Final review of [[Numerical Analysis]] Use the following [[Inner Product]]: $ \huge \inprod{f,g} = \int_{-1}^{2} f(x)g(x)\d x $ Let $g_{1}(x)=x$ and $g_{2}(x)=x^{2}+ax$. Find the value of a such that $\inprod{g_{1},g_{2}}=0$. $\huge \begin{align} \inprod{g_{1},g_{2}} &= \int_{-1}^{2} x(x^{2}+ax) \d x \\ &= \int_{-1}^{2} x^{3}+ax^{2} \\ &= \left. \frac{x^{4}}{4} + \frac{ax^{3}}{3} \right\rvert^{2}_{-1} \\ &= \frac{15}{4} + 3a = 0 \\ a &= -5/ \end{align}$