00:01
So in this question, we want to calculate the taylor polynomials, t2 and t3, centered at x equals a, for the function f of x equals 25 times the natural log of the quantity of x plus 1, and we're centered at zero this time.
00:16
So let's start with our formula for t3 of x, because if we get t3 of x, we will know t2 of x for free.
00:27
So my formula for t3 of x, you may recall, is f of a, which is zero this time, plus f prime of 0 over 1 factorial x, plus f double prime of 0 over 2 factorial times x squared, plus f triple prime of 0 over 3 factorial times x cubed.
00:56
So i'm going to have to figure out three derivatives of the function f of x squared.
01:02
Equals 25 times the natural log of x plus 1.
01:09
Well, let's see.
01:10
My first derivative is just 25 over x plus 1.
01:17
And then my second derivative, remember this guy is 25 times x plus 1 to the negative first.
01:26
This would be negative 25 times the quantity of x plus 1 to the negative second power, or, negative 25 over x plus 1 quantity square.
01:44
And then if i want my third derivative, i've got 50 times the quantity of x plus 1 being raised to the negative third power, or equivalently 50 over x plus 1 being cubed.
02:03
Now if i evaluate all of these at 0, well, f of 0 is 25 times the l n of 1...