Step by step
The question
Question: find the Maclaurin polynomial of degree 3 of e to the -1x.
The Maclaurin series: the nth derivative at 0, divided by n factorial, times x to the n.
Maclaurin means a = 0.
Why: we want a polynomial with the same value, slope, bend and so on at 0 as f.
Take the derivative of c times x to the n, n times: you get n factorial times c.
At 0 every other term is 0 (or gone after n derivatives), so only this one is left.
So c must be the nth derivative over n factorial.
1 The derivatives
We need f and its derivatives at 0.
Start with f: put in x = 0.
f(0) is 1.
The derivative of e to the kx is k times e to the kx.
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
Careful!
Do not forget to divide by n factorial.
The x squared term is f″ over 2 factorial.
3 Write the series
Put them together: each number times x to its power.
Stop at the power 3.
4 Check
Let's check close to 0, at x = 0.1.
Our terms give about 0.90483.
A calculator gives 0.90484 for e to the -1x.
Almost the same.
Right!
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.