let fx ln12x a find the third degree taylor polynomial of fx centered at a1 b find an upper boun

Alright, so let's go through. We know f of x is ln of 1 plus 2x f prime. So this is just chain r

Let fx ln12x a find the third degree taylor polynomial of fx...

Transcript

00:01 Alright, so let's go through.

00:01 We know f of x is ln of 1 plus 2x f prime.

00:10 So this is just chain rule, right? so you're going to get a 1 over 2x because it's an ln.

00:15 I'm sorry, 1 over 1 plus 2x.

00:19 But then whenever we take the derivative of the argument, so the derivative of 1 plus 2x, we're going to get a 2.

00:26 So all of a sudden then you get a 2 over 1 plus 2x.

00:31 I'm kind of assuming if you're doing taylor expansions that derivative process is probably fairly trivial by now, but we can kind of go through it in some amount of detail.

00:40 So anytime, when we take a second derivative, every time you take the derivative of an inverse like this, 1 over 1 plus 2x, you're going to get a negative 2 on the top.

00:52 So you get negative 4, 1 plus 2x squared, and then f triple prime, right? and then here you're going to get 16 over 1 plus 2x.

01:07 Again, this is fairly trivial at that point.

01:09 So remember, the taylor expansion is a summation from n to 0 of the nth derivative evaluated at a over the n factorial times x minus a to the nth power.

01:28 So let's just go through and evaluate these.

01:31 So first we're going to have 1 over 0 factorial, which is just 1 by the way, and then ln of like the original function basically evaluated at 1 at a.

01:41 So 1 plus 2 times 1 is 3 times x minus 1 to the 0th power, and then we're going to get 1 over 1 factorial times 2 thirds now, x minus 1 to the first power now, 1 over 2 factorial times 4 over 9.

02:04 I'm just plugging in 1 here, and you get 2.

02:07 Okay, and then so right now we have a second degree polynomial, and we're going to get that third degree in now...

Question

Let f(x) = ln(1+2x) A) Find the third degree Taylor polynomial of f(x) centered at a=1 B) Find an upper bound error if f(x) is approximated with T3(x) for 0.5 ≤ x ≤ 2.

Question

Let f(x) = ln(1+2x) A) Find the third degree Taylor polynomial of f(x) centered at a=1 B) Find an upper bound error if f(x) is approximated with T3(x) for 0.5 ≤ x ≤ 2.

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