Step by step
The question
Question: find the Maclaurin polynomial of degree 4 of cos x.
f(x) = cos x
The Maclaurin series: the nth derivative at 0, divided by n factorial, times x to the n.
Maclaurin means a = 0.
f(x) = Σ f⁽ⁿ⁾(a)n!(x − a)n
1 The derivatives
Then put in x = 0: f(0) is 1.
f(0) = 1
sin gives cos, and cos gives minus sin.
f′(x) = −sin x
f′(0) = 0
f″(x) = −cos x
f″(0) = −1
f‴(x) = sin x
f‴(0) = 0
f⁗(x) = cos x
f⁗(0) = 1
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
numbers : 10!, 01!, −12! = −12, 03! = 0, 14! = 124
Careful!
Do not forget to divide by n factorial.
The x squared term is f″ over 2 factorial.
Careful: not the x² term = f″(0) × x² ✗
3 Write the series
Put them together: each number times x to its power.
Stop at the power 4.
P₄(x) = 1 − x22 + x424
4 Check
Let's check close to 0, at x = 0.1.
Our terms give about 0.995.
A calculator gives 0.995 for cos x.
Almost the same.
Right!
x = 0.1: the terms ≈ 0.995, cos x ≈ 0.995
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.
Answer
P₄(x) = 1 − x22 + x424