Introduction to Numerical Analysis
Math 104A Midterm 2, Fall 2024
1. (5 points) Attendance-Related Multiple Choice Question
For extra credit assignment, students performed various activities in class.
Which of the following was NOT performed in class?
(A) Demonstrating how to make sourdough bread
(B) Performing a magic show
(C) Presenting a photography gallery
(D) Playing a live guitar performance
(E) Drawing SpongeBob
(F) Presenting an audio synthesizer that uses Fourier transforms
(G) None of these
2. (25 points) Let Πn denote the set of all polynomials of degree at most n. Suppose that {φ0(x), φ1(x), . . . , φn(x)} is a collection of linearly independent polynomials in Πn. Prove that any polynomial in Πn can be writ-ten uniquely as a linear combination of φ0(x), φ1(x), . . . , φn(x).
3. (25 points)
Given the data:
xi 4.0 4.2 4.5 4.7 5.1 5.5
yi 102.56 113.18 130.11 142.05 167.53 195.14
Construct the least squares approximation of the form. bxa
, and compute the error.
4. (20 points) Use the Gram-Schmidt process to construct the Legendre polynomials φ0(x), φ1(x), φ2(x), and φ3(x) on the interval [0, 2]. The weight function is w ≡ 1.
5. (25 points) Find the least squares polynomial approximation of degree two to f(x) on the interval [1, 3] and compute the error E for the approximation.
f(x) = x ln x