Practice question · Fill in the blanks
Complete the statement about where a Taylor polynomial works best.
A truncated Taylor polynomial approximates the function best ______.
Word bank: far from the expansion point · near the expansion point · at every point equally well · only where the function is zero
Hints
- The polynomial is constructed from information gathered at exactly one point.
- Each term is a power of the distance from the centre, and powers of a small number are small.
Show the answer
A truncated Taylor polynomial approximates the function best near the expansion point.
Why
All the information comes from derivatives at the centre, so the fit is best there and decays with distance. Adding terms widens the good region but never makes a finite polynomial match a function everywhere.
Practise Taylor Series and Approximation
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