Why can't a graphing calculator find this type of derivative?

[Infohawk]

If you ask the TI-89 to derive x^2 -24x*y(x) + 16y(x)^2 -400x-300y(x)=0 it will give a "Warning: May produce false equation" and then give the wrong answer. Why is this? I am curious why this type of problem creates problems for the calculator while other seemingly complex problems don't. Also, why does it say "may produce false equation." Won't it either be false or not?

I'm not interested in how to get the solution or why people shouldn't use calculators for math. I'm interested in the technical aspect.

 

[William Gaatjes]

If you ask the TI-89 to derive x^2 -24x*y(x) + 16y(x)^2 -400x-300y(x)=0 it will give a "Warning: May produce false equation" and then give the wrong answer. Why is this? I am curious why this type of problem creates problems for the calculator while other seemingly complex problems don't. Also, why does it say "may produce false equation." Won't it either be false or not?

I'm not interested in how to get the solution or why people shouldn't use calculators for math. I'm interested in the technical aspect.

I never used the calculator, but should you not use brackets or parentheses somewhere ?

 

[MrDudeMan]

Which version of the TI-89 are you using? Also, I'm not really sure what you are trying to say with that equation. Your use of parentheses and the multiply symbol is confusing.

 

[canis]

..

 

[Daedalus685]

y has more than one valid result so you can't solve for it exactly (you being the 89). The calculator gets confused by things like this as it must differentiate an equation in the form of y= such and such. The symbolic math on an 89 is not all that powerful. Try it in mathcad or maple and you will get the correct answer(s).

The 89 does not handle complex numbers very well at times. Depending on the value of x y can be complex in that equation.

To clarify a bit:

When the ti-89 solves something symbolically how you type the symbols makes all the difference. If you simply put 'y' instead of y(x) it will not know if the variable is dependant on x or not, it will assume not and give you the answer as if y were a constant. In this case the answer you get may or may not be a valid expression. For instance, try to differentiate x+5=0; the answer it gives is 1=0 which "May not be a valid equation".

On the other hand, if you put in y(x) it will know that y is itself dependant on x. However, y(x) can't be trivially solved for based on what I said originally (it is a quadratic expression and y has more than one valid result, some of which may be complex functions) so it will simply write that the derivative of y(x) is the derivative of y(x)... not very helpful but the equation will at least be valid . So for this equation you get a variety of y'(x) terms strewn about with the calculator making no attempt to solve for y.

Differentiation doesn't make sense in cases where the variables are not variables. The calculator returns the may not be valid because it assumes y is a constant and thus x is sufficiently defined and differentiation does not make any sense. If you use y(x) then the answer ill be correct but it will not express what y'(x) actually is, jsut give a more or less meaningless equation involving y(x), x, and the derrivative of y(x).

 

[Infohawk]

Thanks Daedalus685. So if mathcad and maple can do it but the ti-89 can't, is it because matcad and maple have access to a more powerful desktop processor or is it a software issue or both?

"For instance, try to differentiate x+5=0; the answer it gives is 1=0 which "May not be a valid equation".

Heh, cool.

 

[Daedalus685]

Matcad and maple are simply more powerful programs (they don't necessarily have more tools, just have a better ability to recognize form as they have many forms preloaded). Mathcad is specifically symbolic and is exceptional at simplifying an equation. It will recognize that the derivative is implicit and put it in the more standard form (most of the time, it takes some pushing). The TI will either not treat y as a variable that could be dependant on x. However, if you take what the TI gives you and solve for y'(x) you will get the more propper form of the implicit derivative just not at all simplified.

The 'not a valid equation' is just a result of the calc assuming everything you put in yet don't explicitly say is dependant on x is independent (an unknown constant). Thus the derivative is meaningless (as x would be sufficiently defined) and will likely result in an expression that is not valid... like 1=0.

 

[Cogman]

Just a note. When the TI, or any math program for that matter, throws the "Answers may not be exact" message, it means that "I wasn't able to solve this symbolically, so I'm guesstimating as close as possible.".

Its like trying to solve for x + 2 = 3 With no knowledge of algebra, you just keep pluging in values for x until you get to a close enough answer. If you notice you are moving away, you change directions.