Apply the central difference formula (4.10) to approximate the first derivatives of the functions (u_{1}(x)=sin (x), u_{2}(x)=sin

Question:

Apply the central difference formula (4.10) to approximate the first derivatives of the functions $u_{1}(x)=\sin (x), u_{2}(x)=\sin (3 x)$, $u_{3}(x)=\sin (10 x)$ at $x=0$. Use the uniform grid with step $\Delta x=0.1$. Compare the results with the exact values of the derivatives, and explain why the error of approximation is different for the three functions.

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