PYTHON

In an earlier lab, we observed how we could have a function that is undefined at some value (such as (sin x)/x at the point x=0), but could come arbitrarily close to it by successively evaluating smaller and smaller numbers (i.e. taking a limit). For example, we might evaluate at x=0.1, x=0.01, x=0.001, etc. until we have come very close to the value. Taking limits like this, numerically, is commonly done when functions are too complicated to evaluate analytically. You will write a program to compute a derivative as a numerical limit. This activity has a few parts:

1. Evaluating a polynomial limit analytically

You should have learned by now the process for finding the derivative of a polynomial (as another polynomial). Write a program that will read in from the user a cubic polynomial f(x) (as a set of 4 coefficients), and use this to compute the derivative polynomial (i.e. compute the three coefficients of the derivative f(x)). Then, read in a value for x from a user, and evaluate the derivative polynomial at that x. Print out that value.

2. Evaluating a polynomial derivative numerically

For a function f(x), the derivative of the function at a value x can be found by evaluating fx+a-f(x)a and finding the limit as a gets closer and closer to 0. Using the same polynomial as the user entered in part (a), and for the same value of x as entered in part (a), compute the limit numerically. That is, start with an estimate by evaluating fx+a-f(x)a using a value for a such as 0.1. Then, repeatedly halve the value of a until the difference between successive evaluations of fx+a-f(x)a is less than some small value, such as 10^-6. Print the result, along with the number of evaluations it took. Calculate how close that result is to the actual answer, computed in part (a).

Challenge: Derivatives can also be estimated by computing the limit fx-f(x-a)a or fx+a-f(x-a)2a . Try computing each of those, and calculate how many iterations you need to converge to the limit. Do you get different results with any of them, or does any of them take fewer steps to get an answer?

Be sure to include appropriate comments in your code, and to use descriptive input and output statements.