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Assignment Score:0/1000ResourcesGive Up?Check AnswerQuestion 2 of 10Find the terms through degree- of the Maclaurin series of f. Use multiplication and substitution...

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Assignment Score:0/1000ResourcesGive Up?Check AnswerQuestion 2 of 10Find the terms through degree- of the Maclaurin series of f. Use multiplication and substitution as necessary:sin(z) f(z) 1-(Use symbolic notation and fractions where needed )fle) ~Question Source: Rogawski Calculus Early TranscendentalsPublisheratout USCateerprivacy polic;terms ofuSecontactmshelp

Assignment Score: 0/1000 Resources Give Up? Check Answer Question 2 of 10 Find the terms through degree- of the Maclaurin series of f. Use multiplication and substitution as necessary: sin(z) f(z) 1- (Use symbolic notation and fractions where needed ) fle) ~ Question Source: Rogawski Calculus Early Transcendentals Publisher atout US Cateer privacy polic; terms ofuSe contactms help



Answers

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f(x)$ and $a$ , and evaluating $L(x)=f(a)+f^{\prime}(a)(x-a)$ . Check your answer using a calculator.
$$
\frac{1}{0.98}
$$

But a fax. He caught two acts to the power of six and hey, you call it true. True. So first refined it's a narrative is equal to six times X to the power file. That's the point two. This is equal to six times two to the Power five, which is equal to 192. Then the linear approximation. It's the call to half to class have from two times X minus two, which is equal to 64 Last 19 two times X minus two. Then you know two point here is there one? It's come true. 64 loss. 192 times points. He was your one minus two, Which is it going to? 64.19

All right. So um for these we need the formula for an infinite geometric series. And let's remind ourselves of this. So a geometric series has a common ratio, which means we're multiplying by the same thing. Every time we start with the number A we multiplied by the number are every time. And if we add that forever Sometimes we can get a finite some if our is big like let's say two. So let's say this number is not zero and it doubles every time Then it's too big. Right? But if our if absolute value of our is less than one then the number gets smaller in value every time. So um the infinite sum is actually going to have a finite value and it's given by this formula and that's what we need to know for this problem. Um So we have the function F. Of X. Is 1/1 plus 10 X. So I'm going to write that as 1/1 minus negative 10 X. And now we have it in this form, So a is one and R is -10 x. So then this is um we can put in one for a and negative 10 X for our And so when we square it, that'll be positive again, 100 X squared -1000 x cubed and so on. So the signs change every time. And that is the MacLaurin series, right? It's an infinite polynomial um centered at X equals zero. And so it makes it a MacLaurin series rather than the tail is serious. And then the interval where it's valid. Remember this formula was only valid for absolute value of are less than one. So this is absolute value of negative 10 X less than one. Which means that the absolute value of X. So this is like 10 times the absolute value of X is not equal to less than Is less than one. So absolute value of X is less than 1/10. That's an interval. And if we want to write it may be an interval notation that will be from negative 1/10 To positive 1/10, not including the endpoints.

Okay, so we need to approximate the value of sign of 0.2 Okay, so 0.2 is approximately or close to a number we already know. So zero. So we're gonna choose our function to be f of X is equal to sign of X with a, uh, equal to zero since 0.2 is very close to zero. Right. So we're going to use a linear approximation. L love X is equal to death of a plus F prime of a times X minus a right. So again, we need to fill in. Ah, three things f of a prime of a and A here so f of a it's gonna be f of zero. All right. Or sign of zero. Remember, from the unit circle zero is this angle here? The white component of that is zero. So f of zero is zero. Well, um, f prime of a right. So first we need to find f prime of X. So if prime of X is co sign of eggs, then we plug in zero there. So we get co sign of zero. So co sign of zero is equal to one and then a, of course, is just zero. So plugging in All those terms in tow are linear approximation. Yet l love X is equal to zero. Plus, um, one times X minus cereal Simplifying further. We just get LF X is equal to X. Okay, so now our approximation l of 0.2 it's just going to be 0.2 So our answer is going to be 0.2 If we take a look at the actual value of sign of 0.2 If we plug that into account later, we get 0.10 points or zero point zero uh, 0.19 99 87 dot, dot dot Okay, So our answer is very close to the actual answer, and then we're done.


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