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Problem 77

True or False? In Exercises $77-80$ , determine w…

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Problem 76

Pattern Recognition Use a graphing utility to compare the graph of
$$f(x)=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right)$$
with the given graph. Try to improve the approximation by
adding a term to $f(x) .$ Use a graphing utility to verify that
your new approximation is better than the original. Can you
find other terms to add to make the approximation even better?
What is the pattern? (Hint: Use sine terms.)

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Video Transcript

all right. Question. Number 76 asks us to use a graphing utility to compare the graph of the following function. F of X equals four over pi times signed pi X plus 1/3 sign three pi X with the graph below, which is just the graph that is Ray here. Um, what we're going to want to do is find a way to improve our approximation formula, This one right here to make it match this graph and the question gives us a hint. And it says, Can you find any other terms that will improve this approximation? So we know that in order to make this function looks like this, we're going tto add some extra terms and there's a pattern. So we're not just going to add them randomly. We're gonna look at the function that they give us and see if there's any pattern we can deduce. Now, when I look right here between the 1st 2 terms were given, we see that the 1st 1 to sign Pi X and the next one is 1/3 sign three pi x. So there's a couple patterns we could think about, um, the first possible pattern could have been that for every new term you divide by three on the outside of her, tricking a metric function and multiply by three on the inside. And that's a reasonable pattern. So the next term in this in that instance, would be won over nine. Sign nine Pi X and so what you would do is you could go online to Dismas Graphing calculator or use your own graphing calculator, and you'd add that new term to this function inside these parentheses. And you do that. Justus. I didn't you'd see that. Well, that doesn't make the approximation any better. It actually makes it worse. So although that is a pattern, it's not the pattern we're looking for. So another pattern you might see is that all right, well, instead of multiplying by three, what if they're just dividing by a number that is two larger than the previous number and so and multiply our adding, adding by that same number on the inside of our children to mention function? So if that were the formula, or that we're the pattern, we see that the third term in this sequence would be 1/5 sign. Five high eggs and, um, I went along with that approximation, and I found or that pattern, and I found that it does a pretty good job of approximating. So over here this is our original function that were given to approximate. And when I add that 1/5 signed five pai ex term. You see that right here at the top form or function where there's all these peaks that a new peak is at and it gets closer to having a straight line. Now, when there's three terms, it looks like this. And then I went along and added two more terms and it gets better and better. And then, for just kicks in giggles, I added three extra terms, and you could see that the more terms we add following this pattern, the closer our approximation gets to becoming a straight line at positive one and negative one. And so, if you were to continue to adm. Or on more terms infinitely, this approximation would get closer and closer to approaching a straight line, and that's how you do it

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