Taylor polynomials can be used to approximate a function around any value for a differentiable function. Taylor polynomials look a little ugly, but if you break them down into small steps, it’s actually a fast way to approximate a function.
Sample problem: Use Taylor polynomials to approximate the function cos(x) around the point x=2.
Step 1: Evaluate the function for the first part of the Taylor polynomial:
p(x) = f(c) ⁄ o! = cos(2)
Step 3: Evaluate the function for the third part of the Taylor polynomial (adding it to the first and second parts from Step 2):
p(x) = cos(2) – sin(2)(x-2) + f”(c)⁄2!(x-c)2 = cos(2) – sin(2)(x-2) – cos(2) ⁄ 2(x-2)2
Step 4: Evaluate the function for the fourth part of the Taylor polynomial (adding it to the first, second and third parts from Step 3):
p(x) = cos(2) – sin(2)(x-2) + f”'(c)⁄3!(x-c)3 = cos(2) – sin(2)(x-2) – cos(2) ⁄ 2(x-2)2 – sin(2)⁄6 (x-2)3
Step 5: Continue evaluating more pieces of the Taylor polynomial, graphing the function periodically to see how well it represents your polynomial.
Tip: Technically, you could go on forever with iterations of the Taylor polynomial, but usually five or six iterations is sufficient for a good approximation
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