We want to approximate a function \(f(x)\) on an interval \([a,b]\). We choose \(n\) nodes \(x_1,\ldots,x_n\) on this interval, e.g. equidistant nodes \(x_j = a + (j-1)\frac{b-a}{n-1}\) for \(j=1,\ldots,n\). We approximate the function \(f(x)\) with the interpolating polynomial \(p(x)\) and obtain the following bounds for the interpolation error: \[ |f(x) - p(x)| \le \frac{1}{n!}\max_{t\in [a,b]}\left|f^{(n)}(t)\right| \,\cdot \, |\omega(x)| \] \[ \max_{x\in[a,b]} |f(x) - p(x)| \le \frac{1}{n!}\max_{t\in [a,b]}\left|f^{(n)}(t)\right| \,\cdot \, \omega_\max \] with \[ \omega_\max := \max_{x\in [a,b]} \left| \omega(x) \right| \] We want to choose the nodes \(x_1,\ldots,x_n\in[a,b]\) so that \(\omega_\max\) is as small as possible.
Drag the nodes on the \(x\)-axis with your finger/mouse. Try to move the nodes so that the blue graph of \(\omega(x)\) is between the two red lines (\(\omega_\max\) for optimal choice of nodes).