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Calculate the Taylor Polynomial Ts for the function f (r) = In(r + 2) centered at a =...

Question

Calculate the Taylor Polynomial Ts for the function f (r) = In(r + 2) centered at a =

Calculate the Taylor Polynomial Ts for the function f (r) = In(r + 2) centered at a =



Answers

Calculate the Taylor polynomials $T_{2}$ and $T_{3}$ centered at $x=a$ for the given function and value of $a .$ $f(x)=\frac{x^{2}+1}{x+1}, \quad a=-2$

For this problem we are asked to calculate the 2nd and 3rd order taylor polynomial is for the function F of X equals one over X squared plus one and the value A equals negative. So to begin, we are going to be taking the derivatives of our F. There. So we need to apply the chain rule. We will get that the first derivative is going to be negative two X over X squared plus one squared Than the 2nd derivative. We're going to have to apply the quotient rule here. So we'll have derivative of negative two times X squared plus or negative two X times X squared plus one elsewhere. That's going to give us a negative two times X squared plus one all square. Then we would have minus the derivative of X squared plus one. All squared times negative two X. So we'll have negative two X. And then the derivative there going to be too times X squared plus one times two X. And then we're dividing by X squared plus one squared or rather X squared plus one. Now all to the power of four rather. So I'm going to pause and simplify this off screen. All right. So we can simplify that down to six X squared minus two divided by X squared plus one. Cute. And then we'll take the third derivative here. I'm just going to report the simplified form. That's basically just another application of the chain rule and quotient rule there. So the result that we get is going to be negative 24 X times X squared minus one divided by X squared plus one All to the power of four. So now we just need to plug in um X equals negative one. Start calculating these out. So fx is just going to be 1/2 an F prime of X. It's going to be positive too divided by ah two all squared. So that's 2/4. So it's going to be just 1/2 than F triple flips. That should be F prime of negative one rather F double prime of negative one. It's going to be well that's six times naked one squared. So it's just 6 -2. So we'll have a four in the numerator divided by negative one squared plus one. So divided by two cube. Uh huh and two cubed 2 4. Then that would be taken care eight. So that's going to simplify down to one half again there and then For the 3rd derivative You'll have -24. Actually positive 24 times one squared minus one. It's going to go just to zero. So that tells us then that T two Is going to equal 1/2 plus one half times x minus one plus 1/2 times two factorial. That's going to be won over four times x minus one squared. And then we would actually, for the T two, we would end there. Well, you can note that that's going to be the same thing as the T three Because of the fact that our 3rd derivative zero.

For this problem we are asked to calculate the 2nd and 3rd order taylor polynomial is for the function F of X equals one over X plus one. In the value A equals two. So to begin, I'll move that A up above because we want to have the function as well as its derivatives and then the function evaluated at A. So we have 1/2 plus one. So everybody is just going to be won over three. Then we'd have f prime of X. The first derivative, we can apply the chain rule here turns out to be pretty simple. We get that is going to be negative one over X plus one squared. Which means then that's F of a. It's going to be negative 1/2 plus one square two plus one is still going to be three. So that's going to be negative 1/9. Or rather that should be a crime of a there Than the 2nd derivative. It's going to be positive too Over X-plus one Cube. So then we'll have F double prime of a. Going to be positive to over three Cube 9 times series 27. So this is going to be to over 27 And the 3rd derivative it's going to be negative six over X plus one to the power of four. So we have our third derivative evaluated at A. is going to give us -6 divided by. Okay, now we can write this negative six is the same thing as negative 3/2. And then we have over three to the power of four. So we can divide out one of the threes from the numerator and denominator. So we're left with negative to over three cube. So we actually end up with a negative -2/27 there. Okay so now that we have all of that, we can construct our first or not. First order polynomial. Our second order polynomial, which is just going to give us 1/3 plus. Actually b minus that would be 1/2 times 1/9. So that's going to be negative 1/18 times X minus To not Cube. Just X -2 there. And we have plus 2/27 times x minus two squared. And our third order polynomial is going to be won over 3 -1/18 times who I just realized slight mistake there. Um Actually that should have been negative 1/9 out front here And then the X -2 would be divided by two. So that should turn into just 1/27. So then this would be negative 1/9 times X -2 And plus 1/27 Times X -2 Squared. Then we'd have -2 Um over 27 times three factorial Times X -2 cubed. And that gives us our 3rd order Taylor polynomial

For this problem, we are asked to calculate um the 2nd and 3rd order taylor polynomial for the function F of X equals sine of X and a equals pi by two. So to begin, we'll want to take our derivatives here. So we know that F prime of X is going to be negative or not negative. Excuse me. That's just going to be coast of X. And then the second derivative with respect to X is going to be negative sine of X. Well, want to go up to the third derivative for that 3rd order Taylor polynomial. So we then note that our third derivative is going to be negative cassette decks. Now we know that equals pi by two. I'll actually move that up here. A equals pi by two. So We want to then calculate out what FFP I buy two is so sine of pi by two. That is going to be just one Then 1 2nd here. Mhm. F. Prime of Pie by two. That is going to be zero. F double prime of pie by two. It's going to be negative one and F triple prime of pie by two Is going to equal zero. So our second order taylor polynomial is going to be just one, then we would have minus. Um Yeah, it would be minus 1/2 times X minus pi by two squared. And that is actually going to do the same thing as the third order polynomial because even when we go up to that third derivative, because that third derivative at pie by two is going to be zero, um The third order polynomial is going to give the exact same remaining terms.

For this problem we are asked to calculate the 2nd and 3rd order taylor polynomial is for the function F. Of X equals E. To the power of two X. And the value A equals lawn of two. We'll begin by taking our derivatives. So the first order derivative or first derivative brother is going to be to either of two X. Second derivative is going to be four E. To the power of two X. And the 3rd derivative is going to be eight E. to the power of two x. He then wants to go through and evaluate these when X is taking on the appropriate value there one of two. I'm going to pause and evaluate those off screen. Alright, so f of Lawn two is going to be four F prime of loan to Is going to be eight F double prime of lawn too is going to be 16 and F triple prime Of Lawn to it's going to be 32. Having that we can write our 2nd and 3rd order polynomial. Our second order is going to be four plus eight times X minus lawn too. Plus 16/2 factorial. So that's just going to be eight times X minus lawn of two. All squared. Then our third order polynomial is going to be very similar up to that squared term. I'm just gonna copy and paste it there. We'll have T3 equal to that. Up to that second Order term. Then we'll add on plus 32/3 factorial times X- Lawn of two. All cute.


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