Consider the piece-wise constant interpolation of the function $$f(x)=\sin(x), 0\leq x\leq 381$$ at points $x_i=ih$ where $h=0.1$. Thus, our interpolant satisfies $$v(x)=\sin(ih), (i-0.5)h\leq x \leq (i+0.5)h$$ for $i=0,1,\dots , 3810.$
How many leading digits in $v(x)$ are guarenteed to agree with those of $f(x)$, for any $0\leq x \leq 381$?
I am not sure what it means by leading digits... Does it mean that the constant interpolant begins at the same spot as $\sin(x)$? I am also not sure how $v(x)=\sin(ih), (i-0.5)h\leq x \leq (i+0.5)h$ is constant... Thank you for any help you can give me.
For example, $\sin(5)\approx 0.08717557$ The interpolation says $v(x)\approx 0.08717557$ for any $x$ such that $4.95 \lt x \le 5.05$, but $f(x)$ will vary over the range. $f(x)$ will be close to $v(x)$ over that range. You are expected to figure out how much they can differ. If they cannot differ by more than $0.001$ you would say that the first three digits agree-the first three after the decimal point. Yes, they can change because of carries, but this is a rough idea of how good the approximation is. In fact the error can be larger than that-I leave that to you.