00:01
So in this question, we have a function f, and f is defined by f of x equals the square of x.
00:08
And we say, what is the approximation for the value of square to three obtained by using a second -degree tailor polynomial for the function f about x equals four? well, we'll remember our formula for a second -degree tailor -polynomial.
00:26
My p2 of x since i'm about x equals four this time is going to look like f of four plus f prime of four over one factorial times the quantity of x minus four plus f double prime of four over two factorial times x minus four being squared and so let's get our derivatives so my f of x is the squared of x, my derivative of that f prime of x is 1 over 2 root x, or equivalently 1 half x to the negative 1 half power.
01:15
And that means that my second derivative is going to be negative a quarter times x to the negative 3 halves power, or equivalently negative 1 over 4 times the square root of x cubed.
01:33
Now i need to evaluate each of these at 4.
01:37
So f of 4 is certainly the square root of 4 is just 2.
01:45
My f prime of 4, that's 1 over, 2 times the square root of 4, and that's just a quarter.
01:57
While my f double prime of 4 is negative 1, over 4 times the square root of 4 cubed, that's the square root of 64.
02:10
Square of 64 is 8, 4 times.
02:14
8 is 32, so that's negative 132.
02:20
And so this means that my p2 of x this time, it is f of 4, f of 4 is 2, plus f prime of 2, a quarter over 1 factorial, times the quantity of x minus 4, plus f double prime of 4, that's negative 1 over 32, divide that by 2 factorial, and i multiply by the quantity of x minus four being squared.
02:57
And so if i clean this up, what is my p2 of x? well, it's two plus a quarter times the quantity of x minus four minus one 64th times the quantity of x minus four being squared...