Problem

Evaluate the given functions by using three terms…

Problem 23 Easy Difficulty

Evaluate the given functions by using three terms of the appropriate Taylor series.
$$e^{\pi}$$

Answer

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Calculus 2 / BC

Basic Technical Mathematics with Calculus

Chapter 30

Expansion of Functions in Series

Section 5

Taylor Series

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Video Transcript

Okay. So for this example we are asked to essentially numerically approximate either the pie. Uh So we are given that our function is either the Acts and access pie. All right. So ah we need to find first three terms that the taylor series and the taylor series is found by using this uh formula here. So let's get started. First Things 1st. We need a value for a. So what are we gonna choose for a? Well, we're going to let a equal three. Just simply due to the fact that the value of A must be nearest to Act such as x minus A should be smaller. Okay, so now that we have a value for a let's get started and let's find the first term of our series. So we can say that F. Of A is equal to E cubed. And the 2nd and 3rd term will be the exact same thing. Just simply do the fact that derivative E to the X. Is E. To the X. So if double private A. Is equal to E cubed. So now we have the first three derivatives evaluate A. Then let's put our series together. So well I. E to the pi is equal to so we'll be cubed us E cubed X minus A. So our excess pie, n. r. a. is three. So plus he three Hi -3 Squared over two factorial plus some additional terms. And as you see we have common factor if he here. So let's just pull that out. So we have eat to the pi is equal to e cubed one plus pi minus three plus hi minus three squared over two factorial plus some additional terms. Okay, so after plugging this into our calculator, we'll get that E to the pi is equal to So well at 23 one 30 eight. And this is our answer.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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