- Topic
- Taylor polynomials
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Taylor polynomials practice question 4
Find the fourth-order Maclaurin polynomial of .
Differentiating four times is slow. Which memorised Maclaurin polynomial can you adapt, and what replaces its variable?
Start from and put . Each power applies to the whole of .
A polynomial in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §9CLP-1 §3.4Key results
Worked solution
Strategy
Substitute into a polynomial you already know rather than differentiating four times: replace u by 3x in the fourth-order Maclaurin polynomial of cos u.
Working
The memorised polynomial: even powers only, alternating signs, each divided by its factorial.
Put u = 3x everywhere; each power applies to the 3 as well as to the x.
(3x)² = 9x², and (3x)⁴/24 = 81x⁴/24 = 27x⁴/8.
A check by derivatives: each differentiation brings out a factor of 3.
Answer
The answer is T four of x equals 1 minus nine halves x squared plus twenty-seven eighths x to the fourth.
Check
At x = 0.1, T₄ and cos 0.3 both come to 0.95534 to five places.
Where this goes wrong
Writing (3x)² as 3x², which gives 1 − 3x²/2 + x⁴/8. The substitution replaces u by the whole of 3x, so the 3 is raised to each power too.
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.