Calc Week 7 Flashcards
Taylor’s Theorem
let I be an interval, and let a be in I
assume that f is a function defined on I such that f ^(n+1) of x exists for all x in I
Then
f(x) = P sub n(x) + R sub n (x)
where P sub n(x) = nth taylor polynomial of f at x=a
and R sub n(x) takes the form
R sub n(x) = f ^(n+1)of z(/n+1)! tiimes (x-a) ^(n+1)
where z is some number between a and x
example on notes
Bound on the remainder
R sub n(x) is almost always small
2nd order Taylor Expansion of f at x=a
n = 2
f(x) = P sub 2(x) + R sub 2(x)
Taylor series
let f be an infinitely differentiable function at the point x=a. the Taylor series of f, centered at a is the series
sigma from n=0 to inf of f^(n) of (a) / n! times (x-a)^n
if a=0, the maclaurin series of f
6 useful Taylor series all at x=a=0
exp(x) = sigma from n=0 to inf of x^n/n! for all x
1/1-x = 1 +x + x^2 + x^3+… for -1 <x <1
log(1+x) = x - x^2/2 + x^3/3 - x^4/4 +…for -1 <x <= 1
arctanx = x - x^3/3 +x^5/5 -x^7/7 +x^9/9 -… for -1<=x<=1
sinx = x - x^3/3! +x^5/5! - x^7/7! +x^9/9! -… for all x
cosx = 1 - x^2/2! +x^4/4! - x^6/6! + x^8/8! -… for all x
connection
use the useful ones to avoid finding too many derivatives
writing a series
- find series you know with y
- plug in y from equation given
- simplify
4.fill in the blanks
if f is infinitely differentiable, then
the function usually = its taylor series
writing a series for an integral function
use the series for the integrand and integrate each term
series converging
write out the terms and take their limit as n -> inf
sequence converging
lim n-> inf and that tells you something about the series (nth term test)