- Topic
- Taylor polynomials
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Taylor polynomials practice question 1
The third-order Maclaurin polynomial of a function is given below. Find .
How is each coefficient of a Maclaurin polynomial tied to a derivative at 0? Find the coefficient that carries .
The coefficient of in a Maclaurin polynomial is . Multiply the coefficient by .
An integer.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Taylor coefficients are derivatives divided by factorials, so a derivative at the base point is a coefficient multiplied back by the factorial.
Working
The x³ coefficient.
3! = 6.
Answer
The answer is negative 8.
Check
Differentiating T₃ three times wipes out 2 − x + 5x² and leaves the constant −8 from −(4/3)x³.
Where this goes wrong
Answering −4/3, the coefficient itself. The factorial is built into every Taylor coefficient and has to be undone.
Leans on
Higher-order derivatives, from Differentiation rules
Try a similar question
The third-order Taylor polynomial of a function about is given below. Find .Saved in this browser only.