This problem considers some of the difficulties interpolating the function (a) If the interpolation interval is 1 ≤  ≤ 10, how many data points are needed for piecewise linear interpolation to...

This problem considers some of the difficulties interpolating the function

(a) If the interpolation interval is 1 ≤
 ≤ 10, how many data points are needed for piecewise linear interpolation to guarantee that the error is less than 10⁻⁶?

(b) Explain why Theorem 5.4 is not so useful if the interval is 0 ≤
 ≤ 1.

(c) One way to deal with the singularity at
 = 0 is to break the interval into two segments, one is 0 ≤
 ≤
 and the other is
 ≤
 ≤ 1, where
 is a small positive number. On the interval 0 ≤
 ≤
 the function is going to be interpolated with a single line. What is the equation for this line, and how small does need to be to guarantee that the approximation error is 10⁻⁶?

(d) Assuming that
 is known, how many data points over the interval
 ≤
 ≤ 1 are needed for piecewise linear interpolation to guarantee that the error is less than 10₋₆?