Casio fx-991CW: Input Output Formats

Casio fx-991CW: Input Output Formats

The Four Formats

There are four input/output formats that the Casio fx-991CW has. With each mode, I will show three example problems and the default format the of the output. If you always want Decimal approximations, I would suggest setting the calculator to either MathI/DecimalO or LineI/DecimalO settings.

To set the input/output:

1. Press [ SETTINGS ].

2. Select Calc Settings and press either [ EXE ], [ OK ], or [ → ].

3. Select Input/Output and press either [ EXE ], [ OK ], or [ → ].

4. Select a format and press either [ EXE ] or [ OK ]. The circle next to the mode selected will be filled.

5. To quickly return the home screen, press [ AC ].

MathI/MathO: Math Input/Math Output

Expressions are entered and displayed as written in textbook fashion. The fx-991CW returns results in exact form whenever possible in terms of π, square roots, and fractions. To get decimal approximations, press [ FORMAT ] and select Decimal.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 47 / 40

√440 returns 2 × √110

cos^-1(0.5) returns 1 / 3 × π

MathI/DecimalO: Math Input/Decimal Ouptut

Expressions are entered and displayed as written in textbook fashion. However, in this mode, the fx-991CW always returns the results as decimal expressions. You can still ask for exact answers, whenever possible, by pressing [ FORMAT ] and selecting Standard.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 1.175

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

LineI/LineO: Line Input/Line Output

Expressions are entered in one-line format, similar to classic computers and calculators. Results, when possible and when fractions are used, are returned using fractions. Otherwise, all other results are returned in decimal approximation. To get decimal approximations on fractions, press [ FORMAT ] and select Decimal.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 47 / 40

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

LineI/DecimalO: Line Input/Decimal Output

Expressions are entered in one-line format, similar to classic computers and calculators. However, in this mode, the fx-991CW always returns the results as decimal expressions. To see if the approximation can be converted to fractions (within the limitation of the calculator) press [ FORMAT ] and select Standard.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 1.175

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

I hope you find this helpful. Until next time,

Eddie

All original content copyright, © 2011-2024. 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.

TI 30Xa Algorithm: Acceleration, Velocity, Speed

TI 30Xa Algorithm: Acceleration, Velocity, Speed

Introduction and Algorithm

Given the acceleration (α), initial velocity (v0), and initial position (p0) of an object on a 2D plane, calculate velocity and position at time T.

Assuming that acceleration is constant, the equations are:

Acceleration: A = α

Velocity: V = α * t + v0

Position: P = α / 2 * t^2 + v0 * t + p0

With some simplification, we can simplify the work flow:

( I )

A = α

( II )

V = α * t + v0

V = A * t + v0 (II)

( III )

P = α / 2 * t^2 + v0 * t + p0

P = t * ( α / 2 * t + v0 ) + p0

P = t * ( α / 2 * t + 2 * v0 / 2 ) + p0

P = t * ( (α * t + 2 * v0) / 2 ) + p0

P = t / 2 * ( α * t + 2 * v0 ) + p0

P = t / 2 * ( α * t + v0 + v0 ) + p0

P = t * (V + v0) / 2 + p0

( I ): A = α

( II ): V = A * t + v0

( III ): P = t * (V + v0) / 2 + p0

Key Strokes:

1. Store the time T in Memory 1 ( T [ STO ] 1 )

2. Store the initial velocity in Memory 2 ( v0 [ STO ] 2 )

3. Enter the acceleration constant, α, press [ = ]

4. Calculate the velocity at time T: [ × ] [ RCL ] 1 [ + ] [ RCL ] 2 [ = ]

5. Calculate the position at time T: [ + ] [ RCL ] 2 [ = ] [ × ] [ RCL ] 1 [ ÷ ] 2 [ + ] p0 [ = ]

Examples

Example 1:

α = 1.3

v0 = 0

p0 = 0

T = 10

1. 10 [ STO ] 1

2. 0 [ STO ] 2

3. Acceleration: 1.3 [ = ]

4. Velocity: [ × ] [ RCL ] 1 [ + ] [ RCL ] 2 [ = ] (Result: 13)

5. Position: [ + ] [ RCL ] 2 [ = ] [ × ] [ RCL ] 1 [ ÷ ] 2 [ + ] 0 [ = ] (Result: 65)

Example 2:

α = -9.80665

v0 = 0

p0 = 10000

T = 30

1. 30 [ STO ] 1

2. 0 [ STO ] 2

3. Acceleration: 9.80665 [ +/- ] [ = ]

4. Velocity: [ × ] [ RCL ] 1 [ + ] [ RCL ] 2 [ = ] (Result: -294.1995)

5. Position: [ + ] [ RCL ] 2 [ = ] [ × ] [ RCL ] 1 [ ÷ ] 2 [ + ] 10000 [ = ] (Result: 5587.0075)

Example 3:

α = -0.05

v0 = 10

p0 = 20

T = 15

1. 15 [ STO ] 1

2. 10 [ STO ] 2

3. Acceleration: 0.05 [ +/- ] [ = ]

4. Velocity: [ × ] [ RCL ] 1 [ + ] [ RCL ] 2 [ = ] (Result: 9.25)

5. Position: [ + ] [ RCL ] 2 [ = ] [ × ] [ RCL ] 1 [ ÷ ] 2 [ + ] 20 [ = ] (Result: 164.375)

Example 4:

α = 4.26

v0 = 1.8

p0 = 0

T = 5

1. 5 [ STO ] 1

2. 1.8 [ STO ] 2

3. Acceleration: 4.26 [ = ]

4. Velocity: [ × ] [ RCL ] 1 [ + ] [ RCL ] 2 [ = ] (Result: 23.1)

5. Position: [ + ] [ RCL ] 2 [ = ] [ × ] [ RCL ] 1 [ ÷ ] 2 [ + ] 0 [ = ] (Result: 62.25)

Hope you find this useful.

Happy Halloween, Everyone!

Eddie

All original content copyright, © 2011-2024. 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.

HHC 2024 Videos are Up

 HHC 2024 Videos are Up

Hi, everyone.  Just want to let you know that the videos of the talks from the wonderful HHC 2024 conference are now up on hpcalc.org's on YouTube.   

As usual, time flew by and we had a great time.  

https://www.youtube.com/@hpcalc

Eddie

All original content copyright, © 2011-2024. 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.

Swiss Micros DM32: Reimann-Louiville Fractional Integral of x^p

Swiss Micros DM32: Reimann-Louiville Fractional Integral of x^p

Introduction

The program presented today calculates the Riemann-Louiville integral of:

f(t) = t^p, where p is a real number.

The formula for this integral is:

cDx^(-v) = 1 / Γ(v) * ∫( (x – t) * t^p dt, t = c, t = x)

= ∫( ((x – t) * t^p) / (v -1)! dt, t = c, t = x)

I covered these type of integrals on my September 14, 2024 blog.

DM32 Program: Reimann-Louiville Fractional Integral of x^p

LBL F

INPUT C

INPUT X

INPUT V

INPUT P

FN= I

RCL C

RCL X

∫ FN d T

RTN

LBL I

RCL X

RCL- T

RCL V

1

-

y^x

RCL T

RCL P

y^x

×

RCL V

1

-

x!

÷

RTN

Here is a text version that can be transferred to a dm32 state file (fractionalintegralm.d32):

https://drive.google.com/file/d/1E-wUq4GW5dX06VZ-5WWwy7KRm3SN_uyq/view?usp=sharing

Examples

Run program F: XEQ F. Make sure that V > 0.

C	X	V	P	Result (FIX 5)
0	5	2	2	52.08333
1	6	3	3	385.41667
2	7	1.5	3	717.69103
0	1	1.75	4	0.05298

Source

Kimeu, Joseph M., "Fractional Calculus: Definitions and Applications" (2009).Masters Theses & Specialist Projects. Paper 115. http://digitalcommons.wku.edu/theses/115

Eddie

All original content copyright, © 2011-2024. 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.

Casio fx-CG 50: Centroid of a 2D spaces

Casio fx-CG 50: Centroid of a 2D spaces

The program CENTROID calculates the center point for an area covered by:

y(x) ≥ 0, x ≥ lower limit, x ≤ upper limit

The center point is calculated by:

x-center = ∫( x * y(x) dx, x = lower limit, x = upper limit) / A

y-center = ∫(1 / 2 * y(x)^2 dx, x = lower limit, x = upper limit) / A

where A = ∫( y(x) dx, x = lower limit, x = upper limit)

Casio fx-CG 50 Program Code: CENTROID

Here is the code, which includes a graphic representation of y(x) and the location of the centroid. In this code, the Y is bold and it comes from the VARS menu. Do not merely use ALPHA+Y. For calculators with monochrome screens, such as the fx-9750G/fx-9860G series, leave out the color commands (Black, Blue, Red). The program assumes that there no preset plots or more than one function to be plotted.

This program works best for functions y(x) ≥ 0 for x ∈ [lower limit, upper limit].

Here is a text-only version:

Examples

Example 1: y = x^2 , lower limit = 0, upper limit = 1

X-Center = 3 / 4 = 0.75

Y-Center = 3 / 10 = 0.3

Example 2: y = 2 * cos( x / 2 ), lower limit = 0, upper limit = π

X-Center ≈ 1.141592654

Y-Center ≈ 0.7853981634

Example 3: y = 3, lower limit = 1, upper limit = 5

X-Center = 3

Y-Center = 1.5

Example 4: y = -4 * x^2 + 2 * x + 6, lower limit = -0.5, upper limit = 1

X-Center = 1 / 4

Y-Center = 307 / 110

Eddie

All original content copyright, © 2011-2024. 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.

Sum of Sequential Integers (featuring Swiss Micros DM32)

 Sum of Sequential Integers (featuring Swiss Micros DM32)

What is the sum of the following:

100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999?

It’s doable on a calculator, but going straight forward will require a lot of keystrokes (unless you have access to the sigma function (Σ)).

Note that:

100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999

= 100 + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)

= (100 + 0) + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)

Notice the sum starts with a base, 100. The sequence goes for 900 terms in the form of 100+n where n = 0 to 899. Let’s look at a general case.

Let x be a base integer, and S be the sum as:

S = (x + 0) + (x + 1) + (x + 2) + (x + 3) + …. + (x + n)

Rearranging the terms leads us to:

S = (x + x + x + x + … + x) + (0 + 1 + 2 + 3 + 4 + … + n)

There are n + 1 pairs:

S = (x * (n + 1)) + (0 + 1 + 2 + 3 + 4 + … + n)

S = (x * (n + 1)) + (1 + 2 + 3 + 4 + … + n)

The sum of Σ( k, k = 1 to k = n) = 1 + 2 + 3 + 4 + … + n = n * (n + 1) / 2

Then:

S = (x * (n + 1)) + n * (n + 1) / 2

Let’s look at some specific examples:

n = 1

S = (x + 0) + (x + 1)

S = (x + x) + (0 +1)

S = 2 * x + 1

n = 2

S = (x + 0) + (x + 1) + (x + 2)

S = (x + x + x) + (0 + 1 + 2)

S = 3 * x + 3

2 * 3 / 2 = 3

n = 3

S = (x + 0) + (x + 1) + (x + 2) + (x + 3)

S = (x + x + x + x) + (0 + 1 + 2 + 3)

S = 4 * x + 6

3 * 4 / 2 = 6

S = 500 + 501 +502 + 503

S = (500 + 0) + (500 + 1) + (500 + 2) + (500 + 3)

S = (500 + 500 + 500 + 500) + (0 + 1 + 2 + 3)

S = ((3 + 1) * 500) + (3 * 4 / 2)

S = 2000 + 6

S = 2006

S = 250 + 251 + 252 + 253 + … + 269 + 270

Base = 250

n = 270 – 250 = 20

Then:

S = 250 * (20 + 1) + 20 * 21 / 2

S = 5460

Let’s go our original problem:

S = 100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999

Base = 100

n = 999 -100 = 899

Then:

S = 100 * (899 + 1) + 899 * 900 / 2

S = 90000 + 404550

S = 494550

The program code calculates the sum:

Swiss Micros DM32 Program (also HP 32SII)

S01 LBL S

S02 INPUT X

S03 INPUT N

S04 RCL N

S05 1

S06 +

S07 RCL× N

S08 RCL N

S09 1

S10 RCL+ N

S11 ×

S12 2

S13 ÷

S14 +

S15 RTN

Eddie

All original content copyright, © 2011-2024. 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.

TI-30Xa Algorithm: Synthetic Division

TI-30Xa Algorithm:  Synthetic Division

Introduction

Synthetic division is a fairly simple division algorithm that divides a polynomial by the nominal (x-a):

p(x) / (x  - c) where

p(x) = a_n * x^n + a_n-1 * x^(n-1) + a_n-2 * x^(n - 2) + … + a_1 * x + a_0

c = a numerical constant

the result, q(x), is a polynomial of order n-1:

q(x) = b_n-1 * x^(n-1) + b_n-2 * x^(n - 2) + … + b_1 * x + b_0 + b_r / (x - c)

b_r = remainder term

where 

b_n-1 = a_n

b_i-1 = b_i  * c + a_i for i = n-2 down to r (“-1”, see the example below)  

For example, for the polynomial:

p(x) = a3 * x^3 + a2 * x^2 + a1 * x + a0

q(x) = p(x) / (x - c) 

p(x) has the order n = 3, so q(x) will have the order n = 3 - 1 = 2:

q(x) = b2 * x^2 + b1 * x + b0 + br / (x - c)

b2 = a3

b1 = b2 * c + a2

b0 = b1 * c + a1

br = b0 * c + a0  (the algorithm stops, this is the remainder term)

If br = 0, the x - c divides p(x) evenly, and x = c is a root of p(x).  

Note: for any “missing” terms, fill the term with 0.   

Example:  x^3 + 4 * x - 5 becomes x^3 + 0 * x^2 + 4 * x - 5.

Calculator Algorithm

1. Store c in one of the memory registers 1, 2, or 3.   We’ll call this memory register m for the purpose of the blog.

2. Note the first coefficient of q(x):  a_n

3. Compute the rest of the coefficients as follows:  [ × ] [ RCL ] m [ + ] a_ni [ = ]

Examples

Remember:  m is memory register 1, 2, or 3, your choice.  

Example 1:  (21 * x^2 + 42 * x + 144) / (x - 12) 

c = 12

a2 = 21

a1 = 42

a0 = 144

b1 = a2 = 21

12 [ STO ] m

Enter 21

b0:

[ × ] [ RCL ] m [ + ] 42 [ = ]

Result:  294

br:

[ × ] [ RCL ] m [ + ] 144 [ = ]

Result:  3672

Stop.

q(x) = 21 * x + 294 + 3672 / (x - 12)

Example 2:  (2 * x^3 + x - 3) / (x - 3) = (2 * x^3 + 0 * x^2 + x - 3) / (x - 3)

c = 3

a3 = 2

a2 = 0

a1 = 1

a0 = -3

b2 = a3 = 2

3 [ STO ] m

Enter 2

b1:

[ × ] [ RCL ] m [ + ] 0 [ = ]

Result:  6

b0:

[ × ] [ RCL ] m [ + ] 1 [ = ]

Result:  19

br:

[ × ] [ RCL ] m [ + ] 3 [ +/- ]  [ = ]

Result:  54

q(x) = 2 * x^2 + 6 * x + 19 + 54 / (x - 3)

Example 3:  (4 * x^3 + 8 * x^2 - 5 * x + 3) / (x + 2)

c = -2

a3 = 4

a2 = 8

a1 = -5

a0 = 3

b2 = a3 = 4

2 [ +/- ] [ STO ] m 

Enter 4

b2:

[ × ] [ RCL ] m [ + ] 8 [ = ]

Result:  0

b3:

[ × ] [ RCL ] m [ + ] 5 [+/-] [ = ]

Result:  -5

br:

[ × ] [ RCL ] m [ + ] 3 [ = ]

Result:  13

q(x) = 4 * x^2 - 5 + 13 / (x + 2)

I think this is a fundamental algorithm for students and math enthusiasts to learn, and it’s fairly simple to get a hang of it. 

Until next time,

Eddie

Quick update: Starting October 5, 2024, my schedule will allow me to blog once a week. Regular posts will go live every Saturday. Thank you for your support and compliments. I wish you all well.

All original content copyright, © 2011-2024. 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.