 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/a/2/c/solution

x^{2}

, the moments are

\int_{0}^{1} x^{j + 2} d x = 1/ (j + 3)

ϕ_{0} = 1

and

ϕ_{1} = x - (1/4) / (1/3) = x - 3/4

For

ϕ_{2} = x^{2} + a x + b

1, x

requires

\frac{1}{5} + \frac{a}{4} + \frac{b}{3} = 0, \frac{1}{6} + \frac{a}{5} + \frac{b}{4} = 0.

Solving gives

a = - 4/3

b = 2/5

. Hence

ϕ_{0} = 1, ϕ_{1} = x - \frac{3}{4}, ϕ_{2} = x^{2} - \frac{4}{3} x + \frac{2}{5},

x_{1} = \frac{10 - 10}{15}, x_{2} = \frac{10 + 10}{15} .

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