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CAS diff() giving incorrect results - Wes Loewer - 09-22-2021 07:10 PM
A couple of years ago I posted about a problem with the diff() command (https://www.hpmuseum.org/forum/thread-13532.html). The problem still exists in the most recent firmware.
I guess I had forgotten about the issue because I ran into the problem again yesterday, only it's worse than I original thought. The problem arises when you try to evaluate a derivative at a value, which is something that frequently comes up in calculus and physics.
For instance, evaluating the 2nd derivative of x^3 at x=4.
diff(x^3,x,2)|x = 4
which should evaluate to 6*x|x=4 which is 24. However, the CAS substitutes 4 in for x BEFORE differentiating, so the expression is interpreted as
diff(4^3,4,2) = diff(64,4,2)
which is meaningless.
Even worse than no answer is giving the wrong answer. If the value of x contains pi, then you get incorrect results.
diff(x^3+π^4,x)|x = π
should result in 3*x^2|x=π which is 3*π^2. Instead, the line is interpreted as
diff(π^3+π^4,π)
and now π is treated as a variable. So the result is 3*π^2+4*π^3.
This issue came to light yesterday when I was checking my test answer key and entered this (using primes which is equivalent to diff).
(x^2*cos(x))''|x = π
evaluated to -2 instead of π^2-2. I assumed that I had made some silly mistake in my work by hand. It took several repeated attempts before I concluded that I was correct and the calculator was wrong.
I realize I could use the d/dx template which works correctly, but there is no template for 2nd or higher order derivatives and
(∂(∂(x^2*cos(x),x),x)|(x = π))
looks really awkward (especially in Textbook mode) and using the ' (primes) is so very convenient. It's really an order of operations issue in which |x=a should be applied AFTER the diff() command has completed, not BEFORE.
And it seems only natural that the HP "Prime" should be able to correctly work with "primes".
RE: CAS diff() giving incorrect results - parisse - 09-22-2021 07:26 PM
I have already tried to make | work as you expect with diff, but that was not successfull, it's too complicated. Please use subst
Code:
subst(diff(x^3,x,2),x = 4)Code:
diff(x^3,x,2)(x = 4)RE: CAS diff() giving incorrect results - Wes Loewer - 09-23-2021 06:20 PM
(09-22-2021 07:26 PM)parisse Wrote: I have already tried to make | work as you expect with diff, but that was not successfull, it's too complicated. Please use subst
Okay, thanks for that info. That's good to know.
Quote:or just parentheses like
Code:
diff(x^3,x,2)(x = 4)
That's an interesting notation. It appears to work for other functions as well. Is there any way to use it with primes? When I try (x^3)''(x = 4) it gives a warning and returns (6*x) = 24, with the expression on the left and the result on the right. Is this the intended behavior?
RE: CAS diff() giving incorrect results - toml_12953 - 09-23-2021 07:26 PM
(09-23-2021 06:20 PM)Wes Loewer Wrote:(09-22-2021 07:26 PM)parisse Wrote: I have already tried to make | work as you expect with diff, but that was not successfull, it's too complicated. Please use subst
Okay, thanks for that info. That's good to know.
Quote:or just parentheses like
Code:
diff(x^3,x,2)(x = 4)
That's an interesting notation. It appears to work for other functions as well. Is there any way to use it with primes? When I try (x^3)''(x = 4) it gives a warning and returns (6*x) = 24, with the expression on the left and the result on the right. Is this the intended behavior?
For that expression, the pipe does work
(x^3) | (x = 4) 64
RE: CAS diff() giving incorrect results - Albert Chan - 09-24-2021 10:48 AM
(09-23-2021 06:20 PM)Wes Loewer Wrote: When I try (x^3)''(x = 4) it gives a warning and returns (6*x) = 24,
When I try same expression in the emulator, above return only 24, without warning.
I beiieve (x^3)''(x = 4) equivalent to subst((x^3)'', x = 4), only less typing.
Quote:That's an interesting notation. It appears to work for other functions as well.
If the expression is not a function, the notation may get confused with implied multiply.
XCAS> (1+x+x^2) (x=0.1)
Warning : using implicit multiplication for (1+x+x^2)(x=0.1)
x*(x^2+x+1)=(0.1*(x^2+x+1))
CAS> (1+x+x^2) (x=0.1) → "Syntax Error"
We can get around the problem (both CAS and XCAS) with identity function.
CAS> id(1+x+x^2) (x=0.1) → 1.11
RE: CAS diff() giving incorrect results - Wes Loewer - 09-24-2021 04:28 PM
(09-24-2021 10:48 AM)Albert Chan Wrote:(09-23-2021 06:20 PM)Wes Loewer Wrote: When I try (x^3)''(x = 4) it gives a warning and returns (6*x) = 24,
When I try same expression in the emulator, above return only 24, without warning.
I beiieve (x^3)''(x = 4) equivalent to subst((x^3)'', x = 4), only less typing.
That's interesting. That's exactly what I was hoping to accomplish.
What version are you using? I tried this on the following versions:
2020-01-16 (HW-A, VC)
no warning, returns 24
2020-09-08 (HW-A), 2021-05-05 (HW-D), 2021-06-09 (VC)
has warning, returns 6*x=24
RE: CAS diff() giving incorrect results - Albert Chan - 09-24-2021 05:23 PM
Hi, Wes Loewer
It was an old version: Build: 2.1.14181 (2018 10 16)
RE: CAS diff() giving incorrect results - Wes Loewer - 09-24-2021 07:59 PM
(09-24-2021 05:23 PM)Albert Chan Wrote: It was an old version: Build: 2.1.14181 (2018 10 16)
Okay, I guess something changed since then.
RE: CAS diff() giving incorrect results - parisse - 09-25-2021 04:04 PM
f(x=...) will work if f is an expression. It will not work with e.g. (x^3)'' because the parser will not detect that it's an expression (and it will do like if it was implicit multiplication because x appears in both).
If you work with derivatives, it's a good idea to save them in variables. First f:=x^3'', then f(x=4)
RE: CAS diff() giving incorrect results - Albert Chan - 09-25-2021 05:03 PM
(09-23-2021 06:20 PM)Wes Loewer Wrote: When I try (x^3)''(x = 4) it gives a warning and returns (6*x) = 24
(09-25-2021 04:04 PM)parisse Wrote: f(x=...) will work if f is an expression. It will not work with e.g. (x^3)'' because the parser will not detect that it's an expression (and it will do like if it was implicit multiplication because x appears in both).
Why does older version detected the expression, but not newer version ?
Does this mean newer versions of HP Prime CAS support implied multiplication ?
It still does not explain Wes Loewer weird result.
Assuming parser get confused, assumed implicit multiplication, we should have:
CAS> (6*x) * (x = 4) → 6*x^2 = 24*x
RE: CAS diff() giving incorrect results - Albert Chan - 09-26-2021 12:04 PM
I thought that expression|(x=a) notation can be simulated with eval(subst(quote(expression), x=a))
But, not quite. subst version is better ! (I skipped eval, because f(x) not defined)
CAS> 'integrate(f(x),x)' (x = t*t
integrate(f(t*t)*(t+t),t)
CAS> 'integrate(f(x),x, 1, 2)' (x = t*t
integrate(f(t*t)*(t+t),t, 1, √2)
CAS> integrate(f(x),x) |x = t*t
x*f(t^2)
CAS> integrate(f(x),x, 1, 2) |x = t*t
integrate(f(x),x, 1, 2)
---
I may have found a quoting bug. If quoted expression is saved to a variable, quote is gone.
CAS> assume( t >= 0)
CAS> m := 'integrate(1/(√x*(x+1)),x)' (x = t*t)
integrate(1/(√(t*t)*(t*t+1))*(t+t),t)
CAS> m // should have matched Ans
2*atan(t)
Update: another bug, if we use notation x = a .. b
CAS> 'integrate(f(x),x = 1 .. 2)' (x = t*t
integrate(f(t*t),(t*t) = (1 .. 2))