In Exercise 26 of Section 1.1 a Maclaurin series was integrated to approximate erf(1), w...

In Exercise 26 of Section 1.1 a Maclaurin series was integrated to approximate erf(1), where erf(x) is

the normal distribution error function defined by

a. Use the Maclaurin series to construct a table for erf(x) that is accurate to within 10⁻⁴ for erf(xi), where x_i = 0.2i, for i = 0, 1, . . . , 5.

b. Use both linear interpolation and quadratic interpolation to obtain an approximation to Which approach seems most feasible?

Reference: Exercise 26 of Section 1.1

A function f : [a, b] → R is said to satisfy a Lipschitz condition with Lipschitz constant L on [a, b] if, for every x, y ∈ [a, b], we have |f (x) − f (y)| ≤ L|x − y|.

a. Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an interval [a, b], then f ∈ C[a, b].

b. Showthat if f has a derivative that is bounded on [a, b] by L, then f satisfies a Lipschitz condition with Lipschitz constant L on [a, b].

c. Give an example of a function that is continuous on a closed interval but does not satisfy a Lipschitz condition on the interval.