116. Taylor Series
Taylor Series (Expansion)
Taylor polynomial
Approximate a complicated function near a point by a polynomial made from the function’s derivatives at that point
Where:
- : Ensures value of the polynomial matches with the value of when
- : Ensures slope of the polynomial matches with the slope of when
- : Ensures rate of change of the polynomial matches with the rate of change of when
Where:
- : -th derivative of evaluated at
Example
Setup
Expand it around the point
Step 1: Compute Derivatives
Step 2: Evaluate at
Step 3: Write the Taylor expansion
Step 3: Expand
It gives back the original function exactly