Step by step
The question
Question: find the Maclaurin polynomial of degree 5 of sin x.
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.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The derivatives
We need f and its derivatives at 0.
Start with f: put in x = 0.
f(0) is 0.
sin gives cos, and cos gives minus sin.
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 5.
4 Check
Let's check close to 0, at x = 0.1.
Our terms give about 0.09983.
A calculator gives 0.09983 for sin x.
Almost the same.
Right!
5 Pick the choice
Our answer is x − x³ over 6 + x⁵ over 120.
That is choice A.
6 Why not the others
B is x + x³ over 6 + x⁵ over 120.
Wrong: the signs alternate.
C is 1 − x² over 2 + x⁴ over 24.
Wrong: that is cos x.
D is x − x³ over 3 + x⁵ over 5.
Wrong: divide by 3!
and 5!, not 3 and 5.
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Then: check.
Then: pick the choice.
Last: why not the others.