Casio fx-991CW: Second Derivative

Casio fx-991CW: Second Derivative

Calculating Higher Order Derivatives

A general formula to estimate derivatives of order n, where n is a positive integer (see Sources, Wikipedia):

d^n/dx^n = lim h→0 (1÷h^n * Σ((-1)^(k+n) * comb(n,k) * f(x+k*h), k=0, n)

The first, second, and third derivatives are derived from the above formula like so:

n = 1:

d/dx

= lim h→0 (1÷h * Σ((-1)^(k+1) * comb(1,k) * f(x+k*h), k=0 to 1)

= lim h→0 (1÷h * ((-1)^(0+1)*f(x) + (-1)^(1+1)*f(x+h))

= lim h→0 (1÷h * (-f(x) + f(x+h))

= lim h→0 (f(x+h) - f(x)) ÷ h

This is the famous forward difference formula.

n = 2:

d^2/dx^2

= lim h→0 (1÷h^2 * Σ((-1)^(k+2) * comb(2,k) * f(x+k*h), k=0 to 2)

= lim h→0 (1÷h^2 * ((-1)^2*comb(2,0)*f(x) + (-1)^3*comb(2,1)* f(x+h) + (-1)^4*comb(2,2)*f(x+2*h))

= lim h→0 (f(x) - 2*f(x+h) + f(x+2*h)) ÷ h^2

n = 3:

d^3/x^3

= lim h→0 (1÷h^3 * Σ((-1)^(k+3 * comb(3k) * f(x+k*h), k=0 to 3)

= lim h→0 (1÷h^3 * ((-1)^3*comb(3,0)*f(x) + (-1)^4*comb(3,1)*f(x+h) + (-1)^5*comb(3,2)*f(x+2*h)

+ (-1)^6*comb(3,3)*f(x+3*h))

= lim h→0 (-f(x) + 3*f(x+h) - 3*f(x+2*h) + f(x+3*h)) ÷ h^3

Using the fx-991CW

- - - - - - - - - -

Start with a note: Commentary and limitations: The functions f(x) and g(x), along with the calculus functions sum (Σ), integral (∫), and derivative (d/dx), has x as variable. It makes it a challenge that x is the only variable the functions f and g can take.

Example:

f(x)=x^2

g(x)=Σ(f(x),x=0 to 5)

Executing g(x) will take the values x=0 through x=5 no matter what value we put for g. The answer, in this example, will always return 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.

- - - - - - - - - -

We can still use f and g to set up specific derivatives in order n. For the second dervative, set up f and g:

f(x) = <function in terms of x>

g(x) = (f(x+2×A)-2×f(x+A)+f(x))÷A²

Store h in A. We can also choose an h and write in the formula directly.

Remember, this will calculate an approximation. With the most appropriate settings for h, we can get the best approximation.

Examples

For all the examples, A is set as 10^-7. The calculator is set to Radians. g(x) is the second derivative approximation.

Example 1:

f(x) = sin(x), f''(x) = -sin(x)

g(x) = (f(x+2×A)-2×f(x+A)+f(x))÷A²

x = 0.6; g(x): -0.564642551 (actual: -0.5646424734)

x = 2.8; g(x): -0.334988057 (actual: -0.3349881502)

Example 2:

f(x) = 2×e^(0.3×x), f''(x) = 0.18×e^(0.3×x)

g(x) = (f(x+2×A)-2×f(x+A)+f(x))÷A²

x = 0; g(x): 9/50 = 0.18 (actual: 0.18)

x = 1.2; g(x): 0.25799928 (actual: 0.2579992946)

Example 3:

f(x) = 1.1×x^3, f''(x) = 6.6×x

g(x) = (f(x+2×A)-2×f(x+A)+f(x))÷A²

x = 0; g(x): 6.6×10^-7 (actual: 0)

x = 2.2; g(x): 14.52 (actual: 14.52)

Sources

McCarty, George. Calculator Calculus. EduCALC Publications. E. & F.N. Spon: London. 1975, ISBN 0- 419-12910-3

Wikipedia "Numeric differentiation" Wikimedia Foundation, Inc. Last Edited June 17, 2025. Last Accessed July 7, 2025. https://en.wikipedia.org/wiki/Numerical_differentiation

Eddie

All original content copyright, © 2011-2025. 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.

The author does not use AI engines and never will.

fx-991 CW: Gamma Function

fx-991 CW: Gamma Function

All screenshots were made with Casio’s classpad.net website.

No Gamma Function? No Problem!

The gamma function is used a lot in advanced mathematics. The gamma functions with a lot of definition functions, but the most common one is for values t>0 (in particular Re(t)>0):

Γ(t) = ∫( x^(t-1) × e^(-x) dx, 0, ∞)

This integral is an improper integral and unless we have a calculator that handles infinite limit, we need to use the following:

Γ(t) =

lim ∫( x^(t-1) × e^(-x) dx, 0, w)

w → ∞

Calculators with the integral function can use the above for estimating the gamma function.

Use Basic Properties for Shortcuts

If t is a positive integer, we can use the factorial function:

Γ(t) = (t – 1)!

Example: Γ(16) = (16 – 1)! = 15! ≈ 1.31 × 10^12

If t is in the form n/2 where n is odd (i.e. 1/2 = 0.5, 3/2 = 1.5, 5/2 = 2.5, 7/2 = 3.5, etc.). we can use the product..

Γ(n / 2) = (n – 2) / 2 × (n – 4) / 2 × (n – 6) / 2 × … × 1 / 2 × √π

Example: Γ(3.5) = Γ(7 / 2) = 5 / 2 × 3 / 2 × 1 / 2 × √π = 15 / 8 × √π ≈ 3.32335097

The following pictures demonstrate the use of the above equivalency along with integral estimate:

I hope you find this helpful,

Eddie

 All original content copyright, © 2011-2025. 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.

The author does not use AI engines and never will.

Casio fx-991CW: Arc of a Quadratic Curve

Casio fx-991CW: Arc of a Quadratic Curve

All screen shots will be taken with the https://www.classpad.app/ website.

Introduction

We are given three points: (x1, y1), (x2, y2), and (x3, y3) which are connected to a quadratic curve:

y(x) = a + b * x + c * x^2

with it’s derivative:

y’(x) = b + 2 * c * x

We assume that all three x-values are unique, therefore x1 ≠ x2, x2 ≠ x3, and x1 ≠ x3.

The arc length is the integral: ∫( √(1 + (b + 2 * c * x)^2 dx, min(x), max(x))

In curve fitting, if we only have three points, the quadratic curve will fit all three points.

Procedure

We can do the entire calculation in the Statistics app of the fx-991CW.

1. Go to the Statistics app by pressing [ Home ], selecting Statistics.

2. Enter the three points. If you need to, you can clear the lists by selecting Tools > Edit > Delete All.

3. Press [OK] (or [EXE] ), select Statistics Calc, y = a + b * x + c * x^2, and then press [OK] (or [EXE]) again. The Statistics Calc will allow us to make calculations using the statistics variables.

4. Calculate the integral:

∫ ( √(1 + (b + 2 * c * x)^2), min(x), max(x))

∫: Catalog > Func Analysis > Integration

b: Catalog > Statistics > Regression > b

c: Catalog > Statistics > Regression > c

min(x): Catalog > Statistics > Regression > Min/Max > min(x)

max(x): Catalog > Statistics > Regression > Min/Max > min(x)

Let’s walk through an example:

Points: (0,0), (5,4), (7,2)

Arc length: 9.97139451

You can see the procedure through the screen shots below:

Quadratic Curve:

Eddie

All original content copyright, © 2011-2025. 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.

fx-991CW: Sums with a Step Not Equal to One

fx-991CW: Sums with a Step Not Equal to One

Step ≠ 1? An Adjustment is Necessary

The summation function on the Casio fx-991 CW calculator, and a lot of other advanced scientific calculators, calculates the sum of f(x) from x = a to x = b, with an increment of 1.

But what if have a sum of following:

log(2) + log(4) + log(6) + log(8): sum(f(x) from x = 2 to x = 8, step = 2)

Clearly, the increment is not 1, but 2 instead. Believe it or not, we can still use the summation function. We will need to adjust both the function and the limits to fit the requirements of the summation function.

1. Adjust f(x) to f(step * x)

2. Adjust the limits to x = a ÷ step and x = b ÷ step

For this particular problem:

1. f(x) becomes log(2 * x)

2. The limits become x = 2 ÷ 2 = 1 to x = 8 ÷ 2 = 4 (this allows the step to be 1)

Which transforms:

sum(log(x) from x = 2 to x = 8, step = 2)

to:

Σ(log(2 * x) from x = 1 to x = 4)

Sum: 2.584331224

A Couple More Examples

Example 1:

sum(x^2 from x = 0.5 to x = 2, step = 0.5), f(x) = x^2

The key here is the step (also known as the increment). The adjustments needed are:

1. f(x) becomes (0.5 * x)^2 = (x / 2)^2

2. The limits become x = 0.5/0.5 = 1 to x = 2/0.5 = 4

Sum: 15/2 = 7.5

Example 2:

sum(e^(x) from x = 1.5 to x = 2.1, step = 0.1), f(x) = e^(x)

The adjustments needed are:

1. f(x) becomes e^(0.1 * x) = e^(x / 10)

2. The limits become x = 1.5/0.1 = 15 to x = 2.1/0.1 = 21

Sum: 43.1994368

In Summary…

For the sum:

sum(f(x), x = a to x = b, step = Δx):

1. Adjust f(x) to f(Δx * x)

2. Adjust the limits: x = a/Δx to x = b/Δx

3. Summation function looks like this: Σ( f(Δx * x), x = a/Δx to x = b/Δx)

Hope you find this useful! Take care,

Eddie

All original content copyright, © 2011-2025. 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.

The author does not use AI engines and never will.

Casio fx-991CW: Generating Graphs

Casio fx-991CW: Generating Graphs

But Wait, the fx-991CW is Not a Graphing Calculator!

Technically, this is correct. We cannot generate graphs of the Casio fx-991CW calculator itself. But through the magic of the QR code, we can use that to generate graphs that appear on our smart phones.

What is needed:

(1) QR Reader. It does not have to be the official Casio EDU+ app, any QR reader app should do.

(2) A smart phone or device that can read QR codes.

(3) Your fx-991CW calculator. I believe that these instructions can apply to the fx-991EX as well, however, the instructions presented today are specific to the fx-991CW calculator.

The fx-991CW has storage for up to two functions: f(x) and g(x).

Steps

Step 1: Determine the number of functions you want to graph. You may either graph f(x) alone, g(x) alone, or both f(x) and g(x). To define functions, press the [FUNCTION] button.

Step 2: Press the [HOME] button, and select the Table app.

Step 3: Choose the number of functions to be evaluated. Press the [ TOOLS ] button, select Table Type, press [ → ], and select from f(x)/g(x), f(x), or g(x). Remember that contents of f(x) and g(x) are not retained when the calculator is turned off.

Step 4: Now we need to give the range. Press the [ TOOLS ] button, select Table Range, and then press [ → ]. Give a start value, an end value, and step. In terms of the graph, the value of step isn’t that important. Think of the start value as Xmin (minimum x value) and the end value as Xmax (maximum x value). Once you are satisfied, scroll down to Execute and either press [ EXE ] or [ OK ]. A table is generated.

Step 5: Next, while the table is shown, generate a QR code by pressing [ SHIFT ] [ x ] (QR).

Step 6: Grab your smart phone or device with your QR reader app open. Scan the QR code. A successful scan will give you a link to the classpad.net website. Some camera apps will read QR codes and give direct access to the link as well, neat!

Step 7: Click on the link and you will see the graph generated. At this point, you can clear the QR code off the calculator by pressing either [ EXE ] or the back key.

I took the following pictures with my smart phone. It’s a challenge to get pictures in a room in a house with two dogs and three cats…

Defining the function

Setting the table type.   Since I'm graphing just f(x), I'm selecting f(x) only for the table type.

Selecting Table Range

xmin =1, xmax = 10

A table is generated

QR code

The graph is generated!

Eddie

All original content copyright, © 2011-2025. 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.

All posts are 100% generated by human effort.  The author does not use AI engines and never will.

Casio fx-991 CW: Scientific Constants in Calculations

 Casio fx-991 CW: Scientific Constants in Calculations

Obtaining the Scientific Constants

I believe we can do this in every mode.

1. Press the [ CATALOG ] button.

2. Scroll to the Sci Constants submenu. It’s faster to scroll up in this case.

3. Press the right button [ → ] to get to the categories.

4. Select the category and press the right button again.

5. Select the constant and press [ OK ].

We can select the back button, the one with the curved arrow, to go back to the previous menu.

I wish the fx-991CW had a custom menu where we could store constants and other commands so we could reduce the number of keystrokes for common constants and commands.

Table of Common Scientific Constants

All scientific constants in the fx-991CW use SI units (meters, kilograms, seconds, Kelvin/Celsius, etc.). I believe the 2018 CODATA values are used, at least with the calculator I bought three years ago in 2022.

Name	Symbol	Category	Value (Norm 1)
Vacuum Permeability (Magnetic Constant)	μ₀	Universal	1.256637062 × 10^(-6) N/A² Exact: (4π × 10^(-7))
Universal Gravitational Constant	G	Universal	6.6743 × 10^(-11) N m²/kg²
Vacuum Permittivity (Electric Constant)	ε₀	Universal	8.854187813 × 10^(-12) F/m
Speed of Light (in a vacuum)	c	Universal	299,792,458 m/s
Planck’s Constant	h	Universal	6.62607015 × 10^(-34) J s
Earth’s Gravity Constant	gₙ (note the n subscript)	Adopted Values	9.80665 m/s²
Universal Gas Constant	R	Physico-Chem	8.314462618 J/(mol K)
Avogadro Constant	Nₐ (the a is capitalized)	Physico-Chem	6.02214076 × 10^23 mol⁻¹
Boltzmann Constant	k	Physico-Chem	1.380649 × 10^(-23) J/K
Stefan-Boltzmann Constant (Constant of Proportionality)	σ	Physico-Chem	5.670374419 × 10^(-8) W/(m² K⁴)
Elementary Charge	e	Electromagnetic	1.602176634 × 10^(-19) C
Mass of an Electron	me	Atomic&Nuclear	9.109383702 × 10^(-31) kg
Volume of a Mole at 0° C	Vm	Physico-Chem	0.02271095464 m³/mol
Atomic Mass Constant (1/12 of mass of a neutral carbon-12 atom)	m_u (the u is a subscript, classically marked u)	Physico-Chem	1.660539067 × 10^(-27) kg

A full list of the 47 constants can be found here:

https://support.casio.com/global/en/calc/manual/fx-570CW_991CW_en/advanced_calculations/scientific_constants.html

Note the manual only lists the symbol.

Examples

Scientific Constants are highlighted in dark blue.

Lorenz Factor

γ = 1 / √(1 – v^2/c^2) (Universal)

γ = time dilation factor

v = 150 × 10^6 m/s, put the entire quantity in parenthesis like this (150×10^6)

γ ≈ 1.154967184

Escape Velocity

v = √(2 * G * m / r) (Universal)

v = velocity required to escape the planet

Radius: r = 7.2 × 10^6 m, again with the ×10^ button, put the entire number in parenthesis (7.2×10^6)

Mass: m = 5.9 × 10^24 kg

v ≈ 10458.69787 m/s

Coulomb’s Law

F = (q1 * q2) / (4 * π * ε₀ * r^2) (Universal)

F = force between two charges

Charge # 1: q1 = 4 × 10^(-5) C

Charge # 2: q2 = 3.3 × 10^(-5) C

Radius: r = 0.24 m (24 cm)

F = 205.9647286 N

Source: Casio fx-1000F/fx-5000F Manual

Solenoid Inductance

L = μ₀ × μr × n^2 × A × h (Universal)

L = inductance in micro henrys (mH)

Relative Permeability: μr = 2.5 (unit-less)

Number of Turns: n = 30/cm

Area: A = 0.2 cm^2

Length: h = 3.5 cm (note the symbol)

L = 0.00197920337 mH

Source: HP 50/49g+/48gII graphing calculator: advanced user’s reference manual

Simple Pendulum

F = -(m * gₙ * x) / l (Adopted Values)

F = force of a pendulum at point x

Position to be analyzed: x = 1 m

Length: l = 1.5 m

Mass: m = 10 lb ≈ 4.535924 kg

F = -29.65481273 N

Source: Casio fx-1000F/fx-5000F Manual

I hope you find this useful, and enjoy the physical constants library of the fx-991CW. Until next time,

Eddie

All original content copyright, © 2011-2025. 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.