**5.1**(a) By using methods we learned in class, we obtain the polynomial of degree two:

(b) Using Lagrange interpolation for the two subintervals, we obtain

(35)

(c) by repeating calculations that we have done in class, we obtain

(36)

**(5.4)**(a) Lagrange Polynomial is

(b) Lagrange Polynomial is

(c) Lagrange Polynomial is

**5.26****(a)**Global polynomial interpolation is

**(b)**Writting spline as

and follwoing steps we discussed in class, we obtain system of equations

We solve this system of equation to finally obtain that

We therefore obtain**c**Result of part

**(a)**does indeed satisfies interpolation and smoothness condition, but result in part**(a)**is*not*a natural spline**(d)**To obtain result in**(a)**one would have to use clamped spline with