Practice question · Put in order
Order the terms of the Taylor polynomial of f about a, from lowest degree to highest.
- f''(a)(x - a) squared, divided by 2 factorial
- f'''(a)(x - a) cubed, divided by 3 factorial
- f'(a)(x - a)
- f(a)
Hints
- Each successive term uses one more derivative and one higher power of the displacement.
- The denominators are the factorials matching the degree.
Show the answer
- f(a)
- f'(a)(x - a)
- f''(a)(x - a) squared, divided by 2 factorial
- f'''(a)(x - a) cubed, divided by 3 factorial
Why
The constant fixes the value, the linear term fixes the slope, the quadratic fixes the curvature, and so on, each term corrects a feature the previous ones could not capture. That correspondence between degree and derivative order is the whole design of the series.
Practise Taylor Approximation
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