[SOLVED] Math 104A Midterm 1 Fall 2024 Introduction to Numerical Analysis Processing

Introduction to Numerical Analysis

Math 104A Midterm 1, Fall 2024

1. (20 points) Let x0 = 1, x1 = 2, x2 = 4, x3 = 6.

(a) Construct the Lagrange polynomials (at most degree).

(b) Find the interpolation of

2. (20 points)

(a) Repeat Problem 1 using the Newton divided di↵erences method for x0 = 1, x1 = 2, x2 = 4.

(b) Use Theorem 3.3 to find an error bound for the approximations.

3. (20 points) (a) Find the Hermite interpolation for

f(x) = sin(x)

given

using divided di↵erences.

(b) Using the Hermite interpolation obtained, estimate f ().

4. (20 points) Determine the natural cubic spline S(x) that interpolates the data f(1) = 1, f(3) = 2, and f(4) = 0.

5. (10 points) Prove Theorem 3.6: Suppose that f ∈ Cn[a, b] and x0, x1,…,xn are distinct numbers in [a, b]. Then a number  exists in (a, b) with