- Topic
- Taylor polynomials
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Taylor polynomials practice question 3
The third-order Taylor polynomial of a function about is given below. Find .
This polynomial is centred at , not 0. Which coefficient carries the information about , and what was it divided by?
About , the coefficient of is . Here , so read the coefficient of .
An integer or a fraction.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Each coefficient of a Taylor polynomial is a derivative at the centre divided by a factorial, so multiply the right coefficient back by its factorial.
Working
The coefficient of (x + 1)², since the centre is −1.
Undo the 2!.
A check: differentiating twice and putting x = −1 agrees. Putting x = 0 instead gives −7, the wrong point.
Answer
The answer is negative 9.
Check
Differentiating T₃ twice gives −9 + 2(x + 1), which is −9 at x = −1.
Where this goes wrong
Answering −9/2, the coefficient itself. The coefficient of (x + 1)² is f″(−1) divided by 2!, so multiply it by 2.
Leans on
Higher-order derivatives, from Differentiation rules
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The third-order Maclaurin polynomial of a function is given below. Find .Saved in this browser only.