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Evaluate the given functions by using three terms of the appropriate Taylor series.$$\tan 46^{\circ}$$
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 given tangent of 46°. So our function will be tangent of X&X is 46°. So we need to approximate this guy here. So we're going to do so by using taylor series expansion. Um So in order to carry this out, we need a value for a. Okay, so the value of A. Must be nearest to X. Such that x minus a should be smaller. So since access 46° Well, at a equal 45° which is biography for radiance. And ah since we are working in radiance and were given Tangent of 46°. Um we can express that as attention of 45 degrees plus one degree. Which is equal to tangent of. I'll be pi over four plus by over 1 80. Okay, so first things first we need to find our function evaluated at a where A. Is pi over four. Okay, so our function F. Of X is tangent of X. So F evaluated at a is equal to one. Okay, So next we need to find this piece here. So first derivative evaluated at A. So F prime of X is equal to see can't squared of acts. Okay. So are uh function are, sorry, the derivative of seconds squared of X evaluated at a Is equal to two. Okay, So now we need to find F double prime evaluated at A. So F double prime of X is equal to two. C Can't squared of X. Tangent uh X. Okay. So F double prime evaluate A Is equal to four. All right? We have everything that we need. So let's just simply plug it into the serious expansion formula. So, we'll have tangent of Pi over 4-plus pi over 1 80 is equal to. So we'll have one plus two Times. Sir, X's pi over four plus pi over 1 80 minus or a was pi over four. Of course these are going to cancel and then the next term is so we'll have pie four multiplied by pi over four Plus pi over 180 minus pi over four squared. And of course these are going to cancel and we'll have some additional terms. Okay, so after throwing this into our calculator, we'll find that tangent of pi over four Close pi over 1 80 Is equal to 12 1.03 for 28 zero 27 zero eight. In this sir, answer.
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