00:01
In this question, we want to find the quadratic approximating polynomial for the function given below.
00:06
We're going to give exact values for our coefficients, and then we're going to use that quadratic polynomial to estimate the value given.
00:13
So i want to find p2 of x for the function f of x, and i believe it's better always to take pictures instead of cut and paste, because i think that the function is f of x equals 1 over x plus 1.
00:28
Without the parentheses in the right place, it's not totally clear, but based on what you have here, i'm pretty sure it's f of x equals 1 over x plus 1.
00:38
So how do i find p2 of x? we're centered at 0.
00:43
So we have a formula for this.
00:45
It's f of 0 plus f prime of 0 over 1 factorial times x plus f double prime of 0 over 2 factorial times x squared.
00:59
So now my f of x is 1 over x plus 1.
01:03
What is that? it's x plus 1 quantity to the negative first power.
01:08
So now i need to get two derivatives of this.
01:11
So first of all, my f prime of x, it is the opposite of x plus 1 quantity to the negative second power, while my f double prime of x is 2 times the quantity of x plus 1 being raised to the negative third.
01:28
And so now, what do we know about f and its derivatives at 0? f of 0 is 1 over 1.
01:37
It is 1.
01:39
F prime of 0, what is that? it is negative 1...