- Topic
- Taylor polynomials
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Taylor polynomials practice question 2
Find the coefficient of in the third-order Maclaurin polynomial of .
Which Maclaurin polynomial that you already know is hiding inside ? Using it saves differentiating three times.
Start from and put .
A fraction.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Substitute into a polynomial you already know rather than differentiating three times: replace u by 2x in the third-order Maclaurin polynomial of log(1 + u).
Working
The memorised polynomial: the uᵏ term is divided by k, and the signs alternate.
Put u = 2x; each power applies to the 2 as well as to the x.
A check by derivatives: f‴(x) = 16/(1 + 2x)³, so f‴(0) = 16.
Answer
The answer is 8/3.
Check
At x = 0.1 the cubic 2x − 2x² + (8/3)x³ gives 0.18267, close to log 1.2 ≈ 0.18232.
Where this goes wrong
Writing (2x)³ as 2x³, which gives 2/3. The substitution replaces u by the whole of 2x, so the 2 is cubed too: (2x)³/3 = 8x³/3.
Leans on
Composing functions, from Functions, graphs and inverses
Try a similar question
Find the coefficient of in the third-order Maclaurin polynomial of .Saved in this browser only.