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HP-28S Advanced Scientific Calculator ._---------.-------------

Reference Manual

rli~ HEWLETT ~~ PACKARD

Edition 2 April 1988 Reorder Number 00028-90068

Notice This manual and any keystroke programs contained herein are provided Has isH and are subject to change without notice. HewlettPackard Company makes no warranty of any kind with regard to this manual or the keystroke programs contained herein, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. Hewlett-Packard Co. shall not

be liable for any errors or for incidental or consequential damages in connection with the furnishing, performance, or use of this manual or the keystroke programs contained herein . Hewlett-Packard Co. 1987. All rights reserved. Reproduction, adaptation, or translation of this manual, including any programs, is prohibited without prior written permission of Hewlett-Packard Company, except as allowed under the copyright laws. Hewlett-Packard Company grants you the right to use any program contained in this manual in this Hewlett-Packard calculator.

©

The programs that control your calculator are copyrighted and all rights are reserved. Reproduction, adaptation, or translation of those programs without prior written permission of Hewlett-Packard Co. is also prohibited. Portable Computer Division 1000 H.E. Circle Blvd. Corvallis, OR 97330, U.S.A.

Printing History Edition 1

October 1987

Mfg. No. 00028-90069

Welcome to the HP-28S Congratulations! With the HP-28S you can easily solve complicated problems, including problems you couldn't solve on a calculator before . The HP-28S combines powerful numerical computation with a new dimension-symbolic computation. You can formulate a problem symbolically, find a symbolic solution that shows the global behavior of the problem, and obtain numerical results from the symbolic solution. The HP-28S offers the following features: • Algebraic manipulation. You can expand, collect, or rearrange terms in an expression, and you can symbolically solve an equation for a variable. • Calculus . You can calculate derivatives, indefinite integrals, and definite integrals. • Numerical solutions. Using HP Solve on the HP-28S, you can solve an expression or equation for any variable. You can also solve a system of linear equations. With multiple data types, you can use complex numbers, vectors, and matrices as easily as real numbers . • Plotting. You can plot expressions, equations, and statistical data. • Unit conversion. You can convert between any equivalent combinations of the 120 built-in units. You can also define your own units. • Statistics. You can calculate single-sample statistics, paired-sample statistics, and probabilities. • Binary number bases. You can calculate with binary, octal, and hexadecimal numbers and perform bit manipulations. • Direct entry for algebraic formulas, plus RPN logic for interactive calculations.

Welcome to the HP-28S

3

The HP-28S Owner's Manual contains three parts. Part 1, "Fundamentals," demonstrates how to work some simple problems. Part 2, "Summary of Calculator Features," builds on part 1 to help you apply those examples to your own problems. Part 3, "Programming," describes programming features and demonstrates them in a series of programming examples. The HP-28S Reference Manual (this manual) gives detailed information about commands. It is a dictionary of menus, describing the concepts and commands for each menu. We recommend that you first work through the examples in part 1 of the Owner's Manual to get comfortable with the calculator, and then look at part 2 to gain a broader understanding of the calculator's operation. When you want to know more about a particular command, look it up in the Reference Manual. When you want you learn about programming, read part 3 of the Owner's Manual. These manuals show you how to use the HP-28S to do math, but they don't teach math. We assume that you 're already familiar with the relevant mathematical principles. For example, to use the calculus features of the HP-28S effectively, you should know elementary calculus. On the other hand, you don't need to understand all the math topics in the HP-28S to use those parts of interest to you. For example, you don't need to understand calculus to use the statistical capabilities.

4

Welcome to the HP-28S

Contents 10 11 11

How To Use This Manual How This Manual is Organized How To Read Stack Diagrams

15

Dictionary

16 16 21

ALGEBRA (Algebraic manipulations) Algebraic Objects Functions of Symbolic Arguments Evaluation of Algebraic Objects Symbolic Constants: e, 1f, i, MAXR, and MINR COLCT EXPAN SIZE FORM OBSUB EXSUB TAYLR ISOL QUAD SHOW OBGET EXGET

25 27

28 33 34

ALGEBRA (FORM)

36 47

FORM Operations FORM Operations Listed by Function

53

Arithmetic

63

ARRAY (Vector and matrix commands) Keyboard Functions purr -ARRY ARRY- PUT GET SIZE RDM TRN ION CON CROSS DOT DET ABS RNRM R-C C-R RE 1M CON]

65 70 75 79

82

GETI RSD CNRM NEG

Contents

5

85 87 89 91 92

6

BINARY (Base conversions, bit manipulations) DEC HEX OCT BIN STWS RCWS RL RR RLB RRB R... B B... R SL SR SLB SRB ASR AND OR XOR NOT

96 96 100 106

Calculus Differentiation Integration Taylor Series

110 111 114 116

COMPLEX (Complex numbers) R-C C-R RE 1M CON] SIGN R-P P-R ABS NEG ARG Principal Branches and General Solutions

124

Evaluation

127 128 132

LIST -LIST POS

133 133 136

LOGS (Logarithmic, exponential, and hyperbolic functions) LOG ALOG LN EXP LNPI EXPM SINH ASINH COSH ACOSH TANH ATANH

139 141 144

MEMORY MEM MENU ORDER PATH VARS CLUSR

145 145 150

MODE (Display, angle, recovery, and radix modes) STD FIX SCI ENG DEG RAD CMD UNDO LAST ML RDX, PRMD

152 152 153 155 155 156 157 160 163 165

PLOT The Display Mathematical Function Plots Statistical Scatter Plots Interactive Plots Plot Parameters STEQ RCEQ PMIN PMAX INDEP DRAW PPAR RES AXES CENTR *W *H

Contents

STO~

LIST... SUB

RCL~

PUT SIZE

COL~

CLLCD DGTIZ PIXEL

GET

PUTI

HOME

SCL~

DRW~

DRAX

CLMF

GETI

CRDIR

PRLCD

168 168 169 169 170 171 174

PRINT Print Formats Faster Printing Double-Space Printing Configuring the Printer PRI PRST PRVAR PRLCD CR PRSTC PRUSR PRMD

176 176 177 178 181

Programs Evaluating Program Objects Simple and Complex Programs. Local Variables and Names User-Defined Functions

183 184 185 186 188 192

PROGRAM BRANCH (Program branch structures) Tests and Flags Replacing GOTO IFERR THEN ELSE END IF START FOR NEXT STEP 1FT IFTE WHILE REPEAT END DO UNTIL END

193 193 195 198

PROGRAM CONTROL (Program control, halt, and single-step operations) Suspended Prqgrams WAIT KEY SST HALT ABORT KILL ClMF ERRN ERRM BEEP ClLCD DlSP

201 201 204 206 211

PROGRAM TEST (Flags, logical tests) Keyboard Functions FS? FC? FS?C SF CF XOR NOT SAME AND OR STOF RClF TYPE

213 214 215 218 219 221

REAL (Real numbers) Keyboard Functions FACT RAND NEG ABS SIGN MANT IP FP FLOOR MIN MOD MAX

RDZ XPON CEll 'l'oT

MAXR

TRAC

FC?C

MINR

RND

Contents

7

236

SOLVE (Numerical and symbolic solutions) Interactive Numerical Solving: The Solver (STEQ, RCEQ, SOLVR, ROOT) Symbolic Solutions (ISOL, QUAD, SHOW) General Solutions

239 239 241 243

STACK (Stack manipulation) Keyboard Commands DUP OVER DUP2 DROP2 ROT LISTROLLD PICK DUPN DROPN DEPTH -LIST

245 246 249 251 254

STAT (Statistics and probability) ~+

~-

TOT

MEAN SDEV CORR COY UTPF UTPN

258 258 262

STORE (Storage arithmetic) STO+ STO- STO* STO/ SCON}

263 264 264 270 273

273 277 280

STRING (Character strings) Keyboard Function -STR STR- CHR NUM -LCD LCDPOS SUB SIZE DISP TRIG (Trigonometry, rectangular/polar and degrees/radians conversion, Hour/Minute/Second arithmetic) SIN ASIN COS ACOS TAN ATAN P-R R-P R-C C-R ARG - HMS HMS- HMS + HMS - D- R R-D

283 285 286 287 295 295

UNITS Temperature Conversions Dimensionless Units of Angle The UNITS Catalog User-Defined Units Unit Prefixes

224 225 234

8

Contents

COL~

UTPC

N~

CL~

STO~

RCL~

VAR LR UTPT

MAX~

MIN~

PREDV COMB PERM SNEG

SINV

III

III C'CI

CJI

C")

0

q)

III

GI

ii:

..

GI

C'CI

III III ~

GI CJI

:E CD

G) ('I

>-

.

C'CI

III III

.S!

a

0

C")

...

Ii I~ 1 '-

8

I~ 1

C") ('I C")

I

II

I

I I II

I I !

I

I~

1 >
': . \') ::; I Gt·l': :'':' ::::TO), Y to have the value 4, and Z to have the value ':'< + T '. We will also assume that Symbolic Result mode (flag 36) is set, so that functions will accept symbolic arguments.

Dictionary

25

... ALGEBRA First consider the expression '::< + ....' '. When we evaluate this expression (' >:: + ...." E',.!AL), we obtain the result 7. Here's why: Internally, , >:: +'l' is represented as X Y +. So when ':": + . . " is evaluated, X, Y, and + are evaluated in sequence: 1. Since ::< is a name, evaluating it is equivalent to evaluating the object stored in the variable :":, the number 3. Evaluating X puts 3 in level 1.

2. Similarly, evaluating level 2.

'y'

puts 4 in levell, pushing the 3 into

3. Now + is evaluated, with the numeric arguments 3 and 4 on the stack. This drops the 3 and the 4, and returns the numeric result 7. Now try evaluating '>( + T ' : 1. Evaluating >( puts 3 in level 1.

2. T is a name not associated with a variable, so it just returns itself to level 1, pushing the 3 into level 2. 3. This time + has 3 and T as arguments; since T is symbolic, + returns an algebraic result, '3 + T ' . Finally, consider evaluating '::< +. . ' +2 '. Internally, this expression is represented as X Y + Z +. Following the same logic as in the above examples, evaluation gives the result '7 + ( ~-( + T::O '. We can evaluate this result again and obtain the new result '7 + ( 3 + T::O ' . Further evaluation makes no additional changes, since T has no value. The values 7 and 3 obtained are not arguments to the same + operator in the expression, and hence are not combined. If you want to combine the 7 and the 3, you can use either the COLCT command for automatic collection of terms, or the FORM command for more general rearrangement of the expression.

26

Dictionary

... ALGEBRA Symbolic Constants: e, MINR

IT,

i, MAXR, and

There are five built-in algebraic objects that return a numerical representation of certain constants. These objects have the special property that their evaluation is controlled by Constants mode (flag 35) as well as by the Results mode (flag 36) .

• If flag 35 or flag 36 is clear, these objects will evaluate to their numeric values. For example: I

*

2 i "E\} AL returns

«(1.,

2).

• If flag 35 and flag 36 are both set, these objects will retain their symbolic form when evaluated. For example: I

2 *i

I

E VAL returns

I

2 *i

I •

The following table lists the five objects and their numerical values. Hp·28S Symbolic Constants Object Name

Numerical Value

e

2.71828182846

7r

3.14159265359

i

(0.00000000000,1 .00000000000)

MAXR

9.99999999999E499

MINR

1.00000000000E-499

Dictionary

27

· .. ALGEBRA The numerical values of e and 11 are the closest approximations of the constants t and Tr that can be expressed with 12-digit accuracy. The numerical value of i is the exact representation of the constant i. t-1A:'

':::0' rather than the expression 'e . . . ::< '. The function EXP uses a special algorithm to compute the exponential to greater accuracy. When the angle mode is radians and flags 35 and 36 are set, trigonometric functions of 11 and IL··· 2 are automatically simplified. For example, evaluating '::; I t·~ 0:: 11::0' gives a result of O.

COLCT

EXPAN

FORM

SIZE

OBSUB

EXSUB

These commands alter the form of algebraic expressions, much as you might if you were dealing with the expressions "on paper". COLCT, EXPAN, and FORM are identity operations, that is, they change the form of an expression without changing its value. OBSUB and EXSUB allow you to alter the value of an expression by substituting new ob jects or subexpressions into the expression.

COLCT

Command

Collect Terms Level 1 , symb 1 '

Level 1 •

'

symb 2

'

COLCT rewrites an algebraic object so that it is simplified by "collecting" I ike terms. Specifically, COLCT: • Evaluates numerical subexpressions. For example: , 1 +2 +LOG 0:: 10::0' is replaced by 4.

28

Dictionary

... ALGEBRA • Collects numerical terms. For example: '1 +>< +2' is replaced by 13+::-:;

I.

• Orders factors (arguments of *t and combines like factors. For example: ':::: ..... 0:: T +:::0 :t:'·,..·····2 ' . • Orders summands (arguments of +), and combines like terms differing only in a numeric coefficient. For example: , ::< +>:: +\' +3:n:: +\' , . COLCT operates separately on the two sides of an equation, so that like terms on opposite sides of the equation are not combined. The ordering (that is, whether X precedes Y) algorithm used by COLCT was chosen for speed of execution rather than conforming to any obvious or standard forms. If the precise ordering of terms in a resulting expression is not what you desire, you can use FORM to rearrange the order.

EXPAN

Expand Products Level 1

Command

Level 1

EXPAN rewrites an algebraic object by expanding products and powers. More specifically, EXPAN: • Distributes multiplication and division over addition. For example: , Al 0:: E: +C::O' expands to 'AlE: +AlC ' ; '0:: E: +C::O . ·. A' expands to , E:./ A +C./ A ' .

. • Expands powers over sums. For example: 'A ..... 0:: E: +C::O' expands to , A·····E:lA····· C ' .

• Expands positive integer powers. For example: '>::···· 5' expands to , >::P(A ) "B)

EXP(AlB)

(D( P (A)

EXP(A ..··· B )

A

HIV ( B) )

Adding Fractions AF

Combine over a common denominator.

Before

( ( ( A l C ) +B ) ". C )

(A+(B / C » ( (A ./B)

+C)

«

+(

A./B)

After

C/O ) )

«A+(BlC) ) / B) «

(AlO)+(BlC»".(BlO »

(A-(B ..·' C»

«(AlC)-B) " C)

« A/B)-C)

« A- (BlC»

«A/B)-(C/O»

«(AlO)-(BlC»".(BlO»

46

Dictionary

". B)

... ALGEBRA (FORM) If the denominator is already common between two fractions, use M+

FORM Operations Listed by Function The following tables show which operations will appear in the FORM menu when a given function is the selected object. The form of the original subexpression and the result is shown for each operation. The operations .ectl1:c*'" '~, ii,Q~~ ;b , , and are available for all functions and variables. These common operations don't appear in the tables. If only the common operations are availc;l.ble for a function, no table appears for that function. (Only the common operations are available for V and SQ; to use other operations, substitute A.S and A2.)

Addition (+)

Operation

After

Before (A"B) (-(A);,t.B)

(B1tA) (B",\A)

~A

( A+( B +C ) ) (A+CB-C»

( ( A+B ) +C) «A+B)-C)

A+

C( A+B ) .. C ) (CA-B)+C)

CAof.< B +C) )

~M

CCA:tB)+CA:tC» CLNCA)+LN(B» CLOGCA)+LOGCB»

(AlItCB+c» LH(A:tB) LOC(A:tB)

1'1+

(CA:tC)+(B:tC) «A/C)+(B/C»

«A+B)*C) «A+B)/C)

(A+B) -(A)+B

·(-(A)-B)

(A+(B/C» «A/8)+(C/O» ( (A/8) +C)

«(A:tC)+B)"C) «CA:tO)+(B:tC»/C8:tO»

-C) AF

(A~(B-C»

~(A-B)

Dictionary

47

... ALGEBRA (FORM) Subtraction (-)

Before

Operation

After

++

(A-B)

(- (B) +A)

+A

(A- (B +C) ) (A-(B-c»

«A-B)-c) ( (A-B) +C)

A+

«A+B)-C) «A-B)-C)

(A+(B-C» (A"CB+C) )

+1'1

«AtB)-(AtC» (LtH A) -UH B» (LOG(A)-LOGCB»

(A*(B-c» LHCA/B) LOGe A/B)

"';.

«AtC)-CBtC) ) «A/C)-(B/C) )

«A-BHc) «A-B)/c)

(A-B) (-(A)-B)

- (- (A) +B) .. (A +B)

(A-(B/C» A/B)-C) «A/B)-(C . ···[J»

«CAtc)-B)/c) C CA-CBtC) hB) «CA*[J)-(BtC»/(B*[J»

-0

AF

«

Multiplication (*)

Operation

48

Before

After

+;.

(A*B) (INV(A)*B)

(B*A) (B/A)

+A

(A*(B*C» CA*(B/C»

( (AtB)*c) «A*B)/C)

«A*B)*C) ( (A/BH:c)

(A*(B*c) (A/(B/C»

+0

( (A+B)*C) (CA-B)*C)

«AtC)+(B*C» ( (AtC) (B*C»

0+

(A*(B+C) ) (A:t CB - C »

«A*B)f(A*C» (CAtB)-(AtC»

Dictionary

... ALGEBRA (FORM) (Continued)

Operation

Before

After

E-M

« A""'B H ( A"'C) ) (ALOG(A)*ALOGCB» (E>

72

r-t?p'"cCllble obJ > ji) HATAjlc; .P~2ow __ C.Oi.. Q( }

l..tef'lo.c.-dlec:.lJec.-t

>,_

Dictionary III)

Re.. t

c) obj, that repIQc.e~ *I1e • ec.· (i -d

{ '

I'\Vt1I. (f.:;.; 1.84467440737E19, the result is # FFFFFFFFFFFFFFFF (hex).

R-B converts a real integer

B-+R

Binary to Rea. Level 1 1* n

Level 1 It

Command

---------1

n

B- R converts a binary integer # 11 to its real number equivalent n. If > # 1000000000000 (decimal), only the 12 most significant deci-

# 11

mal digits are preserved in the mantissa of the result. Avc~Ao6Ui5 6(;.1 mode

R-B. B·~R.

EN";

(t>,B,o.H)Eil--'a(,(~,

tJ€TQt"fonl'1 o .. c;c...,.snno~\;

fA(. ..q Clf'I~O"J

6.'('\10 0(" to'lE'6GEpct 6u6 t:t1f-1c'C'Q.

E.o ..('0"' .s~B~.5~ ei9ht fl I'll E"S

ASR

Arithmetic Shift Right Level 1

#

n1

Command

Level 1 •

#

n2

ASR performs a 1 bit arithmetic right shift on a binary integer. In an arithmetic shift, the most significant bit retains its value, and a shift right is performed on the remaining (wordsize-1) bits.

AND

OR

XOR

NOT

The functions AND, OR, XOR, and NOT can be applied to binary integers, strings, or flags (real numbers or algebraics). This section describes their use with binary integers and strings; see uPROGRAM TEST" for their use with flags. These functions treat binary integers and strings as sequences of bits (a's and l's) . • A binary integers is treated as a sequence of length n, where n is the current wordsize. The bits correspond to the a's and l's in the binary integer's representation in base 2.

92

Dictionary

... BINARY • A string is treated as a sequence of length 8n, where n is the number of characters in the string. Each set of eight bits corresponds to the binary representation of one character code. For AND, OR, and XOR, the two string arguments must be the same length.

AND

And Level 2

Level 1

# n1

# n2

"string1 "

"string2 "

Function Level 1

• •

# n3 "string3 "

AND returns the logical AND of two arguments. Each bit in the result is determined by the corresponding bits (bit1 and bit 2) in the two arguments, according to the following table:

bit 1

bit 2

bit 1 AND bit2

0

0

0

0

1

0

1

0

0

1

1

1

0,

OR Level 2

Level 1

# n1

# n2

"string1 "

"string2 "

Function Level 1

•

•

# n3 "string3 "

OR returns the logical OR of two arguments. Each bit in the result is determined by the corresponding bits in the two arguments, according to the following table:

Dictionary

93

... BINARY biti

bit2

0

0

biti

OR bit2

0

0

1

1

1

0

1

1

1

1

XOR

Exclusive Or Level 2

Level 1

# n1

# n2

II

string 1 II

II

string2"

Function

Level 1

.. ..

# n3 II

string3"

XOR returns the logical XOR (exclusive OR) of two arguments. Each bit in the result is determined by the corresponding bits in the two arguments, according to the following table:

94

Dictionary

biti

bit 2

bit 1 XOR bit 2

0

0

0

0

1

1

1

0

1

1

1

0

... BINARY NOT

Not

Function

Level 1

Level 1

# n1

..

# n2

string1

..

string2

NOT returns the ones complement of its argument. Each bit in the result is the complement of the corresponding bit in its argument.

bit

NOT bit

0 0

Dictionary

95

Calculus The HP-28S is capable of symbolic differentiation of any algebraic expression (within the constraints of available memory), and of numerical integration of any (algebraic syntax) procedure. In addition, the calculator can perform symbolic integration of polynomial expressions. For more general expressions, the J command can automatically perform a Taylor series approximation to the expression, then symbolically integrate the resulting polynomial.

Differentiation a(oz ~I A~' n

' T

}

n

i) d)(. 'Sy~IH ' IEVALl~ step YeSLtlt

i l) IS),mb, ' . ~. 4. _'FULL, result

a (., d/dx I) computes the derivative of an algebraic expression symb 1 with respect to a specified variable name. (Name cannot be a local name.) The form of the result expression symb 2 depends upon whether a is executed as part of an algebraic expression, or as a Ustand-alone" object.

Step-wise Differentiation in Aigebraics The derivative function a special syntax:

a is represented in algebraic expressions with '~name(symb)

"

where name is the variable of differentiation and symb is the expression to be differentiated.

96

Dictionary

... Calculus For example, lOX ( SIN ( Y) ) represents the derivative of SIN ( Y) with respect to X. When the overall expression is evaluated, the differentiation is carried forward one "step"-the result is the derivative of the argument expression, multiplied by a new subexpression representing the derivative of its argument. An example should make this clear. Consider differentiating SIN ( Y) with respect to X in radians mode, where Y has the value X"'2 I

I

0 X( SIN ( y)

I

)

I:

E VAL returns

I

I

*

COS ( y) 0 X(y)

I

•

We see that this is a strict application of the chain rule of differentiation. This description of the behavior of a, along with the general properties of EVAL, is sufficient for understanding the results of subsequent evaluations of the expression:

*(0 X( X>* 2 *X'" ( 2 -

EVAL returns

I

COS ( X" 2 )

EVAL returns

I

COS (X"2) * (2*X)

1) )

I ,

I.

"Jl,.J Fully Evaluated Differentiation When

a is

executed as an individual object-that is, in a sequence I

symb

I

I

name

I

0,

rather than as part of an algebraic expression, the expression is automatically evaluated repeatedly until it contains no derivatives. As part of this process, if the variable of differentiation name has a value, the final form of the expression will have that value substituted everywhere for the variable name. To compare this behavior of a with the step-wise differentiation described in the preceding section, consider again the example expression SIN (Y) where Y has the value X"2 I

I

SIN cn

I,

I

I

X

I

I

0 returns

I

COS ( X 2 ) A

I:

*(2 *X)

I

•

Dictionary

97

... Calculus All of the steps of the differentiation have been carried out in a single operation. The function a determines whether to perform the automatic repeated evaluation according to the form of the level 1 argument that specifies the variable of differentiation. If that argument is a name, the full differentiation is performed. When the level 1 argument is an algebraic expression containing only a name, only one step of the differentiation is carried out. Normally, algebraics containing only a single name are automatically converted to name objects. The special syntax of a allows this exception to be used as a signal for full or stepwise differentiation.

Differentiation of User-Defined Functions When

a is

applied to a user-defined function:

1. The expression consisting of the function name and its arguments within parentheses is replaced by the expression that defines the function. 2. The arguments from the original expression are substituted for the local names within the function definition. 3. The new expression is differentiated. For example: Define F (a, b) «

~

=

2a

+ b:

*

a b '2 a +b' » 'F' S TO.

} Differenti«flcH) USIY',}

:r) Oi" IT) W('4 Y50 (o." neA!- P"'5 e )

Then differentiate 'F (X, X"2) , with respect to X. The differentiation automatically proceeds as follows:

1.

'F(X,X 2)' is replaced by '2*a+b'. A

2. X is substituted for a, and 'X . . . 2' for b. The expression is now '2*X+X . . . 2'.

98

Dictionary

... Calculus 3. The new expression is differentiated.

1-) -

If we evaluated' oX(F(X) X 2»' the result is A

'oX(2*X)+oX(X A 2) '.

o{

1Y,-

If we executed' F (X) (X 2) )' 'X' 0, the differentiation is carried through to the final result '2 +2*X' . A

User-Defined Derivatives If ais applied to an HP-28S function for which a built-in derivative is not available, a returns a formal derivative-a new function whose name is HderH followed by the original function name. For example, the HP-28S definition of % does not include a derivative. If you differentiate '% ( X) Y)' one step with respect to Z, you obtain 'der%(X)Y)oZ(X»oZ(Y»'

Each argument to the % function results in two arguments to the der% function. In this example, the X argument results in X and oZ (X) arguments, and the Y argument results in Y and oZ (Y) arguments. You can further differentiate by creating a user-defined function to represent the derivative. Here is a derivative for %: «

7

X

Y

dx dy

'( x * d y +y * d x ) / 1 (3 (3'

:to

'd e r %'

S TO.

With this definition you can obtain a correct derivative for the % function. For example: , %( X ) 2 * X)' 'X'

0 COL CT returns '. (3 4 * X ' .

Similarly, if a is applied to a formal user function (a name followed by arguments in parentheses, for which no user-defined function exists in user memory), a returns a formal derivative whose name is HderH followed by the original user function name. For example, differentiating a formal user function 'f (x 1) x2) x3)' with respect to x returns 'derf(xl)x2)x3)ox(xl»ox(x2»ox(x3»'

Dictionary

99

· .. Calculus Integration

J

Integrate Level 3

Level 2

Level 1

'symb'

, global'

degree

x

(global a b}

accuracy

'symb'

(global a b}

accuracy

.;;:program» (global a b}

accuracy

(a b}

«program»

accuracy

Level 2

.. . ..

. .

Command Level 1 ' integral'

integral

error

integral

error

integral

error

integral

error

Symbolic Integration

J includes

a limited symbolic integration capability. It can return an exact (indefinite) integral of an expression that is a polynomial in the variable of integration. It can also return an approximate integral by using a Taylor series approximation to convert the integrand to a polynomial, then integrating the polynomial.

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Dictionary

~l c4

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... Calculus To obtain a symbolic integral, the stack arguments must be:

3: Integrand (name or algebraic) 2: Variable of integration (global name) "\

1 :) Degree of polynomial (real integer)

/

/'

~------------------

degree~of

The polynomial specifies the order of the Taylor series approximation (or the order of the integrand if it is already a polynomial), ,_". __ '_"~~_._____

---,

..-

\

.

" ..:,,\"'\; "

t'~ t l,\ '!>~

. Numerical Integration \ c:,1;\\:." ~\~..:> ~_ To obtain a numerical integral, you--mu:srspecify~-'"

.f

1101'

J) • The integrand,

10 • :tJI.). 'fI,).

J:T A,

J

'

I E"")O.j>fes:Sior) •

The variable of integration,

'910 ba l

The numerical limits of integration, {

',.:)

flo w €'v

i i mit I

U

~ fe~ l' mit}

The accuracy of the integrand, or effectively, the acceptable error in the result of the integration.

Using an Explicit Variable of Integration. A numerical integration, in which the variable of integration is named with a name object that (usually) appears in the definition of the object used as the integrand, is called explicit variable integration. In the next section, implicit variable integration will described, in which the variable of integration does not have to be named.

Dictionary

101

... Calculus For explicit variable integration, you must enter the relevant objects as follows:

3: Integrand

2: Variable of integration and limits 1: Accuracy

The integrand is an object representing the mathematical expression to be integrated. It can be: • A real number, representing a constant integrand. In this case, the value of the integral will just be:

number (upper limit - lower limit). • An algebraic expression. • A program. The program must satisfy algebraic syntax-that is, take no arguments from the stack, and return a real number. The variable of integration and the limits of integration must be included in a list in level 2 of the form:

{ name lower-limit upper-limit }, where name is a global name, and where each limit is a real number or an object that evaluates to a number. The accuracy is a real number that specifies the error tolerance of the integration, which is taken to be the relative error in the evaluation of the integrand (the accuracy determines the spacing of the points, in the domain of the integration variable, at which the integrand is sampled for the approximation of the integral). The accuracy is specified as a fractional error, that is,

accuracy r~

1 02

Dictionary

I true value -

computed value computed value

I

... Calculus where value is the value of the integrand at any point in the integration interval. Even if your integrand is accurate to or near 12 significant digits, you may wish to use a larger accuracy value to reduce integration time, since the smaller the accuracy value, the more points that must be sampled. The accuracy of the integrand depends primarily on three considerations: • The accuracy of empirical constants in the expression. • The degree to which the expression may accurately describe a physical situation. • The extent of round-off error in the internal evaluation of the expression. Expressions like cos (x - sin x) are purely mathematical expressions, containing no empirical constants. The only constraint on the accuracy then, is the round-off errors which may accumulate due to the finite (12-digit) accuracy of the numerical evaluation of the expression. You can, of course, specify an accuracy for integration of such expressions larger than the simple round-off error, in order to reduce computation time. When the integrand relates to an actual physical situation, there are additional considerations. In these cases, you must ask yourself whether the accuracy you would like in the computed integral is justified by the accuracy of the integrand. For example, if the integrand contains empirical constants that are accurate to only 3 digits, it may not make sense to specify an accuracy smaller than 1E-3. Furthermore, nearly every function relating to a physical situation is inherently inaccurate because it is only a mathematical model of an actual process or event. The model is typically an approximation that ignores the effects of factors judged to be insignificant in comparison with the factors in the model.

Dictionary

103

... Calculus To illustrate numerical integration, we will compute

~2 exp x dx to an accuracy of .OOODI. The stack should be configured as follows for J:

3: 'EXP(X)' 2: {

X

1

2

}

1: .00001

Numerical integration returns two numbers to the stack. The value of the integral is returned to level 2. The error returned to level 1 is an upper limit to the fractional error of the computation, where normally

error

accuracy

J Iintegrand I

If the error is a negative number, it indicates that a convergence of the approximation was not achieved, and the level 2 result is the last computed approximation.

For the integral of EXP ()-

..

o

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Q

......

co

CD

- C)

r

o

())

II N

ii'

~ ::--;;:- ~ ~ ~ ~

z

""......

)(

,.-..

r


0 and in (y, 0] for y < O.

222

Dictionary

.•. REAL %T

Percent of Tota. Level 2

Level 1

x

y

x

'symb'

'symb'

x

'symb 1 '

, symb 2 '

Function Level 1

•

• • •

100y/x ' %T (x J symb) , ' %T (symb J x) , ' %T (symb 1 J symb 2 ) ,

%T computes the (percent) f!actioJ1, 9Lth~ real-valued argument x in level 2 that is rep!esenteclJ'ythe argument y in levell. That is, %T returns IOOy/x. '.

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(~

liCI

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3

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Ll,:\j"lC'}(;,j

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Dictionary

223

Y\Il\ll\~l"i (

SOLVE \

trW»tev';c. resvit5

( si~bcii( res. v.rts;

RCEQ

SOLVR

.

ro'iJ/N,.. dlf/e v t''142 W,-tL1J--+

o lA t.y

RCL~

;'0. ilA«t ~+ 4 "~.v\~ ", "obj , • ",.i :)~-(; "i'l

displays Resu 1 t = object in line 1 of the display, where object is a string form of an object taken from level 1.

STR ....

-----------]

String to Objects

------::~:T

Command

STR .... is a command form of ENTER. The characters in the string argument are parsed and evaluated as contents of the command line. The string may define a single object, or it may be a series of objects that will be evaluated just like a program. STR .... can also be used to restore objects that were converted to strings by .... STR back to their original form. The combination .... STR STR .... leaves objects unchanged except that .... STR converts numbers to strings according to the current number display format and binary integer base and wordsize. STR .... will reproduce a number only to the precision represented in the string form.

Dictionary

265

... STRING CHR

F

Command

Character Level'

n

Levell

..

"string"

CHR returns a one-character string containing the HP-28S character corresponding to the character code n taken from level 1. The default character ~ is used for all character codes that are not part of the normal HP-28S display character set. Character code 0 is used for special purposes in the command line. You can include this character in strings by using CHR, but attempting to edit a string containing this character causes the C." n ' t Ed i t C H~: 0:: 0) error.

NUM

Character Number Level 1 - ,

Level 1

"string""

n

-----l Command

U _

NUM returns the character code of the first character in a string. The following table shows the relation between character codes (results of NUM, arguments to CHR) and characters (results of CHR, arguments to NUM). For character codes 0 through 147, the table shows the characters as displayed in a string. For character codes 148 through 255, the table shows the characters as printed by the HP 82240A printer; these characters are displayed on the HP-28S as the default character •

266

Dictionary

... STRING Character Codes (0-127)

NUM CHR lJ

.~

NUM CHR 32 3:3

NUM CHR 64 65 66

"-

:34

:3

35

#

67

$

68 69 713 71

4

36

c·

._'

37

%

6 7 8 9 113 11 12

38

Ii.

13

14 15 16 17

18 19 213 21 22 23

39

413 41 42 43 44 45 46 47 48 49 513 51 c·-, "_I

c..

C·" J.J 54 55

24 25 26

c··Jb

27

30

5'3 613 61 62

31

63

28 29

57

* +

,~

96

A 8

97 98

b

C

9'~

c

D

11313 101

d

E F

G

72

H

73

I

74 75

K

-,.-

J

(to

L

77 78

M

/

79

I)

13

Be

p

81

G!

2

82

R

3

83

S

N

4

84

T

5 6

85 86

U

7

87

N

:3

88

i
(GET

C

Type

EXP

Gets a subexpression.

Description

EXGET

Name

HP-28S Operation Index (Continued)

o

...

Co) Co)

CD >C

Q,

:;

:::I

0'

AI

CD

..

"..

Changes the form of an algebraic.

Fractional part.

Tests a user flag.

Tests and clears a user flag .

Gets an element from an object.

Gets an element from an object and increments the index.

Suspends program execution.

FORM

FP

FS?

FS?C

GET

GETI

HALT

Subtracts in HMS format.

Creates an identity matrix.

Symbolic constant i.

i

IDN

Selects the HOME directory.

HOME

Converts from HMS format.

HMS -

HMS~

Adds in HMS format.

HMS +

Sets hexadecimal base.

Begins definite loop.

FOR

HEX

Next smaller integer.

FLOOR

C

At

C

C

C

C

C·

Ct

C

C

C

C

F

C

Ct

F

1, 3 1

1 1

TRIG

1 1

2 1

0, 1

ARRAY

ALGEBRA

1 0 1

MEMORY

0

0

TRIG

TRIG

BINARY

1

2

0

3

2

0

ARRAY LIST

1

2

PROGRAM CONTROL

ARRAY LIST

1

PROGRAM TEST

PROGRAM TEST

REAL

ALGEBRA ALGEBRA (FORM)

PROGRAM BRANCH

REAL

1

1

0

2

1

1

1

75

27

141

280

280

280

87

195

70 128

70 128

204

204

219

28 34

188

219

IC' Z

-

PLOT

Arithmetic ARRAY REAL AL£lEBRA SOLVE ! PROGRAM CONTROL

1 0 0,1

1 1 1 1, 2 0

1 1 0

1 1 2

-. 0 0

F C O· O· A F

C C

Returns the imaginary part of a number or array.

Selects the plot independent variable.

Switches between insert and replace modes; digitizes point.

Deletes all characters to the left of the cursor.

Inverse (reCiprocal).

Integer part.

Solves an expression or equation .

Returns a key string.

Aborts all suspended programs.

IFTE

1M

INDEP

[ill§]

• [ill§]

INV

IP

ISOL

KEY

KILL

C

PLOT

0

F

PROGRAM CONTROL

ARRAY COMPLEX

PROGRAM BRANCH

PROGRAM BRANCH

0

2 3

C

If-then-else function.

PROGRAM BRANCH

If-then command .

0

0

PROGRAM BRANCH

Where

1FT

0

Out

0

In

Ct

Ct

Type

Begins IF ERROR clause.

Begins IF clause.

Description

IFERR

, IF

Name

HP·28S Operation Index (Continued)

195

195

33 234

219

60 69

155

157

82 111

188

188

186

186

"1'1

-

Co» Co» Co»

•

CD

a.

;'

o·:s

III

CD

"....

o

Evaluates the left side at the current equation .

lhEFB

LOGS

LO

LR

.1

LOG

LNP1

MANT

Returns the mantissa at a number.

Replaces log-at-power with product-at-log.

Replaces product-at-log with log-at-power.

Computes a linear regression.

Selects the LOGS menu.

Common (base 10) logarithm.

Natural logarithm at (argument +1).

Natural logarithm.

LN

n+1

REAL

ALGEBRA (FORM) F

ALGEBRA (FORM)

218

45

45

251 STAT

O· _ L - - __

2

133

133 LOGS

LOGS

133

133

LOGS LOGS

128 241

127 LIST STACK

O·

C

O·

A

A

A

C

LIST

O·

Selects the LIST menu.

37

227

SOLVE ALGEBRA (FORM)

264

150

151 240

STRING

MODE

MODE STACK

O·

Moves list elements to the stack.

LIST

1, 2, 3

Displays the level at the selected subexpression.

LIST ...

.1

LEVEL

0

0

Returns display image as a graphics string.

LCD ...

I

C

Switches between upper-case and lower-case modes.

1 LC 1

1

0

O·

i'. LL

"

It

0

O·

i Enables or disabJes LAST mode ....

LAST

0

C

Returns last arguments (it saved).

LAST

Z

-I

,.:I:

>C

:I DCD

o·:I

CD III

"..

o

oil>

w w

Selects the specified built-in or custom menu.

MENU

O·

Enables or disables ML mode.

Modulo .

Selects the MODE menu.

MOD

• 1MODE I

O'

F

C

Finds the minimum coordinate values in the statistics matrix.

MIN~

ML.

A

Symbolic constant minimum real.

MINR

F

Returns the minimum of two numbers.

0

C

C

C

MIN

Switches shifted action and unshifted action of letter keys [EJ through []].

Returns available memory.

MEM

.1 MENUS I

Computes statistical means.

C

Finds the maximum coordinate values in the statistics matrix.

MAX~

MEAN

A

Symbolic constant maximum real.

MAXR

F

Type

Returns the maximum of two numbers.

Description

MAX

Name

MEMORY

1 0

0 1

2

1

STAT

1 0

-.

ALGEBRA REAL

1

0

MODE

REAL

MODE t

Ii

REAL

1

2

MEMORY

STAT

STAT

1

0 1

ALGEBRA REAL

1

0

0

REAL

1

Out

2

In

HP·28S Operation Index (Continued) Where

145

221

150

249

27 215

221

141

141

249

249

27 215

221

&II

Co) Co)

CD >C

S' Do

:::I

O·

CD DI

...

o 'a

~

O'

Advances to next command or unit in a catalog.

Advances to next argument option in USAGE.

Choose not to purge during 0 u t

Logical or binary NOT.

Returns character code.

,Np er-right plot coordinates.

PMAX

POS

PROGRAM .1 BRANCH I • 1CONTRL I .1 TEST I

PRMD

PRLCD

Selects the PROGRAM BRANCH menu . Selects the PROGRAM CONTROL menu. Selects the PROGRAM TEST menu.

Prints and displays current modes.

Prints an image of the display.

Selects the PRINT menu.

Displays the previous argument option in USAGE.

PREV

.1 PRINTI

Displays the previous command or unit in a catalog.

Displays the previous row of menu labels.

• 1PREVI

PREV

Predicted value .

PREDV

Recalls the plot parameters list in the current directory.

Finds a substring in a string or an object in a list.

PPAR

Sets the lower-left plot coordinates.

PMIN

,

Selects the PLOT menu.

.1 PLOT I

.

C

Turns on a display pixel.

PIXEL

O' O' O'

C

C

O'

O'

O'

O'

0

0

0

PROGRAM BRANCH PROGRAM CONTROL PROGRAM TEST

MODE PRINT

PLOT PRINT

PRINT

UNITS

183 193 201

150 174

165 171

168

287

251 STAT

C

160

PLOT

132 270

0

0

157 157

0

PLOT

PLOT

152

165

243

LIST STRING

0

0

PLOT

0 PLOT

STACK

n+1

2

n+1

C

C

C

O'

C

Duplicates the nth object.

PICK

-t

-t

en

1ft

...

o

0' ::s

'a CD III

C

C O· O·

Prints the level 1 object.

Purges one or more variables.

Replaces an element in an array or list.

Replaces an element in an array or list, and increment the index.

Polar-to-rectangular conversion.

Solves a quadratic polynomial.

Exits CATALOG or UNITS.

Exits USAGE display.

PR1

PURGE

PUT

PUTI

P-R

QUAD

QUIT

QUIT

F

C

C

C

C

Prints the name and contents of one or more variables.

PRVAR

C

C

Prints a list of variables in the current directory.

Prints the stack in compact format.

PRSTC

C

Type

PRUSR

Prints the stack.

Description

PRST

Name

2

1

3

3

1

0

1

0

0

0

In

HP·285 Operation Index (Continued)

1

1

2

UNITS

ALGEBRA SOLVE

COMPLEX TRIG

ARRAY LIST

MEMORY ARRAY LIST

0, 1

PRINT

PRINT

PRINT

PRINT

PRINT

0

0

0

0

0

0

Out

Where

287

33 235

114 227

70 128

70 128

140

171

171

174

174

171

-t

"en::a

..

o

CO

w w

•

CL CD

;'

:::I

0'

'a CD III

Redimensions an array.

RDM

REAL

RES

REPEAT

.1

RE

RDZ

Sets the plot resolution.

Part of WHILE ... REPEAT ... END.

Selects the REAL menu.

---- - - - - -

Returns the real part of a complex number or array.

Sets the random number seed.

Enables or disables RDX, mode.

Recalls the binary integer wordsize.

RCWS

RDX,

Recalls the current statistics matrix.

RCL~

I

Recalls the contents of a variable, unevaluated.

RCL

Returns a binary integer representing the user flags.

Recalls the current equation.

RCEQ

~ RCLF

Returns a random" number.

RAND

radians ~ mode.

Sets

RAD

--

C

Ct

O'

F

C

O'

C

C

C

C

C

C

C

C'

PROGRAM TEST PLOT STAT BINARY ARRAY

1 1 1 1 0, 1

0 1 0 0

0 2

1

1

1

0

0

1

0

MEMORY

1

0

1

PLOT SOLVE

1

PLOT

PROGRAM BRANCH

REAL

ARRAY COMPLEX

REAL

MODE

MODE REAL

0

0

160

192

213

82 111

215

150

75

87

163 246

211

140

157 226

215

145

en

m

:D

Real-to-binary conversion.

Real-to-complex conversion.

Radians-to-degrees conversion.

R.. B

R.. C

R.. O

I

Evaluates the right side of the current equation.

RT=

I

F

C

C

0

1

1

280

82 111 279

ARRAY COMPLEX TRIG 1

2

TRIG

89

BINARY

1 1

227

75

89

89

243

240

79

219

SOLVE

ARRAY

BINARY

BINARY

REAL

1 0

1

Computes a correction to the solution of a system of equations.

RSO

3

C

Rotates right by one byte.

RRB C

C

Rotates right by one bit.

RR 1

C

Moves the level 3 object to level 1.

ROT

1

STACK

C

Finds a numerical root.

ROOT

1

SOLVE

1, 3

3

C

Moves the level 2 object to level n.

ROLLO

1

241

STACK

n

n+1 n+1

C

Moves the level n+1 object to level 1.

ROLL

3

STACK

1

n

1

C

Computes the row norm of an array.

RNRM

3

233

ARRAY

1

1

F

89

Rounds according to real number display mode.

BINARY

RNO

1

1

89

C

BINARY

Rotates left by one byte.

1

1

C

Where

RLB

Out

In

Type

Rotates left by one bit.

Description

RL

Name

Hp·28S Operation Index (Continued)

olio

...

Co)

CD >C

s-a.

::s

DI

CD

.o·

"..

o

Conjugates the contents of a variable.

Computes standard deviations.

Sets a user flag.

Resolves all references to a name implicit in an algebraic.

Sign of a number.

SCONJ

SDEV

SF

SHOW

SIGN

Inverts the contents of a variable.

Finds the dimensions of a list, array, string, or algebraic.

SINV

SIZE

Hyperbolic sine.

Auto-scales the plot parameters according to the statistical data.

SCU;

SINH

Sets scientific display format.

SCI

Sine.

Tests two objects for equality.

SAME

SIN

Rectangular-to-polar conversion.

R~P

C

C

A

A

F

C

C

C

C

C

C

C

F

2

0

0

2

0

ALGEBRA ARRAY LIST STRING

STORE

LOGS

TRIG

COMPLEX REAL

ALGEBRA SOLVE

PROGRAM TEST

STAT

STORE

0 0

PLOT

MODE

0

0

PROGRAM TEST

COMPLEX TRIG

28 75 132 270

258

136

273

111 218

33 235

204

249

262

163

145

206

114 277

en N rn

>C

CD

Q.

~

~

o·

DI

CD

"..

o C

Negates the contents of variable.

Selects the SOLVE menu.

SNEG

[SOlV!

Shifts right by one byte.

SR

SRB

C

Stores the current equation.

STEa

0

Ct

Ends definite loop.

STEP

1

C·

Sets standard display format.

STD

0

O·

Selects the STAT menu.

.[STAT!

1

-.

Ct

0

~

PLOT SOLVE

PROGRAM BRANCH

MODE

STAT

PROGRAM BRANCH

STACK

2

Selects the STACK menu.

START

PROGRAM CONTROL

BINARY

0

1

1

BINARY

Arithmetic ARRAY

SOLVE

SOLVE

STORE

BINARY

BINARY

Where

O·

C

1

1

0

1

1

1

1

1

1

Out

1

In

Begins definite loop.

.[STACK!

Single-steps a suspended program.

Shifts right by one bit.

sa

SST

A

Squares a number or matrix. C

0

Selects the Solver variables menu.

SOLVR

O·

C

C

Type

Shifts left by one byte.

Shifts left by one bit.

Description

SLB

SL

Name

Hp·28S Operation Index (Continued)

157 226

188

145

245

188

239

195

91

91

61 70

225

224

258

91

91

I'"

en

STO -

Co) ~ Co)

CD >C

a-

SWAP

SUB

STR-+

STWS

:::I

Swaps the objects in levels 1 and 2.

Extracts a portion of a list or string.

Sets the binary integer wordsize.

Parses and evaluates the commands defined by a string.

Selects the STRING menu.

Stores the current statistics matrix.

STO/

ST0 2:

STRING

C

Storage arithmetic divide.

C

C

C

C

O·

C

C

Storage arithmetic subtract.

:::I

CD I»

'a

I

C

Storage arithmetic add.

STO +

....o· .1

o

C

STORE

Storage arithmetic multiply.

1

STO*

•

O·

C

C

Selects the STORE menu.

I

Sets all user flags according to the value of a binary integer.

,

STOF

..

Stores an object, in a variable.

STO

0 0

2 2

2

3

0

2

2

BINARY

0

STACK

LIST STRING

STRING

STRING

PLOT STAT

STORE

STORE

STORE

STORE

0

0

0

PROGRAM TEST

0 STORE

MEMORY

0

2

2

239

132 270

87

264

263

163 246

258

258

258

258

258

211

139

I'111

en

CD >C

a.

~

~

0'

CD III

. ...

o 'a

O' O' C C O'

O·

Sums the coordinate values in the statistics matrix.

Enables or disables printer Trace mode.

Selects the TRIG menu.

Transposes a matrix.

Returns the type of an object.

Recovers previous stack contents (if saved).

Enables or disables UNDO mode.

'TRIG!

TRN

TYPE

. , UNDO!

O'

Part of BEGIN ... UNTIL ... END.

Displays USAGE for current command in CATALOG.

UNTIL

USE

Ct

Selects the units catalog.

. , UNITS!

UNDO

TRAC

O'

C

Begins THEN clause.

THEN

4II,TOT

Ct

Selects the PROGRAM TEST menu.

O'

C

.'TEST!

Computes a Taylor series approximation .

A

Hyperbolic tangent.

TANH

TAYLR

A

Tangent.

TAN

C

Type

Executes a system object.

Description

SYSEVAL

Name

1

1

1

1

-.

0, 1 1

STAT

1

0

~

PROGRAM BRANCH

UNITS

MODE

M0DE

PROGRAM TEST

ARRAY

TRIG

PRINT

PROGRAM BRANCH

0

PROGRAM TEST

ALGEBRA Calculus

LOGS

TRIG

Evaluation

Where

1

1

1

1 3

0

Out

1

In

Hp·28S Operation Index (Continued)

192

287

150

151

211

75

273

171

249

186

201

33 106

136

273

126

....o·

o

UI

Co)

,.

CD >C

CI.

:;

:::I

"a CD III

Upper-tail normal distribution.

Upper-tail t-distribution.

Computes statisical variances.

UTPN

UTPT

VAR

0

Copies an object to the command line for editing.

Pauses program execution.

• IVISITI

WAIT

YES

.0

Chooses to purge during 0 u t

Executes function SQ. of Memor y .

Returns the exponent of a number.

Logical or binary XOR .

XOR

XPON

Begins WHILE ... REPEAT ... END.

WHILE

0

2 F

O·

A

F

0 0

0

249

STAT

0

Arithmetic ARRAY

REAL

BINARY PROGRAM TEST

PROGRAM BRANCH

PROGRAM CONTROL

61 70

218

92 206

192

195

144

254

STAT

2

MEMORY

254

STAT

3

0

254

254

STAT

STAT

3

2

Ct

C

O·

1

VIEW+

Moves the display window down one line .

1

•

1

VIEWt

Moves the display window up one line .

1

O·

•

C

C

C

C

Returns a list of variables in the current directory.

. VARS

Upper-tail F-disWbution .

UTPF

C

C

Upper-tail Chi-Square distribution .

UTPC

l

O·

Selects the USER menu.

I USER I

en

~

m

o·:I

CD DI

"...

o

J

•

* *H *w

/

~~~m *,rl

~

-

. r+1-1

+

.~ 1/()

.OEJ

Name

A·

Double negates and distributes.

A· A

Divides two objects.

e e

Multiplies by 1.

Adjusts the width of a plot.

Adjusts the height of a plot.

A

A

Subtracts two objects.

Multiplies two objects.

A·

A

O·

A

Type

Adds and subtracts 1.

Adds two objects.

Double invert and distribute.

Executes function INV.

Description

1

0

-1. 2

0

1

1

1

1

Out

1

2

2

2

1

In

HP-28S Operation Index (Continued)

44 58 66

Arithmetic ARRAY

160

160

56 66

43

55 65

45

53 65 127 264

44

69 69

ALGEBRA (FORM)

P4!)T

PLOT

Arithmetic ARRAY

ALGEBRA (FORM)

Arithmetic ARRAY

ALGEBRA (FORM)

Arithmetic ARRAY LIST STRING

ALGEBRA (FORM)

Arithmetic ARRAY

Where

>C

-

...

.ca. .....

Col

>C

~

CI.

0' :s :s

DI

~

..

o '1:1

GB

Selects cursor menu or restores last menu; displays coordinates.

Shift key.

O'

O'

Ft

Greater-than comparison.

>

•

Ft

Ft

Greater-than-or-equal comparison.

Not-equal comparison.

;;"

oF

At

Equals operator.

=

F

Ft

Less-than-or-equal comparison .

~

Equality comparison.

Ft

Less-than comparison.