Homework 1#
Note
You are free to discuss these questions with your classmates and on our class slack, but you must write your own solutions, including your own source code.
All code should be uploaded to Brightspace along with any analytic derivations, notes, etc.
Following the example we did in class to find a difference approximation to the first derivative of \(f(x)\), find an approximation to the second derivative of \(f(x)\).
You should consider \(f(x-h)\), \(f(x)\), and \(f(x+h)\).
What is the truncation error of this approximation?
Write a program to compute the second derivative of \(f(x) = \sin(x)\) as a function of \(h\) and print the absolute value of the error in your estimate compared to the analytic solution for several values of \(h\).
Does the error converge as you expect?
Consider the function:
\[f(x) = \sqrt{x^2 + 1} - 1\]When \(x\) is small, roundoff error from the subtraction will dominate the answer. In the limit of \(x \rightarrow 0\), this expression takes the form:
\[f(x) \sim \frac{1}{2} x^2\]Show that by multiplying and dividing \(f(x)\) by \(\sqrt{x^2 + 1} + 1\) you get an analytically equivalent expression for \(f(x)\) without any subtractions.
Write a program to evaluate \(f(x)\) and the new version of \(f(x)\) for \(x\) with the values
1.e-6,1.e-7,1.e-8.Which version is more accurate?