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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) = e^{-x} \sin x, $ $ a = 0 $

$T_{3}(x)=x-x^{2}+\frac{x^{3}}{3}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

Boston College

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in this trouble. We're us to expend easier native X times sine X around 00 Using Taylor's full meals up to or order three. Now we can see that the orginal function of effects has two functions that are both functions flanks e to the X and sign of eggs. Now we're gonna set a key to the exes or H effects and sign affects our g o X. And we're going to say that t three or fax 40 approximation. To order a Brooks made shin off the origin of function will be a multiplication of tart order approximation off the first function times a turd order approximation off to second function. It means that we're gonna follow the same type of equation. But we need to find the first and higher narratives off H effects and G off ex Wester that we have our h of X as each negative sign of X and B how RG affects as sign of X. So firmness h prime effects would be negative e to the negative X h double prime effects would be each to negative X and age. Triple prime effects then would be negative eater Nate X. Since you effect a sign effect, you prime effects would be co sign effects. G double prime would be negative. Sign X N g Triple prime would be negative coastline effects So well, now let's just plug goes in. Um G three off ex would then be equal to sign of zero plus cosa zero times X minus zero plus negative sign zero times X minus zero squared over two factorial plus negative co sign off zero times X minus zero. Cute over three Factorial. From this, we see that GT off X is that equal to sign of Sierra zero. So these air zeros and co sign of serious one. So we have X minus X cube over three factorial Now let's cock crowed H capital a stereo X. We have e to the negative zero plus negativity to negative zero times X minus zero plus each of the negative zero times X minus zero squared over two factorial plus negativity to negative zero X minus zero cube over three factorial. You know that each of these ear or negative tourism is one. So from here we see that a tree effects is an equal to one minus X plus Ex Craig, or we're too fucked Royal minus X cute over three factorial. So less now, find our, um t three effects. So basically, we're just gonna multiply g three and industry. If it is, always see that 33 off X is equal to X minus X cube. This is minus X minus X cube or T factorial months flight by one minus X plus X grade or two factorial minus X cube over three factorial If you do so be fun. P three effects as X minus X cried plus X cube or three factorial minus extra five case 24 plus extra to six divided by three factorial Times Street factorial here. We're gonna drop these terms. The reason is don't forget that we are interested in turn to order approximation. So the highest power that we could have is Turk power, and this is the expression that we're interested in them. So this would be our tea tree of X. We're asked to graft the function, and it's a proclamation on the same um, graph. If you do so, you could use any online function plotter to do that. If we do so we see that we have our function as agent a negative X which, uh, behaves this such. This is our function f off banks. Now we know turtle order approximation. Oh, are the function Dr Presents Church order approximation and he plotted. We see that it follows such a behavior. It actually does a pretty good job. Approximate thing to function around here. And if you were to capture the behavior of the function outside that region, we would need to add more terms, meaning that he would need to look at high order approximations.

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