If ( ) = sin , then (0) = 0, ( /4) = /2, and ( /2) = 1. Use these data points to answer the following questions. Note that the error that is asked for is the absolute value of the difference between...

If
() = sin
, then
(0) = 0,
(/4) =
/2, and
(/2) = 1. Use these data points to answer the following questions. Note that the error that is asked for is the absolute value of the difference between the exact value
 and the estimated value.

(a) Using piecewise linear interpolation, what is the estimated value of
(/8)? What is the error in this estimate?

(b) Using a global interpolation polynomial, what is the estimated value of
(/8)? What is the error in this estimate?

(c) Using natural cubic spline interpolation, what is the estimated value of
(/8)? What is the error in this estimate?

(d) Using the additional information that
 (0) = 1 and
 (/2) = 0, use clamped cubic spline interpolation to find an estimated value of
(/8). What is the error in this estimate?

(e) Suppose Chebyshev interpolation is used. Determine the three Chebyshev points in the interval, and evaluate
() at these points. What is the resulting estimated value of
(/8)? What is the error in this estimate?