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