1. Apr 27, 2010

### malindenmoyer

Could somebody explain what exactly a "piecewise quadratic approximation" is?

Problem Statement

Find a piecewise quadratic approximation P(x) of f(x), where

$$f(x)=\sin{4x}\; on \; [0,\pi]$$

Plot f(x) and P(x) on $$[0,\pi]$$.

What is the maximum value of the following:

$$|f(x)-P(x)| \; on \;[0,\pi]$$

The problem goes on to say:

Can you find a piecewise approximation to f(x) that is continuous on $$[0,\pi]$$ and each "piece" is a polynomial?

Attempt at Solution

I know that a piecewise is typically a set of linear functions defined at several intervals in the given domain of x. Is this what the problem is asking, except a quadratic approximation at several intervals? If so, how is one supposed to come up with that; it seems rather ambiguous. My thoughts are the same for the next part regarding the "continuous piecewise approximation".

The first part of the problem not listed asked to find a quadratic approximation of f(x), which I can do using a Taylor Series.

I have never heard of the term piecewise quadratic approximation and therefore stumped. If somebody could please give an explanation of what the problem is asking us to find, that would be greatly appreciated.

2. Apr 28, 2010