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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) = \sin x, $ $ a = \pi/6 $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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 polynomi…

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in this problem, we're as to expend sign effects around five or six using terrorist polynomial up to or dropped three. And this is the expression that we're gonna use from here. We see that we need to calculate the first service of secondary to on the turkey with your function. So let's, um, start Dad, Let's ride to function again F or flexes equal sign effects and prime If thanks Dave Itzkoff sign It's co sign effects. Now their ritual. Consign his negative sign, and the original negative sign is a negative of co sign from this lesson, like dysfunction. It's there with you Have to give him points for half off. Pirates six F prime off five or six F double primal fire or sex and F triple prime off Pirates six. You could use your calculators if you plant fire six Into this given expressions, we see that, um, sign elect at that 60.1 health and primal. It at that point is equal to skirt of three or two after primal. It at that point is equal to negative one Health and F triple primal licked at that point is you do negative off skirted off three. Over to now, we have everything that we need. So all we need to do is to take these values to function and say whatever letters that given point and plug it in to this expression here and find 23 or facts as one. Health plus skirt of three or two times explains power sex plus after bill. Prime is negative. One house times X minus five or six squared or two factorial, plus native skirted or three minus two times X once played for six Cubed delight about three factorial from this weekend, right 33 of X. Yes, one Health Plus Skirted a three year old to explain this play over six minus one over four X minus Supply over six. Squared minus skirted or three over 13. Sorry, 12 x minus pi over six. Cute. All right. We're also asked a graft this the function and distort order approximation on the same graph we have. Sign off. Exes are function, so it has a periodic psychotic behavior. All right, this is our function. Effects is a good sign. X less graft. The approximation is well did 30 order approximation follows the sign excreta. Bell and this range. And if you were to get a better proclamation, we even need to add more terms so that we could capture to be here your debts happening in this region and in this region.

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