Finding approximations on an interval?

Question

Calculate the approximation for the given function and interval

L5, f(x)=x^2+3|x| [-3,2]

can someone explain how to do this?

I'd greatly appreciate it. Thanks.

Answer 1

I'm not sure what kind of approximation you are looking for. If I had to guess, I'd use the "L5" as a clue that you are looking for a piece-wise linear approximation.

If so, then you want 5 line segments: one over the interval [-3,-2], the second over the interval [-2, -1], etc.

If you want a continuous piece-wise approximation, then the end of one line segment has to match the beginning of the next, so the end points should be the values of f(x) at the ends of the intervals.

If you go to a discontinuous approximation, then you find the best line segment for each interval whether or not the line segments join. This allows for a significantly smaller error, but requires more work.

The criteria for "best" can vary. Two traditional metrics are minimizing the maximum error over the interval and minimizing the least squares error over the interval.