Find the Taylor series for f(x) centered at the given value of a . [Assume that f has a power se

Find the Taylor series for f(x) centered at the given value...

Key Concepts

Taylor Series

A Taylor series is a representation of a function as an infinite sum of terms, where each term is calculated using the function's derivatives at a specific point, typically denoted as a. This series allows one to approximate or express the function locally around that point by using polynomial terms.

Power Series Expansion

A power series is an expression of the form ? a_n (x - a)^n, where the a_n's are coefficients and a is the center of the series. When a function can be expressed as a power series, its behavior near the center a can be completely understood by these coefficients, and the series converges to the function within a certain interval.

Derivatives in Taylor Series

The coefficients in a Taylor series are given by the formula f^(n)(a)/n!, where f^(n)(a) is the nth derivative of the function evaluated at the center a. This relationship ensures that the Taylor series not only matches the value of the function at the center but also its rate of change to all orders, thereby accurately representing the function locally.

Taylor Series Centered at a Point

When constructing a Taylor series centered at a specific point a, the expansion reflects the function’s behavior in terms of deviations from that point, using powers of (x - a). This shift in variable helps in approximating the function in the vicinity of the point a and is particularly useful when the function's most relevant behavior occurs around a nonzero point.

Transcript

00:01 So we're supposed to find a tator series of this one, this function, right? tater series of this function and is centered about one, right? okay.

00:14 So if we're finding the tator series of this function, then what are we going to do? you know, the tater series formula is, is this same as the mclaurian series.

00:24 It's just that this time is not at zero, right? taylor series formula is this one.

00:32 F of the nth derivative at a this time it's not zero right over n factorial and then x minus a to the power in right so this is the tator series so if you expand this one this is the same as uh f at zero right when n is zero you have this one uh so uh so when n is zero you put zero here you put zero here you put zero here you put zero here this is what you have right and then you continue so when n is one what is happening when in is one you have f prime right of a they have x minus a and then when n is 2 you have f prime prime of a x minus a squared over two factorial right that is that is a taylor series expansion so keep this one here in mind.

01:42 That is what we're going to use to find our taylor series.

01:46 So we have to find the derivatives of this one, right? so the derivatives is going to be, when you take f prime, first of all, we know that the function itself is what? function itself is x to the 4 minus 3x squared.

02:10 Plus one right so we evaluated at a what is a a is one right so we evaluated at one so we evaluated here as one right so if you have one then what is happening then you have uh uh so when you evaluate this one at one if i value this one at one you have in a negative negative one right and then you take the derivative 4x cubic minus 6x right and then you value it that one is at one as well when you do that what are you going to get you're getting uh negative 2 right so this is negative 2 so this is just calculator work and then you take the second derivative this is going to be 12 x squared minus 6 right so and then you evaluate that one at one as well.

03:28 When you do that, what is happening? and you do that, you're going to get negative six, no, positive six, right? so positive six.

03:36 And then we take the fourth, the third derivative, right? the third derivative is going to be 24 x to the power, x to the power one, right? so 24x.

03:54 And then when you evaluate that one, at 1.

04:00 What are you getting? you get in 24, right? so this is 24.

04:05 And then when you take the fourth derivative, right, you're having 24 as well...

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