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 (ie, 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: You may reuse code from activity #1 if it is helpful a) 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 fx) (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. b) Evaluating a polynomial derivative numerically For a function fx), the derivative of the function at a value x can be found by evaluating f (x+a)-f(x) reand 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 ta)-) using a value for a such as 0.1. Then, repeatedly halve the value of a until the difference between successive evaluations o is less than some sma of evaluations it took. Calculate how close that result is to the actual answer, computed in part all value, such as 10 Print the result, along with the number al. 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 (ie, 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: You may reuse code from activity #1 if it is helpful a) 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 fx) (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. b) Evaluating a polynomial derivative numerically For a function fx), the derivative of the function at a value x can be found by evaluating f (x+a)-f(x) reand 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 ta)-) using a value for a such as 0.1. Then, repeatedly halve the value of a until the difference between successive evaluations o is less than some sma of evaluations it took. Calculate how close that result is to the actual answer, computed in part all value, such as 10 Print the result, along with the number al
