Scientific Calculators: Maxima and Minima

Scientific Calculators: Maxima and Minima

Introduction

This blog entry covers four calculators:

Casio fx-991CW

Casio fx-991EX Classwiz

Texas Instruments TI-36X Pro

Texas Instruments TI-30X Pro Math Print

Here is how to find extremum of any general function f(x) and g(x).   This method will tell you where the extremum is located, but not whether the extremum is a minimum or a maximum.  

The extremum of a function can be found by finding the root of the function's derivative:

d/dx f(x) = 0

The steps required are:

1.  Solve d/dx f(x) = 0 to get x.  

2.  Calculate y.  This is easier if we can store the function in f(x) or g(x).

Two methods to find the x point of extremum:

1.  Extremum using f(x) (or g(x))

2.  Extremum using direct entry method

The example used is to find the extremum of e^(sin x) with an initial guess x=1.  We'll be focusing on the x point only.  

Casio fx-991CW 

Either method can be used.   In the Solver Mode, the function button ( [f(x)] ) is available to define f(x) (or g(x)).   Since we are finding a root, we don't have to include the "=0" at the end.

The equation to be used in the solver is:

d/dx( f(x) ) | x = x      (MathIO input)

f(x) can be referenced or typed directly, like:

d/dx( e^(sin(x)) ) | x = x

Casio fx-991EX Classwiz

The direct entry method must be used.  The accuracy may vary for the last digits.  Not as accurate as the fx-991CW. 

Note:  Extremum for quadratic and cubic equations (fx-991CW only) are automatically calculated through the Solver-Polynomial feature of both the fx-991CW and fx-991EX.  

Texas Instruments TI-36X Pro and TI-30X Pro MathPrint

Either method can be used.   f(x) (and g(x) for the TI-30X Pro MathPrint), is edited through the [table] key.   When the function is edited, press [ 2nd ] [ mode ] (quit) to exit the table.    To enter Numeric Solver Mode, press [ 2nd ] [ sin sin^-1 ] (num-solv).  

The solver has the format:

[ ] = [ ]

In this case, we will need to include the 0 on either side of the template.  

Finding extremum is fairly simple.

For the example, the extremum e^(sin x) around x = 1 is x = π/2 (≈ 1.570796327).  

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Integer Division with Scientific Calculators

Integer Division with Scientific Calculators

Integer Division

A popular function of European scientific and fraction-featured calculators is the integer division function (also known as Euclidean division).  Integer division between two numbers (typically positive integers) return both the quotient and the remainder.

Example:  

456 int÷ 78 returns a quotient of 5 and a remainder of 66.

Here's a way to execute integer division with various calculators.

TI-36X Pro and TI-30X Pro Math Print

Integer division can be easily accomplished by using two functions:

Quotient:  Either int(x/y) or iPart(x/y)

Remainder:  mod(x,y)

Example:   456 int÷ 78

int(456/78) = 5,  mod(456, 78) = 66

Casio fx-991EX and Casio fx-991CW

Assuming x and y are positive integers, what we do depends on whether y is prime or composite.  We will assume that MathIO mode is activated.  

If y is prime:  We can divide x ÷ y, and change the result into a mixed format.

Example:  6140 int÷ 47

47 is prime.

Key in 6140 ÷ 47.  Change the result into a mixed format.  The result is 130 30/47.

Then the quotient is 130, the remainder is 47.

If y is not prime, we will not be able to do this because the fractions are always simplified to the irreducible form.

Then we will have to use the following steps:

1.   Divide x by y.   Take note of the integer portion (you may have to change the format to mixed fraction or decimal format first).   

2.  For the remainder, execute this formula:   x - quotient * y.  

Example:  5518 int÷ 32

32 is not prime.

1.  Calculate the quotient:  5518 ÷ 32 = 172 1/4.  Note that the integer part, 172 is the quotient.

2.  Calculate the remainder:  5518 - 172 * 32 = 14.  

Then the quotient is 172, the remainder is 14.  

Hewlett Packard HP 41C/Swiss Micros DM41X

Program Code:

01  LBL ᵀ INTD

02  x<>y

03  STO Z

04  x<>y

05  ST/ Z

06  MOD

07  x<>y

08  INT

09  x<>y

10  END

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Review: Casio fx-991CW (and the lack of multi-statements)

Review:  Casio fx-991CW (and the lack of multi-statements)

Quick Facts

Model:  fx-991CW

Company:  Casio

Years:  Internationally: 2022 - present, United States: April 2023 - present

Type:  Scientific

Power:  Solar with 1 LR44 backup battery

Display:  Multiline, 4-color gray display

Original Price:  $21.00 - $22.99 U.S. Dollars, varies on the specific version

Number of Variables:  9

Operating System: Algebraic

Update on the Classwiz

The fx-991CW is an update in Casio's Classwiz series, specifically on the 2015 fx-991EX Classwiz.  

Check out my review on the fx-991EX Classwiz from November 2015 here:

http://edspi31415.blogspot.com/2015/11/casio-fx-991ex-classwiz-review.html

The modes on the Classwiz are:

Calculate:  the main app for mathematical calculations

Statistics:  1 to 2 variable statistics.  Regressions include:

Linear:  y=a+bx

Quadratic: y=a+bx+cx^2

Logarithmic:  y=a+b*ln(x)

Exponential:  y=a*e^(bx)

Power I:  y=a*b^x

Power II: y=a*x^b

Inverse:  y=a+b/x

Distribution:  Calculations involving the binomial, normal, and Poisson.  The Inverse Normal deal with the lower tail (left) probability only.  

Spreadsheet:  Like the fx-991EX and Casio's current graphing calculators, the fx-991CW has a spreadsheet which has a 5 x 45 cell capacity.  The total memory is increased to 2,380 bytes (from 1,700 bytes on the fx-991EX).  Basic spreadsheet features included are:  labeling cells, sum, mean, minimum, maximum, absolute cell references ($), copy, cut, and paste.  

Table:   Generate a table of one function or two functions.   The functions are defined as f(x) and g(x).  With a QR code, you can generate a graph of these functions.

Equation:   Solve linear systems, up to 4 x 4 equations.  Solve polynomials, up to 4th order.  Real coefficients only but complex roots are calculated.   The general equation solver is moved to this mode.   

Inequality:  Solve inequalities up for polynomials of orders 2, 3, and 4.

Complex Number:  Complex number arithmetic with polar/rectangular conversion, integer powers, real/imaginary parts, conjugate

Base N:  Integer arithmetic with Boolean logic.  Instead of keys, the [FORMAT] key cycles through the four modes: Decimal, Hexadecimal, Binary, Octal.   Binary integers are up to 31 bytes with 1 sign bit.  We can not store or recall varaible values in this mode.

Matrix:  Works with four matrices up to 4 rows and 4 columns.  Functions include transpose, inverse, and determinant.

Vector:  Works with 4 vectors with either 2 or 3 elements.  Functions include dot product, norm, and cross product.

Ratio:  Solves common ratio problems.

Math Box:  This is the a new feature to the Classwiz.   For the United States, the Math Box has two simulations:  Dice Rolls and Coin Toss.   Depending on the country, the CW may include additional features.  

Emulator and Classpad.net

Purchasing a fx-991CW came with a emulator license with ClassPad.net.   The license number can be obtained using the Get Started option from the Settings menu.   Use the QR code.  For my calculator, the license for using ClassPad.net is seven years at no cost.  However, using an emulator without first purchasing an eligible calculator will require an additional cost.  

You can find details about Classpad.net here:  https://classpad.net/intl/features/

Keyboard, What is Added, and What is Subtracted

The keyboard of the fx-991CW is quite different from the previous Casio calculators.   The keys are now round with a lot less labels.   Most of functions are now stored in the catalog and tools.  Let's go over some of these keys:

[ house icon ]: [HOME].   This is where we switch the modes of the fx-991CW.   There are no numerical shortcuts, so we have to arrow and scroll to select the mode we want.  

[ three lines ]: [SETTINGS].  The settings key replaces the SETUP key sequence.   

Also note that the sub menus are selected by either pressing [ → ] or [ EXE ].  If there are radio buttons, select the option desired.  

[ curved arrow ]: [EXIT/EDIT].  This key, on the 2nd row, 2nd key from the left, will be used to exit menus and re-edit expressions.   

[ double up arrow ]/[ double down arrow ]:  This key, top right of the calculator, is used to quickly scroll through menus or lists.  Think of this key as the Page Up/Page Down key.

[ < >x] ]/[VARIABLE]:  This is the variable key.

Here is where we will store and edit values that are stored in each of the nine variables.   This key replaces the STORE key.     If we are operate in a mode that does not allow for editing or store values, there will be a lock icon with the selected variable.  

To store a result:

1.  Execute operation or recalling a calculated stat variable.

2.  Press [VARIABLE].

3.  Select variable, press [ OK ].

4.  Select Store, press [ OK ].

To edit a variable's value:

1.  Press [VARIABLE].

2.  Select variable, press [ OK ].

3.  Select Edit, press [ OK ].

4.  Enter the new value, press [ OK ].

To recall a variable in a calculation:

1.  Press [ SHIFT ].

2.  Press any of the following keys to get the variable:

[ 4 ]:  A

[ 5 ]:  B

[ 6 ]:  C

[ 1 ]:  D

[ 2 ]:  E

[ 3 ]:  F

[ 0 ]:  x

[ . ]:  y

[ x10^ ]:  z

There is no ALPHA key as it was in past Casio calculators.  

[ f(x) ]/[FUNCTION]:  

Here is where we can store and use up to two functions: f(x) and g(x).  The great news is that f(x) and g(x) are no longer limited to the Table, they can be used in other modes such as Calculate.   The equations f(x) and g(x) are retained while switching modes but not retained when the calculator either is turned off or the Input/Output setting is changed.  A missed opportunity for the latter.

A plus is that either f(x) or g(x) can be a composite function.  That is f(x) can contain g(x) or g(x) can contain f(x).  

There are two ways to access the x variable:  its own key [ x ] or the key sequence [ SHIFT ] [ 0 ].  

[book]/[CATALOG]:  This is where all the functions and commands can be accessed.  The menu order of the catalog changes depending on the mode used.  Some submenus include:

Function Analysis:  The calculus functions, that used to be on the keyboard, are now stored in the Function Analysis menu:  derivative, integral, summation, logarithm*, log*, ln* (* also on the keyboard)

Probability:   % (divides the number by 100), factorial, permutation, combination, random number, random integer

Numeric:  Absolute Value, Round Off (round the number to the Fix settings internally)

Angle/Coord/Sexa...:  angle units (degrees, radians, grads, degrees minutes seconds*), polar/rectangular conversions (* also on the keyboard)

Sci Constants:  47 scientific Constants

Unit Conversions

[ three circles ]/[ TOOLS ]:  The TOOLS menu changes dynamically based on the mode.   For example, the Calculate mode will have an Undo function.

[ FORMAT ]:  Instead of the [S<=>D] key, we have the [FORMAT] key, which asks how to change the value:  Standard, Decimal, Improper Fraction, Mixed Fraction,  ENG (Engineering) Notation, Sexagesimal (degrees-minutes-seconds).

In Base mode, the [ FORMAT ] key toggles between the four bases (decimal, hexadecimal, binary, octal).

I did three comparison speed tests between the fx-991CW and fx-991EX  here:  https://www.youtube.com/watch?v=cj0Odnv0Mwk

As I understand, the fx-991CW has a faster processor than the previous fx-991EX.

Now let's talk about what is subtracted in this update.  You read this correctly, several features did not make it from the fx-991EX to the fx-991CW:

The  CALC feature where we could type in a formula, press [ CALC ] and have formula evaluation.  I was not able to find the CALC feature on the fx-991CW.

There is no longer the independent memory M, nor the storage arithmetic functions M+ and M-.  I miss this feature the most.  I really wish scientific and graphing calculators in general embrace store arithmetic like Hewlett Packard and Swiss Micros.

There is no longer the ability to use multi-statement expressions, with each expression separated with a colon.  Even when it was available, (1) storing results immediately terminated the expression (forcing the use of Ans and when available, PreAns to make using results in the next part possible), and (2) when replayed, the statement was broken up into separate parts.  

Addendum 

Note (4/12/2023):  I am wrong when I said that storing results in multi-statements on the fx-991EX (not the fx-991CW) is impossible.  We use the equals key ([ALPHA] ( = )), as in this example:

A = 9 : B = 8 * A

Csaba Tizedes uses the multi-statement and CALC feature on the fx-991EX to create an IF-THEN-ELSE structure.  In this video Tizedes uses this structure to solve equations using the Bisection method.  The IF-THEN-ELSE structure tests whether a number is positive or negative.  Please take a look his video:

https://www.youtube.com/watch?v=umxScZL1V6A

Gratitude to Csaba Tizedes, this video is shared with his permission.  

Final Thoughts

Overall, the fx-991CW is pretty solid calculator with a readable screen and a lot of features.  My favorite part of the updated is the ability to use the functions f(x) and g(x) outside of the table function.  I also like the catalog key and the page up/page down key.

I get that Casio is going for a simpler, non-busy keyboard.  However, I prefer a separate ALPHA key, along with a STORE key: it's the most efficient way to store variables.   A consequence of a non-busy keyboard is that a lot of the commands can only be accessed through menus.  The ALPHA key could have freed nine keys for more common shifted keys such as x!, polar/rectangular conversions, etc.   I would have liked to see a couple of customizable keys where we can store commands, which could be the shift of the multiplication and division keys.  The financial calculator FC-200V has two slots to store commands.

The faster processor and the better screen are pluses.  You will still get a lot for the money spent on the fx-991CW. 

Caveat:   However, if you want, need, or require subtracted features described above (CALC feature, independent memory, multi-statement expressions), you are better off buying the former fx-991EX or a fx-115ES Plus 2nd Edition.  

Unfortunately, removing the multi-statement, Casio sapped it's algorithmic power with the fx-991CW, and it needs to come back.  Furthermore, please put integer part, fraction part, and sign functions.   

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.