Numeric Derivatives Using TI-83, TI-92
Date: 10/17/2003 at 09:41:27 From: Cindy Subject: numeric derivatives What is the difference between a derivative and a numeric derivative? When I needed to graph the sin (x) function and its derivative, I plugged sin(x) in for the original function, then I differentiated and got cos(x) for its derivative. However, when I checked it on my calculator using the nDeriv (numeric derivative) function, it came up as sin(2x)/2x, and obviously graphed differently than cos(x). I am aware that the correct derivative of sin(x) is cos(x), I just don't understand why the numeric derivative gave a different answer.
Date: 10/17/2003 at 12:54:23 From: Doctor Peterson Subject: Re: numeric derivatives Hi, Cindy. You didn't say what kind of calculator you are using, and I think that may be the problem. From my research, it appears that the TI-83 and others have a true numeric derivative function, which gives the derivative as a number for a specific value of x using a difference quotient for a very small difference. You would enter nDeriv( sin(x), x, x ) and get the value of the derivative of sin(x) with respect to x at any specific value x, which it can graph for you by repeating this calculation for each x. The second argument to the function is the variable with which to differentiate, while the third is the value at which to evaluate an approximate derivative. This is done by taking an approximate limit f(a+h) - f(a-h) nDeriv( f(x), x, a ) = lim --------------- h->0 2h You got an expression instead, which suggests that you are using a TI-92 or equivalent. That actually produces a difference quotient expression, rather than just a value, and has a slightly different syntax. You probably entered the same thing you would enter on a TI-83 to graph the derivative, nDeriv( sin(x), x, x ) and got the expression sin(2x) ------- 2x This is because nDeriv( f(x), x, h ) returns the expression f(x+h) - f(x-h) --------------- 2h which is the symmetrical difference quotient. Since you unwittingly entered x for the difference h, this would become sin(x+x) - sin(x-x) sin(2x) - sin(0) sin(2x) ------------------- = ---------------- = ------- 2x 2x 2x This is not the derivative, but a specific difference quotient with a large value of h. You need to make h small to get an approximation to the derivative. Isn't it nice of TI to make two calculators with the same named function that does two entirely different things? To do what you want on the TI-92, drop the third argument, so that it will default to h=0.001, and just graph the resulting function, which will be a good approximation to the derivative. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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