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Evaluate the given functions by using three terms of the appropriate Taylor series.$$\ln 3.1$$

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 5

Taylor Series

Series

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Okay, so for this example we are asked to numerically approximate natural log of 3.1. So it's obvious that our function is natural log of X and X is equal to 3.1. So we're essentially going to numerically estimate natural over 3.1 by using the Taylor series expansion formula. Okay, so in this formula we see a value for a all throughout it. Um So essentially the value of A must be nearest acts Such that X -A should be smaller. So since X is equal to 3.1 and A is equal, we'll just let a equal three. Okay, so let's get started. The first thing that we need to find is our function evaluated at A. So we know that our function is natural log of X and Therefore have evaluated a is equal to natural log of three. Okay, so now we need to find our first derivative and evaluate it at A. So f prime of X is equal to one over X. So therefore F prime of a Is equal to 1/3. Okay, so now we need to find our 2nd derivative and evaluate it at a. Okay, so F double prime of eggs is equal to minus one over X squared. Okay, so now we're going to evaluate it At a given point, so that will just be -1/9. Okay, so now we have all the pieces that we need. So let's just plug them into our series. Okay, so now we'll have natural log of 3.1 is equal to. So we'll have Natural log of three Plus 1/3, 3.1 minus three minus. Okay, so in our taylor series expansion we have a two factorial here and we have a -1/9. So therefore um we are going to have that will just be minus one over 18, 3.1 -3 squared plus additional terms. So after throwing this into our calculator, we'll see that our expression For a natural log of 3.1 is 1.13 13

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