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Find the Taylor polynomials $ T_3(x) $ for the function $ f $ centered at the number $ a $ Graph $ f $ and $ T_3 $ on the same screen.

$ f(x) = xe^{-2x}, $ $ a = 0 $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Find the Taylor polynomial…

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Find the Taylor polynomia…

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Find the Taylor polynomi…

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this problem, whereas to expend x type she's your negative X ray Ron zero using Taylor's polynomial off order three. We know that the expression for tennis polo for off ordered three as given, uh, here and we can see that our function a has to sell functions that are functions off. I said his ex and needs you too negative each of the X And instead of calculating t three effects of proto order approximation directly, were going to say that you threw effects physical to hurt order approximation off age multiplied by third order approximation of G, where this X is our function, age and eater Negative X is our function Jeep. And here we can see that any to calculate first order, second order and third order derivatives. So let's do that separately for both age and G, he had h of eggs as equals. X h crime affects would be one h double primer. Facts would be zero an h triple prime. A fax would also be zero. He also have GI effects as you go to eat negative checks, so G prime effects would be negative to each negative. Two ex g double primer, fax would be four each nation to X and the key triple primal facts would be negative. Eight. Eat a negative to X or it. Now let's first start from third order approximation of function. H. We have to function social f off. Eight Sorry h of X as eggs and since we're trying to let that around zero ff zero would be zero plus one times x 10 plus zero since secondary to Syrian. Third order there with his also zero. So farmers we find a three effects. His eagles act so that is equal to you. Your function itself well, let's find G three effects that is each of the negative. Two times zero plus native to Egypt e zero times X minus zero plus four times each of zero times X minus zero squared over two factorial plus NATO eight each of the zero times X minus zero cubed or t factorial. From this, we find GT effects as one one's two X plus two Expert minus age X cubed over three factorial. So now less confident. E three effects for months playing those two. So we have X times one one's two x plus two Expert on this eight X cubed over three factorial that is equal to X minus two X cried plus two X cubed minus state extra to fort over three factorial. Now the important thing here is that we're interested in third order approximation, so we could have most have third order terms. So we're gonna drop this term and we're going to say that this is the part that we're interested in. So that is our T three of X. Where else asked a graft to function. And that's approximation of the same graph. We have our function. Thanks. As does this is function f effects died Isaac of X times each donated to axe. We also have our church order approx mission where we know the function and that is how it behaves. It fellows, the function pretty good around here and here it overshoots and after this point on the negative cited follows function and now they're shoots. So that would be our third order approximation to that function. TT off X

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