Section 10.16
Taylor Series
Every smooth function is secretly a polynomial if you look close enough.
1
Introduction
If we can find all derivatives of at a point , we can reconstruct the function nearby. This reconstruction is called the Taylor Series.
If , we call it a Maclaurin Series.
Interactive: Sine Approximation
Drag . See how the polynomial "wraps" around the sine wave further and further?
2
The Formula
Taylor Series about
3
Important Maclaurin Series
Exponential
Converges everywhere ().
Sine
Odd powers. Converges everywhere.
Cosine
Even powers. Converges everywhere.
Geometric
Converges for .
4
Worked Examples
Example 1: Deriving e^(-x)
Find Maclaurin series for .
We know .
Let .
.
It's alternating:
Example 2: Taylor Series about a non-zero point
Find Taylor Series for about .
Calculate derivatives at :
- (Pattern!)
Plug into formula:
Simplify :
.
5
Practice Quiz
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