Calculators and numerical differentiation

[Wes Loewer]

(11-01-2020 05:39 PM)Albert Chan Wrote:  How did you deduce there is hidden CAS under non-CAS Nspire ?
May be an example ?

I may have made a leap in my logic, but the fact that examples such as |x| or 1/x at x=0 or at x=0.0001 produce the correct answer in the non-CAS Nspire while these are incorrect on the 84+ lead me to believe that the non-CAS model must be doing something CAS-like under the hood. I had never come across a counter-example. It also made sense to me the it would be very easy to share the same code as the Nspire-CAS for such calculations.

You prompted me to look in the Nspire manual which gives some insight.

Quote:nDerivative()
Returns the numerical derivative calculated using auto differentiation methods.
...
Note: The nDerivative() algorithm has a limitiation [sic]: it works recursively through the unsimplified expression, computing the numeric value of the first derivative (and second, if applicable) and the evaluation of each subexpression, which may lead to an unexpected result.

Consider the example on the right. The first derivative of x•(x^2+x)^(1/3) at x=0 is equal to 0. However, because the first derivative of the subexpression (x^2+x)^(1/3) is undefined at x=0, and this value is used to calculate the derivative of the total expression, nDerivative() reports the result as undefined and displays a warning message.

If you encounter this limitation, verify the solution graphically. You can also try using centralDiff().

(The Npsire has a centralDiff() function as well that behaves like the 84+.)

The part starting with "Note: " is not in the Nspire CAS manual. The Nspire CAS gives the correct answer for this example.

When I first saw the above example, I thought that I must have been wrong about the numeric model having an internal CAS since the numeric model does not give the correct answer while the CAS model does. However, now that I'm reading it again, I'm thinking that I may have been right after all. The fact that it says "the subexpression (x^2+x)^(1/3) is undefined at x=0" means that the calculator must be breaking the expression down into subexpressions and evaluating their derivatives (consistent with the product rule) which means that it must have some CAS capabilities rather than just evaluating the whole expression numerically.

So my current thinking is that the Nspire must have at least some CAS capabilities under the hood, but not to the extent as the Nspire CAS.

Thoughts?