If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Worked example: coefficient in Maclaurin polynomial

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.A (LO)
,
LIM‑8.A.1 (EK)
,
LIM‑8.A.2 (EK)
,
LIM‑8.B (LO)
,
LIM‑8.B.1 (EK)

## Video transcript

and derivative of G at X equals zero is given by so the nth derivative of G who evaluated at x equals zero is equal to square root of n plus seven over N to the third for n is greater than or equal to one what is the coefficient for the term containing x squared in the Maclaurin series of G so let's just think about the Maclaurin series for G so if I were to have my function G of X the Maclaurin series I could say approximately equal to especially if I'm not going to list out all of the terms is going to be equal to well it's going to be equal to G of 0 plus G prime of 0 times X plus G prime prime of 0 divided by I could say 2 factorial but that's just 2 times x squared and that's about as far as we go because we just have to think about what is the coefficient for the term containing x squared I could if they said what's the coefficient for the term containing X to the third I would keep going and go G prime I would take the third derivative evaluated at 0 over 3 factorial I could view this as a factorial 2 but that just evaluates to two I could view this is 1 factorial I could view this as 0 factorial just so you see it's consistent idea here and I could of course keep on going but we just care about they're just asking us what is the coefficient for the term containing x squared so they just want to they just want us to figure out this what is this thing right over here so to know that we need to figure out what is the second derivative of G evaluated x equals 0 well they tell us that over here it's a little bit unconventional where they give us a formula a general formula for any derivative evaluated at x equals 0 but that's what they're telling us here so in this case the N isn't 0 the N is the derivative we're taking and that's going to be our second derivative so this is so if I wanted to figure out G if I am figuring out the second derivative and I could write it like that evaluated 0 or I could write it like this just so the notation is consistent I could write it like that the second derivative evaluated x equals 0 is going to be equal to well our n is two so this is going to be the square root of two plus seven over two to the third power so 2 plus 7 is 9 take the principal root of that it's going to give us positive three over two to the third which is 8 so this part right over here is 3/8 so the whole coefficient is going to be 3/8 that's this numerator divided by divided by 2 which of course is equal to 3 over 16 and we're done they didn't want us to figure out you know a couple of terms of this which we could call the Maclaurin polynomial and nth degree Maclaurin polynomial they didn't want us to find the entire you know keep going with this series they just wanted to find one coefficient right here the coefficient on the second degree term which we just figured out is 3/16
AP® is a registered trademark of the College Board, which has not reviewed this resource.