowthe entry line), then H.

hey Press 2nd, then alpha (pressing the 2nd key locks in the lower-casealphabetic mode, called alpha-lock or a-lock); then press H,then E, and then Y, and then press alpha to turn off thealpha-lock. You can also turn on alpha-lock by pressing alphatwice. (As you turn alpha-lock on and off, note the changes inthe status line.)

H Press the up-arrow key (note the arrow on the status line) andthen H.

HEY Press the up-arrow key , then alpha (to lock in the uppercasealphabetic mode, called ALPHA-lock), then H, then E, then Y,then alpha to turn off the alphabetic mode. (As you turnALPHA-lock on and off, note the changes in the status line.)

H2 Press , then H, then 2.HEY2 Press , then alpha (to lock in the uppercase alphabetic mode),

then H, then E, then Y, then alpha (to turn off the uppercasealphabetic mode), then 2.

Hey2 Press , then H, then 2nd, then alpha (to lock in the lowercasealphabetic mode), then E, then Y, then alpha (to turn offalpha-lock), then 2.

Stored ConstantsThe value of is available on the keyboard (press 2nd and then the keywith printed above it). Some other constants commonly used in physicscan be accessed by pressing 2nd UNITS and then the rightward cursor keyto bring up the submenu of stored constants, listed alphabeticallyoccurring to the TI abbreviations.

Scroll down the submenu to see all the abbreviations. If you wantto use these stored constants, you will have to memorize theirmeanings, though most will be become obvious. You can also use them by


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pressing in the abbreviation as you press in an expression on the entryline. For example, you can use the stored value for the speed oflight c by pressing the green-diamond key, then the MODE key (to getthe underscore symbol, which is the green line above the key), and thenthe C key, to put c on the screen. To see the value, now press ENTER.

However, using the stored constants in this way can be a seriousdisadvantage on an exam, where speedy calculations are needed. Thetrouble is that the stored constants come with units and thus, to beconsistent in any expression you are entering into the calculator, youmust key in the units for the other quantities in the expression. Ifyou dont, the calculator signals an error.

A quicker technique is to store the value of a frequently usedconstant under a name as described below. Then you can use that name ina calculation, with no need to press in units. For example, we couldstore the value of the speed of light under the name c and then usethat name in calculations.

The only constant of frequent use before Chapter 14 is the valuefor the free-fall acceleration g, for which we use a reference valueof 9.8. There is usually no advantage in storing such a simplenumber.

Storing and Recalling a ValueIf you want to store a value, say 5, under a name for later use, putthe value in the entry line (the cursor is then to the right of it),

press STOfor the storage procedure, and then put in a name, using thealphabetic procedures discribed above. The name must begin with aletter but it can also contain numbers.

For example, you might put in glob2 by pressing alpha twice andthen the letters G, L, O, and B, pressing alpha again, and thenpressing 2, so that

5glob2

shows in the entry line. When you then press ENTER, the value is storedunder the name glob2. (You can store values under a name containinglowercase letters, not uppercase letters. If you press in a namecontaining uppercase letters, the calculator automatically changes theletters to lowercase during the storage process.)

Later, if you want to see what that value is, clear the entry line,press in the name glob2, and then press ENTER. The value then appearsin the history area in whatever style you have chosen for displayednumbers via the mode menu. (The displayed value may be rounded off butthe full value is actually stored in the calculator.)

If you want to use the stored value in a calculation, press thecalculation into the entry line, with the name positioned where youwant the value. For example, 3+4glob2/13 adds 3 to 4 times the value of

glob2 divided by 13. To evaluate the expression, press ENTER.

Putting Greek Letters on the Screen

The Greek lowercase letters pi and theta are available on thekeyboard. The pi is reserved for the number 3.1416 and cannot beassigned any other value. The theta is not reserved and can be used asthe name of a variable. To find other Greek letters, press 2nd CHAR andthen press the rightward cursor key to open the Greek submenu. Move toany letter and press ENTER to paste it to the entry line. The only


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Greek symbol of use in the first five chapters ofFundamentals of

Physics is the uppercase delta ().

Solving Quadratic EquationsQuadratic equations occur commonly in Chapters 2 and 4 of Fundamentals

of Physics. The quickest way to solve them, provided that you do notneed to store them, is to use the calculators solve procedure. Firstpress CLEAR and then press F2 for the algebra menu and then press ENTERto choose the solve option in the menu. Enter your quadratic equation(or any other polynomial), then a comma, then the variable, and then aclosing parenthesis. The order of terms in the equation is notimportant. For example, you can put

solve(5x^2-5x-45=0,x)

or

solve(-5x-45+5x^2=0,x)

into the entry line. Pressing ENTER then gives the two solutions to theequation. If, instead, the calculator signals false on the screen,you do not have correct coefficients (numbers or signs) in theequation, because the equation does not have real (as opposed tocomplex) solutions. (Messing up a sign on a coefficient is a commonerror in the homework for Chapters 2 and 4.)

Changing one or more coefficients: If you want to re-solve aquadratic equation with one or more new coefficients, edit theequation. (It is still in the entry line. Press the rightward cursorkey to start at the right end of the expression or the leftward cursorkey to start at the left end. Once you have made the changes, pressENTER again.)

Saving one or both solutions:Here is a slow way but one which keepsall the significant figures of an answer: Press CLEAR to clear theentry line, move the cursor up into the history area to the answerline, press ENTER to paste the answers into the entry line, deleteeverything but the answer you want, put the cursor to the right of the

number, and then press STO. Next, using the alphabetic mode, put in aname, such as bear. (Press the alpha key twice to lock in the lowercasealphabetic mode, then the keys with the letters B, E, A, and R printedabove them, and then finally the alpha key to turn off the alphabeticmode.) Then press ENTER to store the number under the name.

Later, to recall the stored value, first press 2nd RCL and thenpress in the name as you did previously. Then when you press ENTERtwice, the stored value reappears (its appearance depends on the stylefor numbers that you selected in the mode menu, but all the significantfigures are actually kept in the calculator). If you want to use thevalue in a calculation, use the values name instead. For example, todouble the value stored under the name bear, you would press in either2bear or 2*bear and then press ENTER.

Calculator Sample Problem: Solving a Quadratic EquationA hot-air balloon is rising at the rate of 2.0 m/s when, at a height of35 m, a banana is accidentally dropped overboard. How long does thebanana take to reach the ground?


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SOLUTION: The KEY IDEA here is that once the banana is released, it isin free fall at a constant downward acceleration of 9.8 m/s2. Thus wecan use the constant acceleration equations of Table 2-1, for a yaxisthat extends vertically upward through the point of the bananasrelease. We know that the acceleration is a = -g= -9.8 m/s2 and thatthe initial velocity is v0 = +2.0 m/s. We also know that the bananas

displacement from the point of release to the ground is y= -35 m (itis negative because the final position of the banana on the ground isbelow the initial position at the release). To solve for time t, wechoose Eq. 2-15 from Table 2-1 and rewrite it in terms of y:

y= v0t + at2.Substituting known data into this equation leads to

-4.9t2 + 2.0t + 35 = 0,

which is in the form of a general quadratic equation. Press F2 for thealgebra menu and then press ENTER to choose the solve option in themenu. Enter the quadratic equation, then a comma, then the variable

(here it is t), and then a closing parenthesis. The order of terms inthe equation is not important. For example, you can put

solve(-4.9t^2+2t+35=0,t)

in the entry line. (If it is, be sure to use the negation key to put in-4.9; if not, then be sure to use the subtraction key.) Pressing ENTERthen brings up the two solutions to the quadratic equation, in thestyle you have set via the mode menu. For normal notation with twodecimal places,

t = 2.88 or t = -2.48

appears as the answer, in the history area. The positive answer is theone we seek. Thus, the time of fall is

t = 2.88 s 2.9 s. (Answer)

If the calculator refuses to give you answers, then you haveincorrectly entered one or more of the coefficients. Most often, theerror is with the sign of a coefficient.

Changing one or more coefficients: If you want to re-solve aquadratic equation with one or more new coefficients, edit the equationin the entry line and then press ENTER again.

Making and Using a ListIf you need to perform a certain calculation many times with differentdata, you can avoid rekeying the calculation each time by using a listof the data in the calculation. For a simple example, if you need to

calculate 3x+ 5 + 4x for x= 0, 3, 7, 9, and 11, you can first makea list of those xvalues, store the list under the name of, say, qq,

and then key in the calculation as 3qq + 5 + 4 qq. When you then pressENTER, you get a list of answers. The first is for x= 0, the next isfor x= 3, and so on.


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To start a list, press 2nd { (the symbol is printed above the firstparenthesis key). Then press in the values to be listed (such asthe xvalues above), in the order you want, each value separated fromthe others by a comma. End the list by pressing 2nd }. For the x valuesgiven above, {0,3,7,9,11} is then in the entry line.

To store the list under the name of, say, qq, press STO, thenpress alpha, Q, alpha once more, and then Q once more. Then pressENTER. The list appears on the history area in the style you haveselected for numbers via the mode-setting menu. For example, withnumbers in scientific notation with 3 decimal places, the list

{0.000E0 3.000E0 7.000E0 9.000E0 1.100E1}

appears on the screen, extending off to the right (use the rightwardcursor key to scroll the list leftward across the screen). You can now

use the name qq in a calculation. For example, for 3qq + 5 + 4 qq, youget

{5.000E0 2.093E1 3.658E1 4.400E1 5.127E1}

as the answer in the history area. The following Calculator SampleProblem gives another example of making and using a list, in a problemfrom Chapter 2 ofFundamentals of Physics.

If you want to change one or more of the values in a list that isno longer in the entry line, you do not have to rekey everything. Firstpress CLEAR and then press 2nd RCL, and then press in the name of thelist in the box provided (using the same procedure by which youoriginally put in the name). Next press ENTER to put in the name andENTER once more to make the recall. The list then appears in the entryline. Change it as you want. You can then store the new list under theold name or under a new name.

To delete a list from memory, press 2nd VAR-LINK, move down thenames until the marker is at the name of the list, press F1 for Manage

options, and then press ENTER for the Delete option. The calculatorasks if you really want to delete the program. Press ENTER once more tosay yes. The list is eliminated. Press HOME to return to the homescreen.

Calculator Sample Problem: Making and using a listSee Sample Problem 2-7 of Fundamentals of Physicsa pitcher tosses abaseball straight up with launch speed v0. Using a list of launch speedvalues of 9.0, 10, 11, 12, and 13 m/s, generate a list of thecorresponding maximum heights reached by the ball.

SOLUTION: From Sample Problem 2-7b, we know that the maximumheight yof the ball is given by

y=(v02 + v2)/2g

= (v02 0)/(2)(9.8). (1)

We can evaluate this equation on a calculator by evaluating

(launch speed)2/19.6 (2)


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for each of the five given launch speeds. However, we can save time andavoid possible keying errors by making a list of those launch speedsand then using the list in the evaluation on the calculator.

To start a list, press 2nd {. Then press in the launch speedvalues, each value separated from the others by a comma. End the listby pressing 2nd }. The screen should show

{9,10,11,12,13}

as the list. (Keying in 9 instead of the given 9.0 does not matter.)Next, in order to square each value in the list, press ^ and then 2.

Next press the divide key and then press in 19.6 so that

{9,10,11,12,13}^2/19.6

shows in the entry line. Pressing ENTER then evaluates the expressionfor each value in the list. If the calculator mode is set for normalnumber notation (instead of scientific notation) and for two decimalplaces, then you see

{4.13 5.10 6.17 7.35 8.62}

in the history area. This answer list means that

launch speed maximum height

v0(m/s) y(m)

9.0 4.1310 5.1011 6.1712 7.3513 8.62

We could have saved the list of launch speeds under a name and thenused that name in Eq. 2. Such storing would not help with thisparticular sample problem but would be helpful if we had to use thesame list of launch speeds in several other calculations.

For practice, lets store the list under the name v0 (vee zero).To get the list of launch-speed list back in the entry line, pressCLEAR. Then use the up cursor key to snake your way up to that list,and then press ENTER. The list is then duplicated in the entry line.

Now position the cursor to the right of the list and then press STO.Now put in the name by pressing alpha, then V, and then 0 (that is,zero). Next press ENTER to store the list under that name.

We could now evaluate Eq. 2 for each value in the list by firstpressing CLEAR and then pressing the sequence

alpha v 0 ^ 2 19.6

so that

v0^2/19.6

appears in the entry line. Then when you press ENTER, this expressionis evaluated for each launch-speed value. When the list of answersappears, you can store it under another name, such as maxh (for maximum


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height) for latter reference or use by repeating the procedure formoving material from the history area into the entry line and thenstoring it under a name. All the significant figures for those answersare stored, not just the ones in the rounded-off versions that appearon the screen, as set by the mode of the calculator just then.

Graphing a FunctionTo graph any function, you must use the variables yand x. So if youractual equation is, say,

a(t) = 4t + 5t2,

you need to transform it to

y(x) = 4x+ 5x2.

Here, then, are the general steps to set up a graph:

1. Setmode: Check the current graphing mode of the calculator byexamining the right half of the status line. If FUNC (for functionalgraphing) appears there, the mode is already properly set. If not, thenpress MODE and then the rightward cursor key to open up the graphingsubmenu. Press 1 to choose Function and then press ENTER to save themode choices and to return to the home screen.

2. Enter one or more functions: Press the green-diamond key and thenthe F1 key to get the Y= operation of the F1 key. The operation takesyou to the function-list screen and automatically puts y1= in the entryline. If anything is written to the right of the equal mark, pressCLEAR to remove it. Complete the entry line by filling in the function

you want to graph; it will be called function y1. Press ENTER to shiftthe function into the list of functions shown in the main part of thescreen. Repeat this process for y2 if you have a second function tograph.

3. Select the functions to be graphed: When you shift a function fromthe entry line to the list of functions, a check mark automaticallyappears to the left of the function line, to signal that the functionwill be graphed. If the list contains other functions that also havecheck marks, then those functions will also be graphed. If you want toprevent that result without deleting those other functions, move thecursor to one of those function lines and then press F4 to remove thecheck mark. (Note that the F4 option in the screen menu has a checkmark, to tell you the purpose of the F4 key.) F4 is a toggle switch,

putting in or taking out the check mark. Take out all the check marksthat you dont want.

If you do not want to keep a function in the list, move to thatfunction line, press F1 for Tools, and then press 7 for the Deleteoption. The function disappears.

If the calculator has been used to make statistical graphs, as forlinear regression as described below, you must turn off the statisticalgraphing. To do this, press F5 for All, move down to the Data PlotsOff, and then press ENTER.


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4. Set style: Next, move the cursor to the y1= line and then choose theF6 option for Style by pressing 2nd and then F1 (note the gold F6printed above the F1 key). The F6 option allows you to choose how lineswill be drawn on a graph. When you plot only one function, the styledoes not matter much, but when you plot more than one function, usingdifferent styles for them can help you discriminate them on the graph.This time, lets pick Dot by pressing 2.

5. Set range (or window): To set the range of xand yvalues to begraphed, press the green-diamond key and then F2, to get the WINDOWoperation of the F2 key. Then enter the minimum (xMin) and maximum(xMax) values of x, the tick mark spacing on the x axis (xScl), theminimum (yMin) and maximum (yMax) values for y, and the tick markspacing on the y axis (yScl). You may already know these values or beable to guess about them. If the graph does not come out as you want,you can always change these range values.

6. Set graph format: From either the Y= screen (step 2 above) or theWINDOW screen (step 5), press F1 to get the TOOLS menu, scroll or jumpdown the menu until you reach FORMAT and then press ENTER. Lets set

the lines to read

Coordinates RECT for rectangular coordinate systemGraph Order SEQ to draw multiple curves in sequenceGrid OFF to avoid messy grid linesAxes ON to get coordinate axesLeading cursor OFF to avoid a cursor during graphingLabels OFF to avoid the x and y labels

If, later, you want to change one of these features, move to thefeature line and press the rightward cursor key to open up a submenu.There press the line number for the option you want. Then press ENTERto leave the FORMAT menu.

7. Graph: You can now graph the function by pressing the green-diamondkey and then F3, to get the GRAPH operation of the F3 key. The busysignal on the status line disappears when the graphing is complete.

If the graph is not what you want, try changing the range values.To do that, return to the WINDOW menu by pressing the green-diamond keyand then the F2 key. Move to any line and edit it as you want. Then tograph the function with the new range values, again press the green-diamond key and then the F3 key.

Calculator Sample Problem: Graphing and graphical solutionsTwo particle-like rockets Aand B are shot along a horizontal yaxis.The position of rocket A, moving at a constant speed of 5.00 m/s and

located at y= 100 m at t = 0, is given by

yA= 5.00t + 100.

Rocket B, launched at t = 0, accelerates for about a minute, with itsposition given by

yB= t1.5 + 0.100t2.5 0.0100t3


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during the acceleration. In both functions, t is in seconds and yis inmeters.

(a) At what time and where on the yaxis does rocket B reach rocket A?

SOLUTION: We can answer the question by graphing both functions andthen finding the intersection of the two curves becauserocket B reaches rocket Awhen they have the same position at the sametime.

Graphing the Functions:

1. Setmode: Check the current graphing mode of the calculator byexamining the right half of the status line. If FUNC (for functionalgraphing) appears there, the mode is already properly set. If not, thenfirst press MODE and then the rightward cursor key to open up thegraphing submenu. Move to the top line, press ENTER twice to choosethat option and to return to the home screen.

2. Enter the functions: Press the green-diamond key and then the F1 keyto get the Y= operation of the F1 key. The operation takes you to the

function-list screen and automatically puts y1= in the entry line. Ifanything is written to the right of the equal sign, press CLEAR toeliminate it. In functional graphing, a function must be written in theform of yas a function of x. To distinguish functions, we usenumerical subscripts on the y. Thus, our two rocket equations must berewritten as

y1 = 5.00x+ 100.

and y2 = x1.5 + 0.100x2.5 0.0100x3.

Complete the entry line for y1= by pressing in 5x+100. Then press ENTERto move the function up to the list of functions. The entry line nowautomatically shows y2= for the next function. (Clear anything writtento the right of the equal mark.) Complete the line by pressing inx^1.5+.1x^2.5.01x^3 and then press ENTER to move the function up tothe list.

3. Select the functions to be graphed: When you shift an function tothe list, a check mark automatically appears to the left of thefunction line, to signal that the function will be graphed. If anyfunctions in the list besides y1= and y2= have check marks, you need toremove the check marks to prevent the functions from being graphed. Todo this, move the cursor to the line for such a function and press F4,which is a toggle switch for the check mark (putting it in and takingit out).

If the calculator has been used to make statistical graphs, as for

linear regression as described below, you must turn off the statisticalgraphing. To do this, press F5 for All, move down to the line for DataPlots Off, and then press ENTER.

4. Set style: Next, move the cursor to the y1= line in the list offunctions and then choose the F6 option for Style by pressing 2nd andthen F1 (note the gold F6 printed above the F1 key). The F6 optionallows you to choose how lines will be drawn on a graph. When you plotonly one function, the style does not matter much, but when you plotmore than one function, as for the two rockets, using different styles


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for them can help you discriminate them. For rocket A and equation y1,lets pick Dot by pressing 2. Next, move to the y2= line and repeat theprocedure but this time press 4 for Thick.

5. Set range (or window): To set the range of xand yvalues to begraphed, press the green-diamond button and then F2, to get the WINDOWoperation of the F2 key. Moving the cursor down line by line, press inthe following parameters:

0 for xMin35 for xMax5 for xScl (the tick-mark spacing on the x axis)0 for yMin280 for yMax100 for yScl (the tick-mark spacing on the y axis)

You can then return to the Y= screen by pressing the green-diamond keyand then F1.

6. Set graph format: From either the function-list screen (step 2

above) or the WINDOW screen (step 5), press F1 to get a TOOLS menu,scroll or jump down the menu until you reach FORMAT and then pressENTER. Lets set the lines to read

Coordinates RECT for rectangular coordinate systemGraph Order SIM to draw the two curves simultaneouslyGrid OFF to avoid messy grid linesAxes ON to get coordinate axesLeading cursor OFF to avoid a cursor during graphingLabels OFF to avoid the y and t labels

If you need to change the setting on a line, move to that line and thenpress the rightward cursor key to open up a submenu. Then move to theoption you want and press ENTER to make the change. When the settingsare made, press ENTER to leave the format screen.

7. Graph: You can now graph the two rocket functions by pressing thegreen-diamond key and then F3, to get the GRAPH operation of the F3key. The busy signal on the status line disappears when the graphing iscomplete. To stop the graphing procedure (if it is obviously notworking out) before it is complete, press ON.

Whether you wait until the graphing is complete or stop itprematurely, you can return to the function-list screen by pressing thegreen-diamond key and then F1, or return to the window screen bypressing the green-diamond key and then F2, or return to the homescreen by pressing HOME. If the graph did not turn out to be what youwant, rerun the graphing procedure after you change the range values in

the window screen.Finding the Intersection of Two Graphed Curves:

After the functions are graphed, you can find the coordinates atany point on either of the two curves by pressing F3 for Trace. Thecursor is then at some point on the first of the curves (for the firstfunction in the function list, y1= ). The cursor coordinates are listedas xc and yc. Use the right and leftward cursor keys to move along thecurve. Use the up or down cursor key to jump to the other curve, alongwhich you can then move with the leftward and rightward cursor keys.


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You can approximate the coordinates of the intersection of the twocurves by moving along either one to the intersection.

To find the intersection more exactly, use this procedure: Press F5for the Math menu, then press 5 for the Intersection option. The cursorthen sits on one of the curves. Press ENTER to choose that curve. Thecursor then sits on the other curve. Press ENTER to choose that curvealso. Then the calculator wants a lower and upper bound for x, tonarrow the region in which it will search for an intersection. Move thecursor anywhere to the left of the intersection and press ENTER tochoose that location as the lower bound of x. Now move anywhere to theright of the intersection. Press ENTER to choose that location as theupper bound of x. The calculator then searches between the two boundsand soon produces the coordinates of the intersection. Depending on themode you have already selected for numbers, the intersection occurs at

x = 20.944and y = 204.72.

Recall that x is really time t. Thus, we find that rocket B reachesrocket Aat

t = 20.9 sand y= 205 m. (Answer)

(b) Where is rocket B 3.0 s before it reaches rocket A?

SOLUTION: Evaluating a Point on a Graphed Curve. Press F5 for the Mathmenu and then press ENTER for the value option. The calculator asksEval x=? and puts a highlight on the current x coordinate of thecursor. We want to evaluate the curve for rocket B (the noticeablycurved y2 line) for time t = 20.9 s 3.0 s = 17.9 s. So press in 17.9for the requested x value and then press ENTER. The calculator firstevaluates y1 for that time (because y1 is the first in the list offunctions), marking the point on the graphed curve. Use the up or down

cursor key to go to the y2 curve. We find that for x = 17.9, y 154.Thus, at t = 17.9 s, rocket B is at

y154 m. (Answer)

(c) What is the speed of rocket B when it reaches rocket A?

SOLUTION: Finding a Slope on a Graphed Curve. To find the speed ofrocket B at any point on the graph, press F5 for the Math menu, press 6to open up the Derivatives submenu, and then press ENTER to choose thedy/dx option. The calculator then asks dy/dx at ? and highlights thecurrent x coordinate of the cursor, which is on the y1 curve. Because

we want a derivative of the y2 curve, press the up or down cursor keyto move to the y2 curve. Because we want the derivative at theintersection, which occurs at x = 20.944 m, press in 20.944 as theanswer to the question about the x coordinate. Then, when you pressENTER, the cursor moves to the intersection and shows that thederivative of the y2 curve there is dy/dx = 17.668. This slope is thespeed dy/dt of rocket Bat the intersection. Thus, we find

speed 17.7 m/s. (Answer)


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(d) Store the data concerning the graph so that the data can be redrawnlater.

SOLUTION: Storing Data for Graphs. If you go to the trouble of drawinga graph and anticipate that you will need to draw the same graph againor a similar one, store the associated functions and window settings.Then later you can quickly call up all those data and redraw the graphas before or with modifications to the functions or parameters.

For example, if you think you will need to draw a graph of

x= x0 + v0t + at2

later, draw the graph now with assumed values for the equationsvariables and for the window settings. Then store the data. If you doneed to graph that equation later, you can call up the data, modify thevalues of the variables and parameters, and then draw the graph; thisprocedure might be faster than starting fresh.

To store graph data, first produce the graph. Then press F1 for theTools menu and press 2 for the Save Copy as option. The cursor is

then on the Type line, which shows GDB (for graph database). Assumingthat the database is to be stored in the Main folder, move down to theVariable line, where you are to enter a name under which the graphdatabase will be store. You might press alpha twice to lock in thealphabetic mode and then press in the name rocket, by pressing R, O, C,K, E, and finally T. Press ENTER once to store the name and then onceagain to store the database under that name.

Later, if you need the stored functions and window settings, pressthe green-diamond key and then F1 for the Y= operation and then,instead of entering anything, press F1 for Tools and then press Enterfor the Open option. The OPEN screen shows GDB as the type of file youwant and main as the folder. (If the database is stored in anotherfolder, open up the Folder submenu by going to the folder line,pressing the rightward cursor key, and choosing the folder you want.)If you have saved only one database, its name is already on theVariable line. Press ENTER to retrieve the database. If its name is notalready on the Variable line, move to the line, press the rightwardcursor key, and then choose the database from the list of names.

The function lists (y1=, y2=,) that you stored is then displayed.Press the green-diamond key and then F2; the window settings that youstored are then displayed. Change anything you want in either of thosetwo screens and then press the green-diamond key and then F3 to graphthe functions.

To delete a graph database, press 2nd VAR-LINK and move down to thename of the database in the alphabetic listing (to jump down, hold downthe 2nd key and repeatedly press the down cursor key). Then press F1 forthe Manage menu and press ENTER for the Delete option. The calculator

asks if you really want to delete the program. Press ENTER once more tosay yes. The graph database is then deleted. Press HOME to return tothe home screen.


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Linear RegressionIf you take measurements of an independent variable (call it x) and itsassociated dependent variable (call it y), you can fit a function tothe data by using the statistical capability of the calculator. Linearregression is a procedure in which the linear function y= ax+ b isfitted to the data.

Caution: Some books, instructors, and calculators write thegeneric linear function as y= a + bxbut the style on a TI-89is y= ax+ b.

The following sample problem gives a simple example of linearregression.

Calculator Sample Problem: Linear RegressionSuppose that in an experiment, when we set quantity xto be 1.00, 2.00,3.00, 4.00, and 5.00 units, we measure quantity yto be 11.5, 16.1,17.5, 20.0, and 25.3 units, respectively. Suppose also that we assumethat quantities yand xare related by the linear equation

y= ax+ b, (1)where a and b are constants.

(a) With the calculator mode set for numbers in normal notation withtwo decimal places, use linear regression to find the valuesof a and b based on the measurements.

SOLUTION: To check the calculator mode settings, press MODE. If Fix 2is not shown on the Display Digits line, move to that line, press therightward cursor key to open the submenu, and then press 3 for the Fix2 option. If NORMAL is not shown on the Exponential Format line, moveto that line, press the rightward cursor key to open the submenu, andthen press 2 for the NORMAL option. Then press ENTER to save thesechoices and to return to the home screen.

To use the linear regression capability of the calculator, we firstprepare lists of the xand ydata.

Making the lists:

A list begins with the symbol { and ends with the symbol }. Thosesymbols are printed above the parentheses keys. A comma is used toseparate items in a list. Lets enter thexvalues first. Press CLEARto clear the entry line (if it is not already cleared), and then press2nd {. That is, press the 2nd key and then the key with } printed aboveit. Next press in

1 , 2 , 3, 4, 5

and then close the list by pressing 2nd } so that

{1,2,3,4,5}

is in the entry line. (Pressing in the trailing decimal places of 0 in

the data would make no difference.) Next, press the store key STO,then the alpha key, then the X key, and finally ENTER to store the listunder the name x. (The alpha key gives lowercase letters.) Now repeatthe procedure for the ydata, except store the list under the name y.


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Linear Regression:

Press APPS and then press 6 to jump to the line for the Data/MatrixEditor and to open its submenu. Press 3 for NEW (that is, a newcalculation). The top line of the NEW screen has a submenu of Data,Matrix, and List (use the rightward cursor key to open the submenu).Stay with Data by either not opening the submenu or by pressing ENTERwhen the cursor on Data in the submenu.

The second line indicates the folder in which the data will bestored; stay with main. Move to the third line and enter a name underwhich the results of the calculations will be stored. For example,press the alpha key twice to lock in the lowercase alphabetic mode,then press in the name frodo (press F, R, O, D, and then O), and thenpress ENTER twice to save the name and to go to the next screen.

The next screen shows columns c1 and c2 in which we could have putthe experimental data; however, we have already used lists. Press F5for Calc. In the Calculate menu (which has the name frodo printed attop), the top line asks for the Calculation Type. Press the rightwardcursor key to open up the submenu. Press 5 to choose LinReg (linearregression). (Caution: we do not want LnReg.) Then move down to the xline to enter the name of the first list of data, which we previously

chose to be x. To put in the name, press the alpha key and then the Xkey.

Next, move to the y line to enter the name of the second list ofdata, which we previously chose to be y. Press the alpha key and thenthe Y key. Now move to the Store RegEQ line, which allows you to storethe regression equation under a name. (We need this storage to plot theresults of the linear regression, but if you dont want such a plot,skip this step.) To store the equation, open the submenu on the Storeline, move to y1(x) and then press ENTER to store the regressionequation (Eq. 1) under the name y1.

Press ENTER again to start the regression calculation; almostimmediately the results pop up on the STAT VARS screen. We find that tofit y= ax+ b to the data,

a = 3.15 and b = 8.63,

and thus

y= 3.15x+ 8.63.

(b) Plot the experimental data and include the result of the regression(the regression line).

SOLUTION: After you see the results on the STAT VARS screen, pressENTER and then press F2 for Plot Setup, where we decide how theexperimental data will be graphed. The plot 1 line has a check mark at

the left and the x and y names of the data lists at the right. Thecheck mark indicates that this plot will be made. (F4 turns the checkmark on and off.) Press F1 to make choices about the graphing.

We want the experimental data to be graphed as scatter dots, ratherthan connected dots. If Scatter does not appear on the Plot Type line,open the submenu and then press ENTER to choose that type. Move to theMark line to choose how the data dots are to appear. Open the submenuthere and press the number for your choice of style: 1 gives an openbox at each plotted value, 2 gives a cross, 3 gives a plus sign, 4gives filled-in squares, and 5 gives small dots (which can be hard to


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see on a graph). Once your choice is made, examine the next two lines;they should show that the x data and y data are in the main calculatorfolder.

Next press the green-diamond key and then Y= (printed above the F1key). The linear regression line appears in the y1= line. (The checkmark to the left of the equation means that this equation will begraphed when we eventually start the graphing; there is also a checkmark to the left of Plot 1, as we previously saw.) To choose the stylein which the regression equation will be graphed, choose F6 by pressing2nd and then F1 (note the gold F6 printed just above F1). For now,choose the line style by pressing ENTER.

We next must set the range over which the scatter dots and theregression line will be plotted. Press the green-diamond key and thenF2 to get WINDOW. Based on the experimental data given in the problemstatement, here are some reasonable values to put in for the minimumand maximum xand yvalues and the tick-mark spacing (xscl and yscl) onthe axes:

xmin =0xmax=6

xscl=2ymin=0ymax=35yscl=10

Now graph the scattered values and the regression line by pressingthe green-diamond key and then F3 for GRAPH. If you want to see thegraphing once more, press F4 for ReGraph. To return to the home screen,press 2nd Quit.

Trigonometric FunctionsThe trig functions sine, cosine, and tangent are available as

2nd

operations on the keyboard. They can operate on an angle expressedin either degrees or radians, as set by you in the mode menu. Yourchoice is always on display as DEG or RAD in the status line. To changethe setting, press MODE and then move the cursor to the Angle line,open up the submenu by pressing the rightward cursor key, and then moveto the measure that you desire, press ENTER to make your choice, andthen press ENTER again to save the choice and to return to the previousscreen.

If the mode is set for degrees and for answers in scientificnotation with two decimal places, then pressing in

2nd SIN 30

and then pressing ENTER yields 5.00E-1, which is equivalent to 0.5 (the

sine of 30is 0.5).Caution: A common error in exams is to assume that the

calculator is in, say, degree mode when it is actually in radian modethe calculations are then quite wrong and the error can be difficult todetect. For example, sin 30 is -9.88E-1, not 5.00E-1, when thecalculator is in radian mode.

If you do notice that the angle mode is wrong after you evaluate anexpression, you usually do not have to re-enter the expression. Firstchange the mode to what you want and return to the home screen. If the


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evaluation at fault is the last one performed, the expression is stillin the entry line, so just press ENTER to re-evaluate with the newangle measure. If the evaluation at fault is not the last oneperformed, use the up cursor key to reach the expression in the historyarea, press ENTER to duplicate the expression in the entry line, andthen press ENTER again to re-evaluate the expression with the new anglemeasure.

The inverse trig functions sin-1, cos-1, and tan-1 are available onthe keyboard as green-diamond operations. Here too you must be carefulabout the mode setting for the angle. For angles in degrees, we canevaluate sin-1 0.5 by pressing in

green-diamond sin-1 .5

and then pressing ENTER. With the calculator set for answers inscientific notation with two decimal places, it then shows 3.00E1,

which means 30.

Vector NotationsFundamentals of Physicsuses two styles of vector notation, as shown inthe following table:

Magnitude-angle notation:Textbook style:

4 units at 30Calculator style:

entered as [4,30] in entry lineappears as [4 30] in history area

Unit-vector notation:Textbook style:

2i 4j + 6kCalculator style:

entered as [2,-4,6] in entry lineappears as [2 -4 6] in history area

The textbook style includes hats on the symbols i, j, and k; the hatsare not used here in these notes. The textbook also uses an overheadarrow for vector symbols; here a bold symbol is used. The bracketsneeded for the calculator styles are the second operations of the comma

and divide keys. The angle symbol is the second operation of the EEkey.

Caution: Using a calculator to process vectors can save muchtime and avoid careless errors. However, it can also allow you to blowoff understanding what you are doing. For example, even if you canprocess vectors quickly on a calculator, you may still fail a exam that

involves vectors. Because vectors are used in most of Fundamentals ofPhysics, understanding what you are doing with them is essential.

Vectors in Magnitude-Angle NotationThe angle of a vector can be entered in either degrees or radians, asset in the mode menu and displayed in the status line. To change theangle measure, go to the MODE menu as explained above. For the examplesthat follow, choose degrees and also set the mode for answers inscientific notation with 2 decimal places.


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To enter a two-dimensional vectorAthat has a magnitude of 4 m and

an angle of 30(measured counterclockwise from the positive directionof an xaxis on an xycoordinate system), press in

2nd [ 4 , 2nd30 2nd ]

to put [4, 30] in the entry line. (Be sure to press in the comma.) Ifyou now press ENTER, the calculator will display vectorAin thehistory area in either magnitude-angle notation or unit-vectornotation, as set in the mode menu.

Because we entered the vector in magnitude-angle notation, letsset the mode menu to display the vector in that same style. Press MODE.If the listing on the Vector Format line is Spherical, the calculatoris already set to show vectors in magnitude-angle notation. So, pressENTER to return to the home screen. If either Rectangular (which givesunit-vector style) or Cylindrical (which we do not use) is listed, movedown to the Vector Format line, press the rightward cursor key to openup a submenu, and then press 3 to choose SPHERICAL. Then press ENTER tosave the choice and to return to the home screen. Again pressing ENTERmakes the calculator show

[4.00E0 3.00E1]

in the history area (remember that we set the mode menu for scientificnotation and two decimal places).

A negative measure for the angle of a two-dimensional vector meansthat the angle is measured clockwise from the positive direction ofthe xaxis. For example, a vector with a magnitude of 4 units and

directed 40clockwise from the positive direction of the xaxis can beentered as either [4,-40] or [4,320] because a full circle contains360.

Caution: When you enter a value for an angle in magnitude-angle notation, you should first check whether the calculator is setfor degrees or radians by noting which is displayed in the status line.If you enter data in, say, degrees when the calculator is in radianmode, the data will be interpreted as being in radians and anycalculations made with the data will be quite wrong. This error is verycommon, especially during exams.

Vectors in Unit-vector NotationTo enter a two-dimensional vector B with components of -4 m onthe xaxis and 2 m on the yaxis of an xycoordinate system, press

2nd [ (-) 4 , 2 2nd ]

to put [-4, 2] in the entry line. (Remember, the symbol (-) representsthe negation key.) If you now press ENTER, the calculator willdisplay B in the history area in either magnitude-angle notation orunit-vector notation, as set in the mode menu.

Because we entered the vector in unit-vector notation, lets setthe mode menu to display the vector in that same style. Press MODE,then move the cursor to the Vector Format line, press the rightwardcursor key to open a submenu, move to Rectangular, press ENTER to makethat choice of measure, and then press ENTER again to return to the


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preceding screen. Pressing ENTER once more causes the calculator todisplay the vector in unit-vector notation. We see

[-4.00E0 2.00E0]

in the history area, which means -4i + 2j.To enter a three dimensional vector, with components -4 m on

the xaxis, 2 m on the yaxis, and 3 on the z axis, press in

2nd [ (-) 4 , 2 , 3 2nd ]

to put [-4,2,3] in the entry line. If the mode is set for scientificnotation with two decimal places, pressing ENTER puts

[-4.00E0 2.00E0 3.00E0]

in the history area.Note the general rule: You can enter a vector in either

magnitude-angle notation or unit-vector notation regardless of thecalculator setting. That setting determines only the way a vector

answer is displayed.Caution: Vectors cannot have a single dimension. Thus, if a

vector has, say, only an xcomponent of 2 m, you cannot enter it as[2]. Instead you must enter it as

[2,0] or [2,0,0].

Switching between Magnitude-angle Notation and Unit-vectorNotationSuppose that you have pressed in a vector in the entry line in one ofthe two types of vector notation and now you want to switch the vectorto the other type. For example, you might be given a vector in unit-

vector notation and then requested to find the magnitude and angle ofthe vector. Or you might be given a vector in magnitude-angle notationand then requested to find the vector in unit-vector notation, to findthe components of the vector. Here are two ways to make the switch innotation:

Quick way, if you dont mind switching the calculator mode ifnecessary:

Press MODE to check the current vector format setting. If the VectorFormat line already shows the requested format, press ENTER to returnto the previous screen, and then press ENTER again to display thevector in the history area, in the requested notation. If the VectorFormat line does not already show the requested format, move down tothat line with the down cursor key, press the rightward cursor key toopen up the submenu, move to the requested vector format, and thenpress ENTER twice to return to the previous screen. Then press ENTERonce more to display the vector in the history area in the requestedformat.

Slower way, which does not switch the calculator mode:Press 2nd MATH and then press 4 to open up the Matrix submenu. TheVector ops (for vector options) is at the bottom of the long menuthat extends down and off the screen.


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1. To move slowly to that option, hold down the down cursor key. Thenopen the submenu by pressing the rightward cursor key.2. To jump quickly to that option, hold down the 2nd key and press thedown cursor key twice. Then open the submenu.3. To jump immediately to that option and also automatically open the

submenu, press the up-arrow and then L.

In the submenu, press 5 for the transform Rect if you want to switchthe vector to unit-vector notation. Or press 7 for the transformSphere if you want to switch the vector to magnitude-angle notation.Either way, the transform appears in the entry line. Then press ENTERto make the transformation, which appears in the history area.

Caution: From home screen, return to the Vector ops submenu.The first option in that submenu is called unitV (for unit vectors).That option produces a vector in a type of unit-vector notation, butnot the type used in Fundamentals of Physics. (The vector componentswill be divided by the magnitude of the vector.)

Angles for Three-dimensional VectorsA three dimensional vector can be expressed in magnitude-angle notationin which

[magnitude, , ]

is the general form. In a right-handed xyz coordinate system, angle ismeasured, as usual, in the xyplane, from the positive direction of

the xaxis (is positive if measured counterclockwise and negative ifmeasured clockwise). Angle is measured from the z axis; it ranges from0(in the positive direction of z) to 180(in the negative directionof z) and can never be negative. Thus, with the calculator set fordegrees, a vector entered as

[2, 45,90]

has a magnitude of 2 units and is directed in the xyplane, at

45counterclockwise from the positive direction of the xaxis.The vector is probably easier to picture mentally if we switch it

to unit-vector notation. Using one of switching procedures discussedabove, with the calculator set for scientific notation with two decimalplaces, we find

[1.41E0 1.41E0 0.00E0]

displayed in the history area. This result means that the vector is

1.41i + 1.41j + 0k.

Adding VectorsYou can add two or more vectors on screen provided that they all havethe same number of dimensions.


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These combinations of two vectors will work:Two-dimensions:[2,0] + [4,5]

[5,40] + [4,5][5,40] + [2,30][2,3] [4,5]

[2,3] [2,30]

Three-dimensions:[2,3,5] + [5,7,8]

[2,3,5] + [2,30,40][4,55,76] + [5,45,21][4,6,-7] [8,0,9][5,21,60] [8,0,9]

These combinations[2,3,6] + [4,5]

[3,30] + [3,1,2]will not work, but we can fix the trouble by rewriting them as

[2,3,6] + [4,5,0][3,30,90] + [3,1,2]

In the first will not work example, we fixed the trouble byexplicitly showing that the third component of the second vector iszero. In the second example, the first vector is directed inthe xyplane; so we fixed the trouble by explicitly showing that its

angle relative to the z axis is 90.

Storing a VectorYou can store a vector under a name, so that you might use it later.

For example, enter [3,4] and then press the STOkey, then the alphakey, and then the key with A printed above it. The vector 3i + 4j is

then stored under the name a (lowercase). You can then set up anaddition of that vector to another one, say, [4,5], by pressing in

alpha a + 2nd [ 4 , 5 2nd ]

so that the entry line shows a+[4,5]. To complete the addition, pressENTER; the answer appears in the history area.

Overriding the Mode Setting for AnglesYou can enter angles in degrees, radians, or both degrees and radiansregardless of the mode setting by explicitly indicating degrees with

the degree symbol and radians with the elevated-r symbol r. The degreesymbol is a 2nd operation on a key about midway along the left side of

the keyboard. To get the radian symbol, press 2nd MATH and then 2 toopen up the Angle submenu. Then press 2 again to paste the radiansymbol in that submenu into the entry line. For example, to make theaddition

cos(0.7854 rad) + cos 60,

press in


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2nd cos 0.7854

and then paste in the radian symbol using the above procedure, and thenpress in

+ 2nd cos 60 2nd

so that

cos 0.7854r + cos 60

appears in the entry line. Then pressing ENTER to evaluate the additionyields 1.21 as the answer in the history area, regardless of the modesetting for angles.

Calculator Sample Problem: Entering and Displaying VectorsIn a memory experiment, a trained mouse runs through a horizontal mazeon which the experimenter has superimposed an xycoordinate system. The

mouse takes three straight-line runs: 0.25 m at 36

, 0.38 m at 120

,and 0.15 m at 210, with each angle measured counterclockwise from thepositive direction of the xaxis. At the end of the three runs, what isthe net displacement of the mouse from the starting point? Express theanswer in both magnitude-angle notation and unit-vector notation.

SOLUTION: We first check the mode settings. Although we can tell if thecalculator is using the angle measure of degrees by the display on thestatus line, we need to go to the mode menu anyway, so press MODE.There the angle measure is indicated on the Angle line. If the lineindicated radians as the measure, go to that line, press the rightwardcursor key to open the submenu, move to DEGREE, and then press ENTER tochoose that angle measure.

To get the answer in magnitude-angle notation, we want the

calculator to be in spherical coordinates. If Spherical is not listedon the Vector Format line, move to that line, press the rightwardcursor key to open the submenu, move to SPHERICAL, and then press ENTERto choose that vector format.

Lets also set the mode for scientific notation with two decimalplaces. If Scientific is not listed on the Exponential Format line orFIX 2 on the Display Digits line, move to those lines, open thesubmenus, and make the choices. Finally, press ENTER to save all thechoices and to return to the home screen.

We can represent a vector in the generic form of

[magnitude, angle in degrees or radians]

where the brackets indicates a vector. We get the brackets

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