Chapter 3.1, Problem 3ES

Use Theorem 3.3 to find an error bound for the approximations in Exercise 1.

1. For the given functions f(x), let x₀ = 0, x₁ = 0.6, and x₂ = 0.9. Construct interpolation polynomials of degree at most one and at most two to approximate f(0.45) and find the absolute error.)

1. a. f(x)  = cos x
2. b. f(x) = ln(x + 1)
3. c. $f(x)=\sqrt{1+x}$
4. d. f(x) = tan x

Theorem 3.3 Suppose x₀, x₁, ... , x_n are distinct numbers in the interval [a, b] and ∈ C^{n + 1}[a, b]. Then, for each x in [a, b], a number ξ(x) (generally unknown) between min{x₀, x₁,..., x_n}, and the max{x₀, x₁,..., x_n} and hence in (a, b), exists with

$f(x)=P(x)+\frac{f^{(n+1)}(\xi (x))}{(n+1)!}(x-x_0)(x-x_1)\cdots (x-x_n),$    (3.3)

where P(x) is the interpolating polynomial given in Eq. (3.1).