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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) = x^{2/3}, $ $ a = 1, $ $ n = 3, $ $ 0.8 \le x \le 1.2 $

(a) $T_{3}(x)=1+\frac{2}{3}(x-1)-\frac{1}{9}(x-1)^{2}+\frac{4}{81}(x-1)^{3}$(b) 0.000097(c) From the graph, the error is less than 0.0000532845 on the interval.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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in this problem, whereas too approximate given function effects using Taylor's polynomial off Order three and we want to expend it around, given points or one A is equal to one. Ah, we know that Taylor's polynomial off order tree will have a form like this. Obviously, we need to like to function at a given point, the first derivative secondary to on the 30 over to solo start by First Catholic and Der Bitters. So we have a flex as actually ah to third F prime of X f double prime effects and F triple primary fights is what we need. Using the function, we see that the first separatists equal to two or three extra negative 1/3 2nd ever to physical too negative to over nine extra negative for over three and the third relativist eight over 27 extra, the negative seven over three. And if you plugged a given point and we see that F one is equal to one at a primal one is equal to two or three af double prime of one is equal to negative to over nine, and a triple prime one is equal to eight over 27 so we need to do is now to thank those under, plug it into the sea question and find the third order. Taylor Shmona mellitus for its function as one plus two over three times X minus one minus one over nine X minus one Squared plus four or 81 X minus one. Cute. All right. Now, this was part in part B, whereas two sailors and they felt it Thio decide on me to determine d accuracy off the separate summation. Since we're interested in ecru curious you were interested in and plus won't turn Ah, that would be since and is given a street and plus one would be four. So we're actually interested in for Terry throughout dysfunction now using third driver told dysfunction And if you take their it off that one more time you find 1/4 or bitter as negative 50 six over 81 extra t negative 10 over three that, in order to see how this function is behaving what it is increasing or decreasing even need to look at the dirt off that So that would be the 5th 3rd toe off the initial function and they're better off. This would be positive. How do I know that it is positive? Well, if you look here, we see that the function itself is a constant and that it's positive. The first, Sarah Attar, is positive. Second, narrative is negative. 30. Operative is positive for charity was negative, so the conservative will be positive. That's why should be greater than zero. That's an alternating ah Siri's. Now, since 50 relative is greater than zero, it is positive it means that they functions are for charity. L F F four fax will be then increasing. Right now we know that four effects is increasing and because off this negative sign right here and since X is positive and miss given interval, we know that the function itself sort of four derivative is increasing. However, it is negative everywhere. Now imagine a function that is increasing but negative everywhere. So let's set of you have functioning towards quadrant. It is increasing. However, it is always negative, and since we're interested in absolute value of that function while determining the accuracy, it means that the function of suitability off it will be maximum at the lower end. So when it Zico to point eight all right now, it means that F four of 0.8 will be less than or equal to end. So from here we see that done. M will be greater than or equal to 56 over the negative six or eight and one times 0.8 to the power of 10. A word three. Yeah, Taylor's nickel tells us that the accuracy for Aunt Order pull no meal will be less than or equal to and divided by and plus one factorial. What's like by absolute value of X minus eight times and plus one that is equal to keep like everything. And we know that Emma's 56 over 80 on times point into the 10 or three. What is that? I'm plus one, Will industries and plus one is four factorial x this 0.8 A s, one absolutely off down to the power of three. From here we see that, then our tree off eggs. So do it yourself. This will be about 0.0 097 So this is just an approximation or estimation about the acres in I mean parts us to graft the air and find the actual, um, accuracy. Now the Arab will be the difference between the deduction itself, up effects and the third order approximation to that. We know the function we also found in part night throat order Taylor. Pull them of that represents dysfunction if we look at the difference and if he graft absolute Sally off that this is what we get. Don't forget we were interested in the area. Uh, lower end. So one into Mexico 10.8. And from that we see that the area's about points. You're 000 53. So it means that the actual error is less than the estimate that we found in part B. That is 0000 night, seven.

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