Step by step
The question
Question: find the Taylor polynomial of degree 3 of √x at x = 4.
f(x) = √x
The Taylor series: the nth derivative at the point a, divided by n factorial, times x minus a to the n.
f(x) = Σ f⁽ⁿ⁾(a)n!(x − a)n
1 The derivatives
Then put in x = 4: f(4) is 2.
f(4) = 2
Take the derivative again and again.
f′(x) = 12√x
f′(4) = 14
f″(x) = −14x^(3/2)
f″(4) = −132
f‴(x) = 38x^(5/2)
f‴(4) = 3256
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
numbers : 20!, 14 ÷ 1!, −132 ÷ 2! = −164, 3256 ÷ 3! = 1512
Careful!
Do not forget to divide by n factorial.
The x squared term is f″ over 2 factorial.
Careful: not the (x − 4)² term = f″(4) × (x − 4)² ✗
3 Write the series
Put them together: each number times (x − 4) to its power.
Stop at the power 3.
P₃(x) = 2 + (x − 4)4 − (x − 4)264 + (x − 4)3512
4 Check
Let's check close to 4, at x = 4.1.
Our terms give about 2.02485.
A calculator gives 2.02485 for √x.
Almost the same.
Right!
x = 4.1: the terms ≈ 2.02485, √x ≈ 2.02485
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.
Answer
P₃(x) = 2 + (x − 4)4 − (x − 4)264 + (x − 4)3512