[48] how to simplify derivative?

07152023, 09:11 PM
(This post was last modified: 07152023 10:41 PM by grbrum.)
Post: #1




[48] how to simplify derivative?
I need help to simplify derivative of sin(x)/(1+cos(x))
Try it out. 

07162023, 08:53 AM
Post: #2




RE: [48] how to simplify derivative?
I assume that you are familiar with calculating the derivative on the HP48G:
\( \begin{matrix} \frac{\mathrm{d} }{\mathrm{d} x} \frac{\sin(x))}{1 + \cos(x)} = \frac{\cos(x)}{1 + \cos(x)}\frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)^2} \\ \end{matrix} \) We have to cheat a bit and use the following identities: \( \begin{matrix} \sin(x)^2 + \cos(x)^2 &=& 1 \\ \left(1 + \cos(x)\right)^2 &=& \left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right) \end{matrix} \) Therefore we put everything on the stack for the later substitutions: 3: 1 2: '(1+COS(X))*(1+COS(X))' 1: 'COS(X)/(1+COS(X))SIN(X)*SIN(X)/(1+COS(X))^2' Now we can enter the equation editor by pressing the ▼ key. Select the expression \(\left(1 + \cos(x)\right)^2\) and use REPL to replace it with \(\left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right)\). Select this whole expression: \( \begin{matrix} \frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right)} \end{matrix} \) Use ←A to associate left which leaves us with: \( \begin{matrix} \frac{\cos(x)}{1 + \cos(x)}\frac{\frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)}}{\left(1 + \cos(x)\right)} \end{matrix} \) Use ←→ to commute the arguments: \( \begin{matrix} \frac{\cos(x)}{1 + \cos(x)}\text{inv}\left(1 + \cos(x)\right) \frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) We do the same on the left expression as well: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \cos(x) \text{inv}\left(1 + \cos(x)\right) \cdot \frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) This allows using ←M to mergefactorsleft: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \left(\cos(x)  \frac{\sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)}\right) \end{matrix} \) Now we can use AF to add the fractions: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{\left(\left(1 + \cos(x)\right) \cdot \cos(x)  \sin(x) \left(\sin(x)\right)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) We use ←D to distributeleft the expression: \(\left(1 + \cos(x)\right) \cdot \cos(x)\) to end up with: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{1 \cdot \cos(x) + \cos(x) \cdot \cos(x)  \sin(x) \left(\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) With COLCT we can clean that up a bit: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{\cos(x)^2 + \sin(x)^2 + \cos(x)}{\left(1 + \cos(x)\right)} \end{matrix} \) Now we substitute \(\cos(x)^2 + \sin(x)^2\) by \(1\) using the REPL command again: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{1 + \cos(x)}{\left(1 + \cos(x)\right)} \end{matrix} \) We can now use COLCT on the fraction and then on the whole expression and end up with: \( \begin{matrix} \frac{1}{1 + \cos(x)} \end{matrix} \) The result is then: 1: '1/(1+COS(X))' Conclusion The equation writer is severely limited:
This makes it close to useless. 

07162023, 10:46 AM
(This post was last modified: 07162023 10:46 AM by Gilles.)
Post: #3




RE: [48] how to simplify derivative?
(07162023 08:53 AM)Thomas Klemm Wrote: The equation writer is severely limited: Hi, with the 4950g, its easy, you select the portion of the equition that you want manipulate and then : 1/ expand \((a+b)^2\) to \((a+b)(a+b)\) In the equation writer juste push soft menu FACTO 2/[*] it doesn't know about trigonometric identities In the EQW choose Right Shift TRIG and then there are a lot of possibilité (TCOLLECT, TEXPAND etc.) 

07162023, 09:57 PM
Post: #4




RE: [48] how to simplify derivative?
Thank you. I love your explanation. I will study it carefully. I guess the 48 can’t see the sin(x)2+cos(x)2 =1. And the answer is super crazy.
Thank you. (07162023 08:53 AM)Thomas Klemm Wrote: I assume that you are familiar with calculating the derivative on the HP48G: 

07172023, 07:44 AM
(This post was last modified: 07172023 07:45 AM by Gilles.)
Post: #5




RE: [48] how to simplify derivative?
There is no native CAS with the 48 serie (4950 got it). I never used it, but I think that if you want to do this kind of thing on a 48, you can use erable library


07182023, 07:54 PM
Post: #6




RE: [48] how to simplify derivative?  
07182023, 08:35 PM
Post: #7




RE: [48] how to simplify derivative?
(07182023 07:54 PM)BruceH Wrote:(07162023 08:53 AM)Thomas Klemm Wrote: … the hard way! Bruno Sanyo CZ0124 ⋅ TI57 ⋅ HP15C ⋅ Canon X07 + XP140 Monitor Card ⋅ HP41CX ⋅ HP28S ⋅ HP50G ⋅ HP50G 

07192023, 12:59 AM
Post: #8




RE: [48] how to simplify derivative?
(07182023 07:54 PM)BruceH Wrote: You can use the MATCH↓ and MATCH↑ commands to create your own functions to carry out these transformations. The second argument would be for instance: {'(&Z)^2' '&Z*&Z'} This can be done before entering the equation editor. Not sure about the trigonometric identity though. It seems easier to replace the sum by 1 manually in the editor than writing a pattern. But then you'd have to know in advance that you will need this value and put it on the stack. 

07192023, 11:33 PM
Post: #9




RE: [48] how to simplify derivative?
The MATCH commands return a test result to say whether a match occurred or not. So for trigonometric identities you would, I presume, create a program that tries several identities in succession, noting which succeed, and then, based on that information, decide which one or ones were the appropriate ones to apply.


« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)