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(a) Approximate $ f $ by a Taylor polynomial with degree $ n $ at the number $ a. $

(b) Use Taylor's Inequality to estimate the accuracy of the approximation $ f(x) \approx T_n(x) $ when $ x $ lies in the given interval.

(c) Check you result in part (b) by graphing $ \mid R_n(x) \mid . $

$ f (x) = \sin x, $ $ a = \pi/6, $ $ n = 4, $ $ 0 \le x \le \pi/3 $

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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

Idaho State University

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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(a) Approximate $f$ by a T…

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(a) Approximate $ f $ by a…

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in this problem. Worst approximate effects is he could sign effects using Taylor's phone number off Order four. And we want to expend dysfunction using Kalish Bonneville around Pie over six. So we know that be a court order to eligible New meal will have formed given on right. So obviously we need to talk like to function, and it's deputised at the given point. Okay, so let's start by finding the devotees and calculating derivatives so we have affects us in it. Sign off. Excessive prime effects would be co sign effects. F double prime would be negative side effects after pill. Prime effects would be negative of co signer packs. Enough for territo effects will be equal to sign up x 34 oh thanks. Will then be signed pie or six plus co. Sign off by over six times, explains Priore. Six Minus sign high or six times X minus five or six squared over two factorial minus co. Sign pi or six times X minus five or six. Cute divide by three factorial and then we have finally signed pie or six times X minus five or 6 to 4. Theo, I'd buy four factorial now emulating those. We find t four effects as one. Health plus Scurried off three over two times X minus five or six minus 1/4 off X minus five or six squared minus Skirted a three or 12 x minus five or six cute plus one over 48 x minus pi over six to the fourth power. Now this is part of a in part you were asked to use telegenic alloted to determine the accuracy Officer approximation sensitizes girls who were interested in n Plus one term. Why are interested in that? Because we're expending dysfunction. So basically we're approximating dysfunction. And does this use representation of function? And we truncated the Siri's at the fourth term. So the next sermon, which is the 5th 3rd will determine the accuracy office approximation. So then it means that we're interested in at five off X. What is that Final effects? Well, the five affects is equal to co sign effects and to determine the behavior off 50 over two effects, we would need to look at 60 writable so f six if X is equal to negative sign X. And we know that since in this given intervals between zero and pi or three Sinus positive. But because of that negative sign, it means that this F six will be negative. So we know that F sex will be negative. Now send six. Charity was negative, which means that 5th 32 well, then be decreasing. And in this given in trouble, co Sinus positive. So it will be positive. So it is always positive entities dig raising. So since this is the case, the maximum oh absolute value of five effects will be at the, uh, left bound points or left on dhe point. So it means that absolute failure at five off thanks will be less than or equal to em. So f five off the given point, it's Pirates six. Well, actually, we're calculating that at the left endpoint, so that would be a five off. Zero will be less than I am now at five Cisco side effects, a co signs your is one said. From here we found that M will be greater than or equal to one from tells Nicole, too. We know that our four off accents and it's four will be less than him. Divide by and possible, so that is four plus one factorial multiplied by X, which in this case is zero minus. What does A well, A is given in pi over six times in plus one sort is four plus one. Here. We know that and we will take them as one since Diana's warp. You count here from this we found D accuracy to be less than or equal to 10.0 zero 3 to 8. So our Fort order of population is this accurate? And actually, it's quite a crit. And it makes sense since we're including for terms, don't forget, the more times you have, the more accurate it is. Now in party were us too, Uh, graph Tierra and find the exact curiously already area What? We're gonna define air as a difference between the approximation sortie for effects and the actual function. Wife ex Um, this is our approximation. Taylor spell Nebula Fort order. And here we have the exact function side of ex defeat plot that into given in terrible zero pirate three. And if you look at the lower end point if see that the actual air is 30.0.39 and this is less than the air we find using Taylor's inequality, that is 0.0 three to eight. Don't forget. Taylor's inequality gives us an estimation about the accuracy, and this part is the exact air or D security.

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