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\noindent {\bf Problem statement} ~a) For $x$ near $0$, $\sin x$ is well-approximated by its tangent line
at $x=0$. What is this tangent line?

\medskip

\noindent b) Approximation over an interval is preferred over
approximation near a point for many purposes. One criterion for
assessing the accuracy of such an approximation is {\it mean-square
error}.  The mean-square error between a straight line $y = Ax$ going
through the origin and the function $\sin x$ over the interval $[0,
1]$ is given by the definite integral $ \int_0^1 \left( \sin x - Ax
\right)^2 \, dx$.  Find the $A$ which $\underline{\rm minimizes}$ this
integral.

\medskip

\noindent {\bf Hint} Expand the integrand, compute the integral, and
find the $A$ minimizing the result.

\medskip

\noindent c) Sketch $\sin x$ and the straight lines found in a) and b) on the
unit interval $[0,1]$.










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