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I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is:

Consider the function $f(x) = \sin(x)$ on the interval $[0,\pi]$. Evaluate $f(x)$ in enough points to find an interpolating polynomial $p(x)$ and a natural cubic spline approximation $s(x)$ such that the error functions $|(f-p)(x)|$ and $|(f-s)(x)|$ satisfy

$|(f-p)(x)| \leq 0.005$ for $0 \leq x \leq \pi$

$|(f-s)(x)| \leq 0.005$ for $0 \leq x \leq \pi$

Give the number and the location of the data points for $p$ and $s$. This can be different for $p$ and $s$. Why is the second derivative of the spline function in begin and endpoint equal to zero a good choice?

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up vote 2 down vote accepted

Wherever you get the formulas for constructing the interpolating polynomial and spline should give you an error estimate. For the polynomial, Wikipedia gives an error limit $f(x)-p_n(x)=\frac {f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^n (x-x_i)$ where the $x_i$ are the interpolating points. For $\sin x$ the derivatives are limited to $\pm 1$, so you can use that for your error limit. To minimize the product, you can use Chebyshev nodes. For the last question, what is the derivative of $\sin x$ at $0$ and $\pi$?

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