Calc Week 7 Flashcards

1
Q

Taylor’s Theorem

A

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

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2
Q

Bound on the remainder

A

R sub n(x) is almost always small

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3
Q

2nd order Taylor Expansion of f at x=a

A

n = 2
f(x) = P sub 2(x) + R sub 2(x)

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4
Q

Taylor series

A

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

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5
Q

6 useful Taylor series all at x=a=0

A

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

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6
Q

connection

A

use the useful ones to avoid finding too many derivatives

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7
Q

writing a series

A
  1. find series you know with y
  2. plug in y from equation given
  3. simplify
    4.fill in the blanks
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8
Q

if f is infinitely differentiable, then

A

the function usually = its taylor series

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9
Q

writing a series for an integral function

A

use the series for the integrand and integrate each term

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10
Q

series converging

A

write out the terms and take their limit as n -> inf

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11
Q

sequence converging

A

lim n-> inf and that tells you something about the series (nth term test)

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