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# Degree 3 Taylor on a interval

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In this example:
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{0} = 1$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v'(\tfrac{1}{2})$$

$$\displaystyle \phi_{1} = x - \frac{1}{2}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto v'(\tfrac{1}{2})$$

$$\displaystyle \phi_{2} = \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto v'(\tfrac{1}{2})$$

$$\displaystyle \phi_{3} = \frac{x^{3}}{6} - \frac{x^{2}}{4} + \frac{x}{8} - \frac{1}{48}$$

This DOF is associated with edge 0 of the reference element.