Chapter 3.5, Problem 8E

1. Construct a natural cubic spline to approximate $f(x)=e^{-x}$ by using the values given by $f(x)$ at $x=0,0.25,0.75,$ and 1.0.
2. Integrate the spline over [0,1], and compare the result to ${\displaystyle {\int }_0^1{e}^{-x}}dx=1-1/e$
3. Use the derivatives of the spline to approximate $f^{\prime }(0.5)$ and $f^{\prime \prime }(0.5),$ and compare the approximations to the actual values.