Plus42 Solver: Derivatives

Plus42 Solver: Derivatives 

Introduction

The Technical Applications book for the HP 27S and HP 19B (and can apply to the HP 17B outside of trigonometry) shows the numerical first and second derivative can be calculated by the formulas:

f ' (x) = (f(x+h) - f(x-h)) / (2 * h)

f ' ' (x) = (f(x + h) - 2 * f(x) + f(x + h)) / h^2 

where h is sufficiently small, like 10^-5 to 10^-12.

Legacy Formulas vs. Plus42 Formulas 

The formulas suggested by the Technical Applications Book are:

First Derivative:

Second Derivative:

Depending on the function FX, this above can turn the above into long equations.  With the ability of user functions, this allows us to use the original definitions.

FX(X): f(x) (insert f(x)

First Derivative:

F'X=(FX(X+H)-FX(X-H))÷(2×H)

Second Derivative:

F''X=(FX(X+H)-2×FX(X)+FX(X-H))÷SQ(H)

SQ:  press by the key sequence [(shift)] (x^2)

':  (ALPHA) [ ↓ ] (PUNC) [ ↓ ] ( ' )

Note:  Radians mode 

Examples 

FIX 5 mode is set.

Example 1:

f(x) = 0.5 * cos(3*x)

x = π/4

FX(X):0.5×COS(3×X)

f'(x) ≈ -1.06066

f''(x) ≈ 3.18198

Example 2:

f(x) = (x^2 + 3*x + 5) / (4*x - 1)

x = 2

FX(X):(X^2+3×X+5)÷(4×X-1)

f'(x) ≈ -0.22449

f''(x) ≈ 0.54227

Functions with Variable Constants

It is easy to expand the user function FX to include variable constants.  For example:

f(x) = -ln(cos(√(a*x)))

Calculate the value and first derivative at x = 0.11 and a = 0.46

Attach variable constants at the end of FX:

FX(X:A):-LN(COS(SQRT(A×X)))

F'X=(FX(X+H:A)-FX(X-H:A))÷(2×H)

f(x:a) ≈ 0.02552

f'(x:a) ≈ 0.23396

The user function makes the calculating numerical derivatives easier.  

Source:

Technical Applications: Step-by-Step Solutions for Your HP-27S or HP-19B Calculator Hewlett Packard.   Edition 2.  Corvallis, OR.   November 1988.  pg. 44

Have any Halloween plans?  Wishing you a great day,

Eddie 

All original content copyright, © 2011-2022.  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 DM41X and HP 41C: Numeric Derivatives

Swiss Micros DM41X and HP 41C:  Numeric Derivatives

Introduction

The program DFX calculates one of three types of derivative:

1.  Normal Derivative   (DX)

2.  Logarithmic Derivative (LN)

3.  Power Derivative (PWR)

Where:

Normal Derivative:  f'(x)

Logarithmic Derivative:  d/dx( ln f(x) ) = f'(x) / f(x)  

Power Derivative:  d/dx( f^n(x) ) = n * f^(n-1)(x) * f'(x),  n doesn't have to be an integer

where f'(x) is estimated by:

f'(x) ≈ ( f(x + h) - f(x) ) / h

Variables:  x, h

The program uses a subroutine FX, see the examples for details.  

DM41X and HP 41C Program:  DFX

Uses program FX as a subroutine as f(x).

01 LBL ^T FX

02 RCL 01

03 ^T X?

04 ARCL 01

05 PROMPT

06 STO 01

07 RCL 02

08 ^T H?

09 ARCL 02

10 PROMPT

11 STO 02

12 RCL 01

13 XEQ ^T FX

14 RCL 03

15 RCL 01

16 RCL 02

17 +

18 XEQ ^T FX

19 RCL 03

20 -

21 RCL 02

22 /

23 STO 04

24 ^T 1 DX 2 LN 3 ↑N

25 PROMPT

26 INT

27 STO 05

28 GTO IND 05

29 LBL 01

30 RCL 04 

31 LBL 02

32 RCL 04

33 RCL 03

34 / 

35 GTO 04

36 LBL 03

37 RCL 03

38 ^T N?

39 PROMPT

40 1

41  - 

42 Y↑X

43 LASTX

44 1

43 +

44 *

45 RCL 04

46 *

50 LBL 04

51 RTN

The function FX:

x is loaded in the X stack register (and on display)

01 LBL ^FX

02  execute f(x)

...

##  RTN

For DFX, do not use R01, R02, R03, R04, and R05 in FX.  

Notes:

Sequences such as:

02 RCL 01

03 ^T X?

04 ARCL 01

05 PROMPT

Puts the prompt as X? [contents of R1].   If you want the previous value, press R/S.  Otherwise enter a new value, then press R/S.

This program uses indirect goto statements.  R05 is used to hold the person's choice and uses it to direct which label is executed.

Examples

Example 1:  f(x) = x * sin x

Set FX as:

01 LBL^T FX

02 RAD

03 ENTER

04 ENTER

05 SIN

06 * 

07 RTN

Setting H to 10^-6  (1E-6):

DF:   f'(x)

x = 0.5, Result:  0.9182; x = 1.6, Result: 0.9530

LN:  ln f'(x)

x = 0.5, Result:  3.8305; x = 1.6, Result:  0.5959

PWR: with n = 3,   (f'(x))^3

x = 0.5; Result:  0.1583; x = 1.6;  Result: 7.3128

Example 2:  f(x) = x^2 + 3 * x + 1 = x * (x + 3) + 1

Set FX as:

01 LBL^T FX

02 ENTER

03 ENTER

04 3

05 +

06 *

07 1

08 + 

09 RTN

Setting H to 10^-6  (1E-6):

DF:   f'(x)

x = 3.2, Result:  9.4000; x = 6.8, Result: 16.6000

LN:  ln f'(x)

x = 3.2, Result:  0.4511; x = 6.8, Result: 0.2454

PWR: with n = 1.5,   (f'(x))^1.5

x = 3.2, Result:  64.3677; x = 6.8, Result: 204.7864

Until next time, 

Eddie

All original content copyright, © 2011-2022.  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. 

HP 17B and HP 27S: Derivatives to the Nth Order (and TI-84 Plus CE Python 5.7 update)

 HP 17B and HP 27S: Derivatives to the Nth Order

With the Solver of the HP 17B family, HP 27S, and the HP 19B calculator, we can calculate derivatives of any order.  Four derivatives are presented here.  You can use any name you want other than those presented here.  Each derivative is to the kth order. 

Derivative 1:   d^k/dx^k a × x^n 

N and K must be positive integers, D is the value of the derivative  

DER1: D=A×X^(N-K)×PERM(N:K)

Example:

Input:  N = 2, K = 1, A = 3, X = 1.5

Result:  D = 9

Derivative 2:   d^k/dx^k e^(a × x) 

K must be a positive integer, D is the value of the derivative

DER2: D=A^K×EXP(A×X)

Example:

Input:  A = 1.8, K =3, X = 3

Result:  D = 213.4409

Derivatives 3 and 4 will require trigonometric functions, which are not available on the HP 17B family.   It is recommended you set the calculator to Radian angle mode.

Derivative 3:  d^k/dx^k sin(a × x)

DER3: D=IF(MOD(K:2)=0:(-1)^(K÷2)×A^K×SIN(A×X):(-1)^((K+3)÷2)×A^K×COS(A×X))

Examples:

Input:  A = 0.75, X = 0.66, K = 2

Result: D = -0.2672

Input:  A = 0.75, X = 0.66, K = 3

Result: D = -0.3712

Derivative 4:  d^k/dx^k cos(a × x)

DER4: D=IF(MOD(K:2)=0:(-1)^(K÷2)×A^K×COS(A×X):(-1)^(K÷2+1÷2)×A^K×SIN(A×X))

Examples:

Input:  A = 0.75, X = 0.66, K = 2

Result: D = -0.4950

Input:  A = 0.75, X = 0.66, K = 3

Result: D = 0.2004

Eddie

All original content copyright, © 2011-2021.  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. 

Derivatives with Surprisingly Imaginary Results

Derivatives with Surprisingly Imaginary Results

Here are three derivatives of functions where complex numbers are involved with further algebraic simplification.  

d/dx √(a - x)^(1/2)

d/dx √(a- x)^(1/2)

= 1/2 ∙ (a - x)^(-1/2) ∙ -1

= -1/2 ∙ 1 ÷ (√(a - x))

Going a step further...

= -1/2 ∙ 1 ÷ (√(-1) ∙ √(x - a))

With √(-1) = i ,  1/i = -i

= i ÷ (2 ∙ √(x - a))

d/dx  arcsin(x + a)

d/dx arcsin(x + a)

= 1 ÷ √(1 - (x + a)^2)

= 1 ÷ √(1 - (x^2 + 2 ∙ a ∙ x + a^2))

= 1 ÷ √(-x^2 - 2 ∙ a ∙ x + 1 - a^2)

Factoring out -1 in the denominator: 

= 1 ÷ √((-1) ∙ (x^2 + 2 ∙ a ∙ x - 1 + a^2))

= 1 ÷ (i ∙ √(x^2 + 2 ∙ a ∙ x - 1 + a^2))

= -i ÷ √(x^2 + 2 ∙ a ∙ x - 1 + a^2)

d/dx e^(√(a - x)) 

d/dx e^(√(a - x)) 

= e^(√(a - x)) ∙ d/dx √(a - x)

= -e^(√(a - x)) ÷ (2 ∙ √(a - x))

With:  √(a - x) = i ∙ √(x - a) and e^(i ∙ Θ) = cos Θ + i ∙ sin Θ

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

=  i ∙ e^(i ∙ √(x - a)) ÷ (2 ∙ √(x - a))

= (i ∙ (cos √(x - a) + i ∙ sin √(x - a)) ÷ (2 ∙ √(x - a))

= (-sin √(x - a) + i ∙cos √(x - a)) ÷ (2 ∙ √(x - a))

Eddie

All original content copyright, © 2011-2021.  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. 

Book Review: Calculus for Middle Schoolers by Serena Swegle

 Book Review: Calculus for Middle Schoolers by Serena Swegle

Just The Facts

Calculus for Middle Schoolers

Author:  Serena Swegle

Publisher:  Sunhut Publishing

Cost:  $26.50 for Paperback, $9.99 for Kindle (as of 10/23/2020)

Link:  https://www.amazon.com/Calculus-Middle-Schoolers-Serena-Swegle/dp/057871275X/ref=sr_1_3?dchild=1&keywords=Calculus+for+Middle+Schoolers&qid=1603471676&sr=8-3

Topics Covered

The number e (2.718281828...)

The common logarithm  (base 10)

The natural logarithm (base e)

Trig Functions (sine, cosine, tangent)

Sums 

Limits

Derivative - the derivative of a polynomial

Integral - the integral of a polynomial

The Derivative and Integral of e^x

An Introduction to Calculus 

The target audience is middle school students.  However, book serves as a great introduction to calculus for high school and college students who are taking calculus for the first time.  The book gives a simple, concrete introduction to various subjects, in an easy-to-read narrative.  Calculus is a complex subject, and this book allows readers, who may be intimidated about the subject, to develop a understanding.   

I would recommend this book to be read prior to the student's first calculus class.   The book can be read in one or two days, but I feel it was meant to read as one chapter a time per day or week.   

Verdict

Swegle's book is well written, in a concise language.  The chapter covers one concept at the time, which serves as a great introduction to a rich subject.   The examples are simple and apply closely to the text.   I wish Swegle put a summary of all the topics covered at the end of the book as a wrap up.  Otherwise, I recommend this book for educators and parents.   For those who have the Kindle app, $9.99 is a good price point.  Recommended.

Thank you, Serena for recommending this book for me to review.  

Eddie

All original content copyright, © 2011-2020.  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. 

Calculus Revisited #5: Derivatives

Welcome to entry #5 of 21 in our Calculus Revisited series. Today, we tackle derivatives!

First the formal definition.

Derivative:

df/dx = f'(x) =

lim ( f(x + δx) - f(x) ) / δx
δx → 0

Note that both symbols for derivatives, the quotient-like symbol and the use of the prime symbol are used interchangeably.

Use the derivative to:
1. Find the instantaneous rate of change at point x0 using the function f(x).
2. Find the slope at given point.
3. When given an equation dealing with the position of an object, f(t), you can find the velocity of an object by calculating f'(t).

But we are going to jump into doing the derivatives. Here is a basic of derivatives:

d/dx ( f(x) + g(x) ) = f'(x) + g'(x)
d/dx ( f(x) * g(x) ) = f'(x) * g(x) + f(x) * g'(x)
d/dx ( f(x) / g(x) ) = (g(x) * f'(x) - g'(x) * f(x))/(g(x)^2)
d/dx a = 0 (a is a constant)
d/dx x = 1
d/dx x^n = n * x^(n-1)
d/dx sin x = cos x
d/dx cos x = -sin x
d/dx tan x = sec^2 x
d/dx e^x = e^x
d/dx a^x = a^x ln a (a is a constant)
d/dx ln x = 1/x
d/dx asin x = 1/√ (1 - x^2)
d/dx acos x = -1/√ (1 - x^2)
d/dx atan x = 1/(x^2 + 1)

TIP: For d/dx ( f(x) / g(x) ), let N = f(x) (numerator) and D = g(x) (denominator). Then d/dx N / D = ( D * N' - D' * N)/(D^2)

Higher Order Derivative: Repeat the derivative operation on f(x).

f''(x) = d^2/dx^2 is the second derivative: take the derivative of f(x), twice.
f'''(x) = d^3/dx^3 is the third derivative: take the derivative of f(x), thrice (three times).
f^(n)(x) = d^n/dx^n is the nth derivative: take the derivative of f(x) n times.

Problems
1. Find the derivative of f(x) = x^2 + 2x + 3.

We can use the addition property to help us.

d/dx (x^2 + 2x + 3)
= d/dx (x^2) + d/dx (2x) + d/dx (3)
= 2x + 2 + 0
= 2x + 2

Remember, the derivative of a constant is 0.

2. Find the slope of f(x) = -x^5 + 2x - 1 at x = 1

First find the derivative
d/dx (-x^5 + 2x - 1)
= d/dx (-x^5) + d/dx(2x) - d/dx(1)
= -5x^4 + 2 - 0
= -5x^4 + 2
The slope is determined by calculating f'(1).
f'(1) = -5(1)^4 + 2 = -3

The slope of f(x) at x = 1 is -3.

3. Find the derivative of f(x) = x^2 * ln x.

Use the product rule: x^2 multiplied by ln x.

d/dx (x^2 * ln x)
= d/dx (x^2) * ln x + x^2 * d/dx (ln x)
= 2x * ln x + x^2 * 1/x
= 2x * ln x + x

4. Find the derivative of f(x) = (x^2 -1)/(x + 2)

Use the division rule: numerator is x^2 -1 and denominator is x + 2.

d/dx ((x^2 - 1)/(x + 2))
= ((x + 2) * d/dx(x^2 - 1) - (x^2 - 1) * d/dx(x + 2))/(x + 2)^2
= ((x + 2) * 2x - (x^2 - 1) * 1)/(x + 2)^2
= (2x^2 + 4x - x^2 + 1)/(x + 2)^2
= (x^2 + 4x + 1)/(x + 2)^2

5. Higher order derivatives: Find the first, second, and third derivative of f(x) = 3x^6 + 7x and g(x) = sin x respectively.


f(x) = 3x^6 + 7x
First Derivative
f'(x) = 18x^5 + 7
Second Derivative
f''(x) = 90x^4
Third Derivative
f'''(x) = 360x^3

g(x) = sin x
g'(x) = cos x
g''(x) = -sin x
g'''(x) = -cos x

6. An interesting property: d/dx( asin x + acos x)

d/dx(asin x + acos x)
= d/dx asin x + d/dx acos x
= 1/√(1-x^2) + (-1)/√(1-x^2)
= 0

This implies that the function asin x + acos x for all x. Bonus question: what is that constant?

Thanks for joining us. Next time we will work with an important rule in calculus: The Chain Rule.

This blog is property of Edward Shore. © 2012

By the way, the answer to the bonus question: asin x + acos x = π/2