Step by step
The question
Question: find the Maclaurin polynomial of degree 3 of e to the x.
f(x) = ex
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
The derivative of e to the kx is k times e to the kx.
f′(x) = ex
f′(0) = 1
f″(x) = ex
f″(0) = 1
f‴(x) = ex
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!, 11!, 12! = 12, 13! = 16
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 3.
P₃(x) = 1 + x + x22 + x36
4 Check
Let's check close to 0, at x = 0.1.
Our terms give about 1.10517.
A calculator gives 1.10517 for e to the x.
Almost the same.
Right!
x = 0.1: the terms ≈ 1.10517, e^x ≈ 1.10517
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.
Answer
P₃(x) = 1 + x + x22 + x36