So, we are continuing to learn more and more about who Joe Biden will be putting in his administration, and unsurprisingly, most of his picks, with a few exceptions, are completely terrible people. Yeah. Ronald Klain is someone who we found out about last week. He will be Joe Biden's chief of staff, and this individual has had a long career at a venture capital firm. Oh, goodie. But this week, he announced more picks for key roles, including Tony Blinken as Secretary of State, Alejandro Mayorkas as DHS Secretary, Linda Thomas Greenfield as UN Ambassador, John Kerry as Climate Envoy with NSC seat, Avril Haynes as Director of National Intelligence, and Jake Sullivan as National Security Advisor. Now not all of these names are terrible. I think that Linda Thomas Greenfield as UN Ambassador, that doesn't necessarily concern me. I think that Alejandro Mayorkas, he is the individual who basically is the architect of DACA during the Obama era. So I don't have any problems with him being DHS Secretary. However, what I do have a problem with is Tony Blinken, and to a lesser extent, Jake Sullivan, who is basically the same type of person. Tony Blinken is an individual who has repeatedly defended the atrocities of the apartheid state of Israel, and he bragged about defending them. He bragged about the United States being the only country to defend Israel when everyone else in the world condemned their behavior. Take a look. That unprecedented support is true in terms of our vigilance to protect Israel's legitimacy on the world stage and fight for its full and equal participation in UN institutions. We help secure Israel's permanent membership in the Western European and others group, as well as its membership in the like-minded Human Rights Caucus, from which it had long been excluded in New York. Last year, the United States opposed 18 resolutions in the UN General Assembly that were biased against Israel. On five occasions last year, the U.S. cast the only no vote against unfair, anti-Israel measures in the UN's Human Rights Council. We will continue to stand with Israel and against one-sided biased resolutions, even if we are the only country on earth to do so. So someone who is pro-apartheid, who's literally racist against Palestinians and Muslims in the Middle East, that individual I don't think is qualified to be Secretary of State. But when it comes to his foreign policy, he is arguably to the right of Joe Biden on foreign policy. So as Jake Johnson of Common Dreams reports, Blinken served as Deputy National Security Advisor and Deputy Secretary of State in the Obama Administration, and as the Washington Post reported Sunday, has been described as having a centrist view of the world and has also supported interventionist positions. Oh, goodie. He once broke with Biden and supported military action in Libya, for example, the Post noted, referring to the Obama White House's catastrophic decision to join with NATO to bomb that country, an armed intervention that helped unleash a violent civil war that is still ongoing. When it came to Syria policy under Obama, Blinken is also reported to have supported aggressive military measures against the government of President Bashar al-Assad, and more recently has indicated that the Biden administration would opt for leaving U.S. troops in the war-torn country. When Biden, then a Senator and Chairman of the Powerful Senate Foreign Relations Committee, voted in 2002 to authorize the Bush administration's disastrous invasion of Iraq, a decision he has since described as a mistake, Blinken was the Democratic staff director of the committee. President intercepts Ryan Grimm reported last July that Blinken helped craft Biden's own support for the Iraq War. Speaking to the New York Times earlier this year, Blinken characterized the vote to invade Iraq as a vote for tough diplomacy. That's not diplomacy, that's the lack of diplomacy. So we will have a president who supported the invasion of Iraq and a Secretary of State Tony Blinken who supported the invasion of Iraq, tweeted Medea Benjamin, co-founder of anti-war group Code Pink. In the U.S., there is no accountability for supporting the worst foreign policy disaster in modern history, only rewards. So this is the individual who will be Secretary of State. Now speaking of ghouls who are hawkish, John Kerry will also play a key part in Biden's administration as the climate envoy. And as Andrea Germanos of Common Dreams explains, food and water action called the choice in alarming red flag and the group's executive director, Winona Hodder, accused Kerry of being a longtime apologist for fossil fuel fracking and a reliable promoter of false climate solutions like market-based carbon trading schemes. Kerry's proposals are tired ideas from years past that will do little or nothing to address our climate crisis and will actually continue to place a disproportionate unjust burden on vulnerable communities that have borne the brunt of fossil fuel pollution and climate impacts for decades now, she said. As such, we have our work cut out for us, added Hodder. Now I will say that even though food and water action are speaking out against John Kerry, there isn't universal condemnation within progressive circles because Greenpeace USA as well as Sunrise, they are actually coming out and saying we're not too angry about this decision here to make John Kerry the climate envoy. In fact, the co-founder and executive director of the Sunrise movement, Varshini Prokash tweeted out support for Kerry saying, I served with Secretary Kerry this summer on the Biden-Sanders Task Force and one thing is clear, he really does care about stopping climate change. That's something we can work with and encouraging move from the Biden team. Now I'm keeping my eyes peeled for a domestic equivalent. Now on top of that, John Kerry did say that the United States has to do more than just getting back into the Paris Climate Accord, which is encouraging. However, I'm not as optimistic as my fellow progressives here because John Kerry is someone who has been in government forever, and he is a centrist, arguably a moderate Republican. And if he truly believed that climate change was the national emergency that he said it was when this position was announced, then don't you think he would have opted to endorse someone other than Joe Biden? He endorsed Joe Biden in the Democratic Party primaries when there were other options available with more solid climate change policies, Bernie Sanders. He could have even endorsed Elizabeth Warren or Jay Inslee. I mean, the bar is super low. But he endorsed Joe Biden, someone with a bad plan when it comes to climate change. I mean, better than Donald Trump, of course, but in comparison with Bernie Sanders, climate organizations rated him, what was it, a D, D minus.  John Kerry knows that if he endorsed Bernie Sanders, he wouldn't get a spot in Bernie's administration and endorsing Joe Biden would secure him another job.  So that's why I'm not optimistic. But here's what I say to John Kerry. Prove me wrong. Prove to me that my cynicism is unwarranted. Don't think I'm going to be proven wrong, but I would love to stand corrected here. Now, when it comes to Avril Haines, she's going to be appointed as director of national intelligence and there's some red flags, to say the least. I think I'm being incredibly charitable to describe my concerns about her as red flags because as Alex Koch pointed out, Avril Haines helped create the legal justification for Obama's drone wars, helped cover up the CIA's torture program and then supported Gina Haspel to lead the agency and consulted for Palantir. Haines is credited with moderating the drone policy. People say without her there would have been even more drone bombings. She also worked to make sure the drone bombings they did were legal. Sounds kind of like being a proponent of clean coal, but people die instantly. And he's exactly right. It's not just immoral, but illegal. Not only under domestic law, but international law as well. This individual should be nowhere near power. But yet she's going to get a spot in Biden's administration. And what has the mainstream media said about this? Well, that we should be happy because a woman is serving in this position. And this is historic. I mean, I'm sure that the people in Pakistan, Yemen and Somalia, who are having bombs dropped on their heads by Bannon's administration will take comfort knowing that it's a woman who's bombing them instead of a man. So there's that despicable. Now, on top of that, Janet Yellen will be Treasury Secretary. Let me just take a moment to laugh at Elizabeth Warren, who sold out Bernie Sanders to get a spot in Biden's administration, only to be shunned. But Janet Yellen is not someone who I am enthusiastic about. Wall Street is excited about her. That tells you everything you need to know. And additionally, someone who is being considered by Joe Biden is someone who should be nowhere near power. We're talking about Rahm Emanuel for Transportation Secretary. And the fact that Joe Biden would even consider someone this morally bankrupt. I mean, it speaks to how shitty Joe Biden is as a human being. And I think that AOC put it best. What is so hard to understand about this? Rahm Emanuel helped cover up the murder of Laquan McDonald. Covering up a murder is disqualifying for public leadership. Now, there's good news and bad news with regard to Rahm Emanuel. The good news is that Biden hears our complaints and concerns and he's trying to respond to that. The bad news is that his response is to placate us. And rather than making Rahm Emanuel Transportation Secretary, now he wants to put Rahm Emanuel in a less visible position. Why do you have to have Rahm Emanuel in your administration? Are we not going to draw the line anywhere? He covered up the murder of Laquan McDonald. Does that not mean anything to you? Does that not like strike a chord? I don't get it. Why do you have to put him in your administration? And the thing about Joe Biden that you're going to see is that there's this ongoing theme that the people who he is appointing to his administration, it's all Obama era alone. So I mean, this is this is really concerning and it should be concerning to people. It's not surprising. None of this is surprising to me, but here's why you should be concerned about this. If Joe Biden conducts business as usual and we just basically see a continuation of Obama's administration, a third Obama term, if you will, what's going to happen? Well, Obama's failures is part of the reason why we got a Donald Trump presidency. So if Joe Biden doesn't actually make meaningful changes and materially improve people's lives in 2024, do you honestly expect him to win again? I think he probably would have lost had it not been for Joe for Donald Trump's bungling of COVID-19.  I mean, if you are worried about right wing extremism in this country, then you shouldn't be worried about the fact or angry about the fact that I'm criticizing Joe Biden. You should join me in criticizing Joe Biden, because if you are afraid of Republicans taking power again, then understand that if Joe Biden changes nothing, they will be emboldened and more powerful than ever in 2024. And I get that that's a ways away. We can't necessarily think that far ahead. Sure. But do you honestly think that just having someone who's not Trump in power will suffice, that that's enough? If so, then you are in an incredibly privileged position. So it's frustrating because if you tune into mainstream media, they talk about how diverse all of these picks are and how historic it is that, you know, Biden is appointing a woman to this position and that position.  Joe Biden has a unique opportunity. We are we're at a crossroads in American history. He can actually make change. And all of this tells us right now, the people who he's choosing to surround him during this presidency. It tells us that he's not up to the challenge. This is what progressives had warned you about.


So, what do we have? We have that Hume is trying to say that moral beliefs are justified by the passions, and they're justified by the passions as opposed to the understanding. Justified by the passions as opposed to the understanding.  What's true or what's false? Well, that's something different. The passions deal with desire. They deal with what you want. The understanding deals with what's true or what's false. Okay, so we can say this, and it's very easy to say this, but let's try practicing a little bit. So, let's take a look at a collection of statements, and let's ask ourselves. Are these statements from the passions, or are they from the understanding? Are these statements from the passions, or are they from the understanding? Okay, well, let's try one. The capital of Texas is Austin. All right, the capital of Texas is Austin. Well, yes, this is from the understanding. That should be pretty straightforward. That should be pretty obvious. All right, no matter what you might want, maybe you want the capital of Texas to be in Corpus Christi. That way we can have beach days with the state legislature. No, no matter what you might want, that's not going to change the fact that the capital of Texas is Austin. Where the capital is, is not determined by what you want. All right, let's try another one. Chocolate is good. Chocolate is good. Well, yeah, according to Hume, this is from the passions. And this is from the passions. Whether you want chocolate, whether it's good or whether it's bad, that determines by, excuse me, whether the chocolate is good or whether it's bad, that's determined by what you want. If you want chocolate, then yes, chocolate is good. If you don't want chocolate, I mean, first of all, I feel sorry for you because who could not want chocolate. But if you don't want chocolate, then that determines that chocolate is bad. Not some fact about the world, not some truth that's independent of what you want or believe. The only thing that determines whether chocolate is good or bad are your passions. So if you want chocolate, then chocolate is good. Water is composed of dihydrogen monoxide. And so it's composed of one oxygen molecule and there's one oxygen atom and two hydrogen atoms. And it's very, very happy. Well, whether water is composed of dihydrogen monoxide or something else, that is not determined by what you want. That is determined by some fact of the world. That is true. That belief is from the understanding, not from the passions. That belief is from the understanding, not from the passions. You can want water to be composed of something else, but it's not going to change the fact that water is composed of something else. So water is composed of dihydrogen monoxide. That is from the understanding. All right. Now, so according to Hume, a Republican is the current President of the United States. That is determined by the understanding. That is determined by what is true in the world. Now, you may want it to be otherwise. You know, whether this is true or false, you might want it to be otherwise, but you don't get to change the political party of the current President of the United States. So a Republican is the current President of the United States. That's from the understanding, if it's true. That's from the understanding, if it's true. How about this? A Democrat would be a good President. You know, whether a Democrat's President or not, right? The statement is a Democrat would be a good President. Well, for Hume, that's from the passions. That's determined by what you want. Whether a Democrat or Republican is currently President is a different question of which one would be a good President according to Hume. So yeah, according to Hume, whether a Democrat would be a good President, that's determined by the passions, not by the understanding. Okay, let's try another one. A diet of hamburgers and french fries is an unhealthy diet. That is determined by the understanding. This is something that's true in the world. This is not true about what you desire. Plenty of people desire hamburgers and french fries, but that's not going to change the fact that if that's all that you eat, you will likely die very quickly. So yeah, a diet of hamburgers and french fries, that's an unhealthy diet. That's from the understanding. That's what's true or false. You can desire hamburgers, but that doesn't make them healthy. There is no life on Mars. Well, this is true in virtue of the understanding. You may want there to be life on Mars, but whether there's life on Mars or not, that is true in virtue of, well, what's happening on Mars, not in virtue of what you want. So this is from the understanding, not from the passions. Wait, did you guys see that? Hmm. Okay. So we've had a little practice. We've had a little practice with the difference between the passions and the understanding. And Hume says, passions justify moral beliefs. What you want determines what's moral or immoral.


So, technology, because the technology is subjective, now the technology is that you are handling or objective technologies, now I am talking about a subjective technology, it's about you. Here you are the software that you are trying to handle. Fundamentally, if you look at yourself, right now you are operating according to the software that you have become, please see what kind of family you were born in, what kind of parents, what kind of school, what kind of friends, what kind of atmosphere, what kind of religion, what kind of society, that's the kind of person you are right now, isn't it? So, software was written into you haphazardly by anybody and everybody that you came in touch with, not by trained professionals, just about anybody who came, they put in something.  You had no choice from where to receive, where not to receive. If you say, I don't like this man, I don't want to receive anything from him, you will receive much more from him than anybody else. That's how this one is. So, right now the very way you think, feel, understand and act in your life is just the way your software is, isn't it? Yes? The very way you move your body is the way your software is. I want you to understand this. It is so deeply conditioned. If you are, somebody you know very well, somebody whom you are very familiar with is walking one mile away. You can't recognize anything about that person, but just the way he moves, you know it is this person, isn't it? Because the very way his body moves, only moves whichever way his software is written, but this software was written unconsciously.  You're too identified with it, you're just lost in it, that's all you become. Now you have to create a little distance. You've heard of a word called Buddha. If I say Buddha, probably you always think of Gautama the Buddha. Gautama is not the only Buddha, that's not his name. He became a Buddha. There have been thousands of Buddhas and they still are. What Buddha means is, Bu means Buti or the intellect. Dha means Dada, one who is a bow. One who is a bow is intellect, is a Buddha.    Yes, isn't it so? People are capable of suffering just about anything in the world, please see. Very, very easy. If you're not educated, you will suffer that. If you get educated, you suffer that. If you don't find a job, you'll suffer that. If you find a job, you suffer that. If you're not married, you'll suffer that. If you get married, you'll suffer that. If you don't have children, you suffer that. If you have children, you suffer that, isn't it? People are capable of suffering just about anything. This is the nature of the intellect. Once you're into the intellect, this is how life is, If you are below the intellect, you won't suffer so much. Animals don't suffer as much as you suffer. Physical things, if they are taken care of, they are quite okay. They don't have the kind of suffering that human beings know, isn't it? So if you are below the intellect, you don't suffer so much. Like you asked that question. If you are above the intellect, you are a Buddha. If you are below the intellect, maybe we can call you Buddha. If you are in the intellect, non-stop suffering. So to avoid suffering, people are inventing many ways to go below the intellect. Excessive eating, alcohol, you know, excessive indulgence in physical pleasures. These are all the ways you are trying to avoid the torment of the intellect for some time. You get drunk and you feel good for a few hours but then life will catch up with you with more intensity. It won't let you go. Now this being which was below the intellect somehow evolved into the intellect. The only way out for this is to evolve beyond the intellect. So the fundamentals of yoga, the technology is how to go beyond this. So how to do it now? Subjective technologies cannot be explored in uncommitted atmospheres. You need a very committed, focused atmosphere. Otherwise it will not be in your grasp. It's not a teaching. It's a method. To implement a method, you need to be in a certain way, in a certain level of commitment, in a certain level of receptivity. Otherwise it doesn't work. Because, see now if you go into a chemistry laboratory, you take this chemical, this chemical, if you put it into it, something happens. But if you are not willing to put it into it, if you sit here and think and think and think, what will this do? What will that do? You will come to all kinds of wrong conclusions. Till you put it there, you won't know. Maybe it will explode in your face. But till you put it there, you will not know what it is about.  Everything is you. Now you need a different kind of laboratory. Otherwise it doesn't work. If it was objective, we could have just written a book and given it to you. But because it's subjective, you have to be taken step by step because it's a completely different dimension of perception and experience altogether. But is it possible for me? It is possible for every human being. Internally, all of us are equally capable. We are not capable in different ways. All of us are capable of joy, isn't it? Yes? If you have been joyful for one moment, that means you are capable of joy, isn't it? It is just that you are unable to stay there. That's the whole thing. There is no human being on the planet who's never been joyful. Every human being has been joyful at some point. That means he's capable of joy. It is just that he's unable to stay there. So he needs a technology as to how to maintain the atmosphere within himself. It's possible.


Hi everybody, Sean with jaspyscasebreaks.com here doing 2020 Bowman draft baseball eight box jumbo case break pick your team number nine Where at the end we're gonna also give away a hundred fifty dollars in break credit We're gonna take all customers names randomize them top three get fifty dollars in break credit each so Some details no paper ship except for Kirsten v in Torkelson Hassell no other exceptions all paper donated charity all chrome ship Just some details about the boxes So here is our list You can see the case over my shoulder Last spot mojo Ryan Redmond with a brace Finally saw a box topper this one I think it's the second one we've pulled this one was not out of draft First one we pulled in store or in on here was autographed so for anybody watching this video you can check the The description of the video and you'll see a little timestamp for a recap and randomizer You can click on that and it should take you to where the randomizer and the recap are at the end of the break For anybody watching live next 25 minutes is pretty much going to be me opening up the packs If you're watching the video and you want to see everything But don't watch this portion this portion Just forward into the video probably about 25 minutes. That's cool How much is your sketch going for? So are these like wrecks are these guys that collected like the old-school version and they collect the new school version or what I Want to know more about this Like I know some people that there's somebody That regularly whenever we get it in would buy it all off our personal just have it shipped straight to them So I'm sure they know all about that group. That's all profit wrecks if uh if you got a chrome Could you could you put like your finger just a big finger for a smudge on it and post the picture Just be like, oh my god guys. Look what I fold. Is this any good? I don't like that That also seems like the kind of group wrecks that would overly complain about almost anything Just from the way it sounds. That's so good Okay, so now they actually sound like really funny people I Might have to find those group If I said listen though Listen my my sports car collecting is enough if I if I start bringing home It's sold out right sweet. We'll do that after this If I start bringing home Garbage pill kids though. I feel like my girlfriend's gonna kill me So rex if I find blasters out in the wild Should I pick them up no matter what and just even if you want them Figure out a way to get into you at cost Plus shipping see that kind of what kind of Barbie cards Brendan I didn't even know they made Barbie cards. I've heard of a channel of a girl who looks like a Barbie who breaks cards Never seen it Okay, well, I'll keep an eye out for him Maybe just let the boxes sit. I Think I think Montgomery Club did an exclusive release on them Rex for And like I think my buddies bought a couple of boxes each and like they were saying how ridiculously like the same thing They were like honestly like I don't know anything. I'm not gonna do anything with them But I couldn't pass it up because of the price on what they were reselling for Yeah, I hadn't looked into it at all nice little palette cleanser Brendan I'm not the sports. Here's Barbie Was it like new product Brendan or was this like old like vintage Barbie stuff that they found I'm sure that there's probably a huge market for that Target and Walmart have them They have the Barbie Rex or are you talking GPK still? Well, when I'm not trying to find some prism, maybe I'll take a look the GPK That's kind of cool. I can appreciate that When we were doing like personal stuff with Pokemon people were buying that stuff up like crazy So Brendan, what's the biggest you're in Australia, right? It's back at the basketball the biggest collection down there your biggest collectible It's the most in-demand Did it kind of like blow up because of the mellow playing there and stuff and all the other That didn't want to go college and stuff Well, so you had RJ Hampton down there, right? So do you think like Lamello and playing down there and stuff has made it even bigger or Just in general growing the same as like the hobby Yeah, yeah, I mean, I think it put a lot of people in front of their computers and TV screens trying to find You know ways to keep themselves entertained and keep themselves You know, especially sports with no sports going on They were trying to find anything that was sports adjacent. So I think a lot of people kind of have that same story Orange or red? Yeah We didn't do that here. Yeah, that's kind of cool Yeah, so how do you find us Brendan?  It's like hey You like these 10 videos Maybe you'll like this And then you end up having to stare at my face No, it's cool being able to do this late night though because I get people from all over the world like you Regularly will have a guy in here that's from Austria hanging out and you get people in Hawaii All over the US Canada Pretty cool. So you collect a lot of baseball? Yeah, I mean if it's if it's a product that comes out, we're gonna get it 99% of the time I like that. We always try to go for older stuff, too We'll try to get like restocked of like a couple of years ago and stuff instead of just kind of beating it over The top with brand-new stuff over and over and over again. I was actually a customer before I went to Before I started working so I was on the other side of the monitor watching and chatting and stuff You have a three out of five triple threads the deca relic nice From this year or what year? What players last box guys? No box copper this time from 2020. It's like 25 minutes right at 24 minute mark That's cool good names there really good names very nice Four boxes other four right there Jordan Westberg to 499 for the Orioles Thomas Reigns. Oh, and like we said the four papers that ship are Robert Hassel for the San Diego Padres, which is Mark Bassett Zach vene for the Colorado Rockies, which is Jack vane Spencer Torkelson for the Detroit Tigers, which is Aaron Pettit and Heston Kirstad for the Baltimore Orioles, which is Thomas Reigns So you're gonna hear me say those names a lot and not the customer. It's going to but those are the four people Zach vene Rockies Like right there Heston Kirstad Orioles Casey Martin to 150 For the Phillies Kayshawn Chang Spencer Torkelson for the Tigers Robert Hassel for the Padres Yeah, I feel like That second part Brendan you might be able to email us about I don't know have you tried that? You can use the support button and just ask I Don't know if that how that works. That's kind of a tease Spencer Torkelson greatness For the Tigers Aaron Pettit Alika Williams sky blue for the Rays John Samuelson Joey Cantillo purple to 250 for the Indians Mark Bassett Heston Kirstad greatness going to the Orioles Thomas Reigns Gage workmen Sky blue for the Tigers Aaron Pettit. Is there is there such a thing Carlos? I'll go check again. I didn't think they had image variations in this Now that you're pointing it out. I think it might be Robert Hassel nice start first auto of the break blue wave to 150 for the Padres Mark Bassett read Detmer's sparkle for the Angels Jorge Mir Will Klein refractor auto to 499 for the Kansas City Royals Darren McKenzie Heston Kirstad for the Orioles Thomas Reigns And our third auto Evan Carter for the Texas Rangers and Mark Bassett, let me let me set it aside Carlos and see I now that you're pointing it out I think it might be Jason have you seen anything about an image variation inside of draft? Okay So there are image variations. I we just hit one and I was like Yeah, okay, so Carlos good catch Yeah, I was like Carlos is like I think that's an image variation There you go, yeah, it is different because the other one's him batting All Chrome ships anyways, but obviously There we go Nice start full box to Spencer Torkelson Tigers Casey Schmidt to 99 for the Giants That goes to Mark Bassett. I don't know. Let me think Count them all up Zach vene for the Rockies Shane draw hand to 250 for the Red Sox Eric Angel CJ Van Eck To 499 for the Blue Jays Jeffrey who are you? Spencer Torkelson for the Tigers They pulled this right now earlier and I was like I had to buy it before it gets too expensive What do you would you do on it? Okay, I'm starting almost six hundred dollars. Are they really I have a buddy that just got one the other day $150 last week He got a retail pack obviously not that but retail pack very first cars Burrow silver yeah second card the burrow base and then like I was like, okay Like you think that makes a big difference down the bottoms, but I guess it could damage it wait. What do you mean the corners? Oh How they're Who'd you get it from? Yeah, that's cool It's a Jalen Hertz Silverado Massive yeah hassle for the pod race. I'm hoping he does Heston Kirstad for the Orioles Hassle for the Padres a salacy sparkle going to the Royals and Darren McKenzie Zach mean for the Rockies Jack Vane is Nick trying to buy him yet or no Yeah Torgelson for the Tigers Aaron Pettit Brady McConnell sky blue for the Royals Darren McKenzie. Are you doing Jason? Little guys. Have a good night and good weekend. Ed Howard. That's another good one Chicago Cubs Greg S. Vang Class of 2020 to 250 There you go Nolan Gorman to 150 for the Cardinals That is Thomas Reigns Adam Seminaris Auto for the Angels Jorge Mier. We got some gold. It looks like peeking out the back and Otto Alec Burleson Gold auto 250 for the Cardinals Thomas Reigns And guys, we still have a couple more breaks of this in the store that are single digits pretty low actually And single digits and so have the break credit giveaway Get your spots. We can keep doing these all tonight Still in search of our first Super Fracture Maybe it's gonna be you can't get you can't get you a super if you don't get a spot Spencer Torgelson for the Tigers Trey Cruz to 250 for the Tigers That's Aaron Pettit Heston Kirstad for the Orioles almost said Oilers Robert Hassel Padres Torgelson for the Tigers Bailey Horn to 499 for the White Sox Heston Kirstad for the Orioles almost said Oriol or Oilers again Zach Veen for the Rockies. Oh, we'll see Brendan We'll see Or maybe it's gonna be the biggest one that we've pulled yet Austin Hendrick Sky blue for the Reds Christopher Gerard Zach Daniels Sparkle for the Houston Astros G. Jia Chen Ryan Jensen sky blue for the Cubs Greg S. Vang Robert Hassel Going to the Padres Mark Bessette Nick Sweeney for the San Francisco Giants Mark Bessette Bobby Whit Jr. Glimpses of Greatness Green to 99 Kansas City Royals Darren McKenzie Will Klein also green to 99 Darren McKenzie in the Royals Chris McMahon to 499 for the Colorado Rockies and Jack Veen box for Chris McMahon to 499 for the Rockies and Jack Veen Heston Kirstad for the Orioles Zach Veen for the Rockies Spencer Torgelson for the Tigers Robert Hassel for the Padres Joe Boyle to 250 for the Cincinnati Reds Christopher Gerard Kirstad for the Orioles and Chrome Jeter Downs to 150 for the Red Sox and Eric Ainge Xavier Warren For the Milwaukee Brewers going to David Lewis Tyler Freeman sky blue going to the Indians Mark Bessette Spencer Torgelson for the Tigers Aaron Pettit Max Meyer to 250 for the Marlins Walter Lutz Carson Tucker refractor to 499 for the Indians Mark Bessette CJ Van Eck Blue Wave for the Toronto Blue Jays and Jeffrey Woo For you Jeffrey that'll end the first four boxes next four over here on the cart coming up Alika Williams to 150 for the Rays John Samuelsen Heston Kirstad for the Orioles Dane Aker to 499 for the A's Joe Maderos Robert Hassel for the Padres Zach Veen for the Rockies Spencer Torgelson for the Tigers Heston Kirstad for the Orioles and Thomas Reigns Zach Veen For the Rockies Jack Vane Nick Sweeney Sky blue Going to the Giants and Mark Bessette will climb to 150 for the Royals and Darren McKenzie Zach Veen for the Rockies going to Jack Vane Robert Fawson sky blue A's Joe Madero's Robert Hassel Padres Mark Bessette all three autos in the very end Max Meyer, there you go third overall pick Miami Marlins Walter Loutz You go Walter Alexander Canario purple To 250 for the Giants Mark Bessette Bryce Ed elder to 499 for the Braves last spot mojo Ryan Redman and Hunter Barnhart blue wave auto to 150 the Tampa Bay Rays and John Samuelsen. I have not run I'd love to go though someday Take the girlfriend Ben Hernandez to 99 for the Royals Darren McKenzie Spencer Torgelson for the Tigers Bobby Miller to 499 for the Dodgers and Jorge Amir Heston Kirstad for the Orioles Let's see in a lot here What's going on? Come on Big stuff Zach Veen for the Rockies Robert Hassel for the Padres Robert Hassel Padres box six chrome Heston Kirstad for the Orioles Thomas Reigns Robert Hassel Refractor for the Padres Mark Bessette Heston Kirstad Refractor For the Orioles Thomas Reigns Sam Weatherly sky blue for the Rockies That's going to Jack Vane and how about a Robert Ploss in green team color matching to 99 for the A's and Joe Maderos Zach Veen Refractor For the Rockies Jack Vane Mike Siani sky blue for the Reds and Christopher Gerrard Spencer Torgelson greatness For the Tigers Aaron Pettit all those are saying staying till the end again Zach Veen For the Rockies Jack Vane cage workman Auto to 99 for the Tigers Aaron Pettit Robert Hassel Padres Mark Bessette Isaiah Green the 250 for the Met Eric Ainge Ian Seymour auto to 150 for the Reyes and John Samuelson and Robert Hassel Refractor auto for the Padres Mark Bessette Second Hassel of the break very nice box seven Landon Knack to 499 for the Dodgers Jorge Muir Robert Hassel for the Padres Zach Veen for the Rockies Heston Kirstad for the Orioles Spencer Torgelson for the Tigers Jordan and wogu to 150 For the Cubs Greg S Vang Robert Hassel Padres DL Hall Sparkle for the Orioles Thomas Reigns Joe Boyle for the Cincinnati Reds Christopher Gerrard Carmen Wadzinski sky blue for the Pirates Nicoba Connor Phillips to 99 for the Seattle Mariners David Lewis and AJ Vukovic to 71 for the Arizona Diamondbacks Mark Bessette I feel like I pulled a couple of his Not a lot out there Jaspie's corner in the market on AJ Vukovic sparkles Will Klein blue to 150 for the Royals and Darren McKenzie Nice team color matching on the blue It kicks off next week Brendan nice Are you a fan of any specific team or is there a team that?  Well, maybe I'll piggyback on yours Tanner Burns to 499 for the Indians mark Bessette Zach vene for the Rockies Melbourne aces Spencer Torgelson for the Tigers any good Robert Hassel Padres. I think I did hear about that actually I didn't know that he signed but I heard that he was trying to go there to play Well, you know what I'm team I'm team aces going with Melbourne Melbourne Is it Melbourne or burn? Which way it makes me sound less dumb Bryce Bonin to 150 for the Reds Christopher Gerard But I'm aces all the way. I don't know. I forgot which one I said the second one So it sounds like the blue socks are pretty loaded No, Sydney Not the born Melbourne. Okay. Okay. Okay. So I'm all I'm all Melbourne Melbourne aces all the way No, blue socks. No, Sydney blue socks Heston Kirstad with the Orioles Blue socks are trying to buy themselves a championship Dylan Dingler to 499 Detroit Tigers Aaron Pettit Zach vene for the Rockies Robert Hassel for the Padres Spencer Torgelson for the Tigers. Oh We got orange in here Gerard and Carnacion sky blue for the Marlins Walter Lutz and Orange Tanner Burns for the Cleveland Indians Mark Bissette AJ Vukovic refractor auto to 499 Diamondbacks Mark Bissette Robert Poisson gold Not a bum in first, but still big Oakland a is Joe Medeiros out of 50 Casey Schmidt auto for the Giants Mark Bissette Christian Robinson sky blue for the Diamondbacks Mark Bissette Heston Kirstad for the Orioles Thomas Reigns and There you go Robert Poisson auto kind of had a feeling that kept popping up 43 out of 99 Oakland a is Joe Medeiros. I Think that was our only non Bowman first auto of the break Very nice You're not gonna be mad when you get a Robert Poisson auto Very cool. So is this a variation? Carlos if you're listening, I'm assuming that's why it's numbered out of 99 is the auto So it's a variation auto I'm gonna say I was like normal refractors wouldn't be numbered out of that So Jason has one in this too. Are they the only non first Shit Nice though, okay Jason's out of 25 cool, so let's do the break credit giveaway and the recap, okay, so break credit randomizer coming up So top three names $50 and break credit each here's the original customer list from Mark Bissette all the way down to David Lewis dice roll and and top three names after Ever it rolls two and a four six times six in final time Hope everybody had a good break, but congratulations a little bit of extra money going to Jack Vane Darren McKenzie and Jeffrey Wu or you You three will get $50 and break credit sent to you guys after I wrap all this up and our recap Here is the non auto Tanner burns out of 25 for the Indians Robert Plosson gold out of 50 for the A's Robert Hassel refractor going to the Padres Heston Kirstead refractor going to the Oros Zach Vane refractor going to the Rockies and a Jason Dominguez image variation going to the Yankees as well our autos Casey Schmidt For the Giants AJ Vukovic to $4.99 for the Diamondbacks and a Robert Plosson image variation auto out of 99 Very nice Joe Boyle for the Reds AJ Vukovic sparkle auto to 71 for the Diamondbacks We'll climb to 150 for the Royals gauge workmen to 99 for the Tigers Robert Hassel to $4.99 for the Padres Ian Seymour to $1.50 for the Rays Bryce elder to $4.99 for the Braves Max Meyer for the Marlins Hunter Barnhart to $1.50 for the Rays Carson Tucker to $4.99 for the Indians Xavier Warren for the Brewers CJ Van Ek to $1.50 for the Blue Jays Dick Swinney for the Giants will climb the 99 for the Royals Crispic man to $4.99 for the Rockies Ed Howard to $2.50 for the Cubs Adam Seminaris for the Angels Alec Burleson to $5.00 for the Cardinals Evan Carter for the Rangers and Robert Hassel to $1.50 for the Padres plus we'll climb to $4.99 for the Royals. There you go overall very nice break Got some nice image variations in there And our first non bowman first auto that I've seen so far. So that was 2020 Bowman Draft baseball eight box jumbo case break pick your team number nine from jaspyscasebricks.com. I'm Sean jaspyscasebricks.com. Thanks for hanging out and we'll see you next time


We're at lecture 51 math 241 We had an interesting problem yesterday It kind of started from a web assigned question although we went kind of beyond what the web assigned question was but we did a problem two different ways and the answers were off by a factor of one-fifth and That kind of bothered me Actually, it woke me up this morning about 5 13 And I didn't get back to sleep. So I kind of like that stuff that tells you how geeky and nerdy I am that If you do a problem two different ways for goodness sakes you ought to get the same answer both ways In fact, I did it a third way Yesterday and I think I have justified why one has The factor one-fifth in it and the other one did not and in fact, I think I've got all three ways of doing this problem Justified now in my mind if I can convey that to you. I don't know but I will make an attempt So let's rectify that situation first and then we'll go on further with Taylor and McLauren series So we had previously Come up with a power series for inverse tangent of x And we decided that we were going to use that result For inverse tangent, I think it was x over five I looked it up in the book and there's a problem similar to this in the book But it's x over three, but I think the web assigned homework that we dealt with was x over five inverse tangent of x On its interval of convergence looked like this we had developed that from Integrating one over one plus x squared no correction factors are necessary there We decided that everywhere there was an x on the left side if we replaced that with x over five then Basically that compels us to do the same thing on the right side, which this should be the correct answer Okay, this substitution. So we should get this factor of one-fifth that we did the problem another way Was missing that factor of one-fifth, and I think we can we can justify this So I think this is the correct answer and let's look at it two other ways that we looked at it We went ahead and instead of integrating one over one plus x squared with respect to x Which gets us inverse tangent? We decided to use the same approach that got us that formula that's listed at the top of this page Now without getting back to the same potential problem that we had yesterday When you have something here that is not x The quantity squared and then the derivative of x we decided that we needed a For this to integrate to this We needed a one-fifth DX So this should integrate to inverse tangent of x Here's what we actually did to get there to the to the best of my recollection Which by the way, couldn't I take that one-fifth that's present and move it out in front This is what we actually did We got a power series for one over one plus x over five the quantity squared Then we integrated it and that's how we got the other answer, which was instead of x over five. It was x there Instead of x over five Cubed all over three. We were missing a factor of five there in the denominator Basically, we got the same series, but we got it times five There is what we did not do in that series. So we got an answer. Let me see if I can find that answer well, I was trying to Kind of say well, we didn't really have that We didn't we didn't here's what we had because of how we went about it and we didn't have that correction factor, so Let's go back and do that Okay, here's what we did and we called that one minus negative x squared Negative the negative of x squared x over five the quantity squared it's a lot easier to deal with when it's just x squared But we've got x over five So we decided that that was the first term and that Was the ratio So we got that notice no extra one-fifth in there no extra five in there just that And we decided that was the first term was one and the ratio was negative x over five the quantity squared if that's not what we had it should have been what we had now We decided that's not what we wanted what we wanted was the integral of that So therefore we had to integrate this here's where we had the problem where we got an x and we got a That five squared is a one over twenty five and then we integrated x squared which was x cubed over three this was a One over five to the fourth which was six twenty five and we integrated x to the fourth Which is x to the five over five this is what we compared to the other one and it was very clearly Here we had an x and on the other one we had an x over five so it seemed like we were missing that factor one-fifth Well, this isn't really what we found the first time. It's one-fifth of What we found the first time because in order for this to be the inverse tangent of that of x over five It has to have a one-fifth in there. We never did deal with the one-fifth. So what we really found was one-fifth So here's what we found Okay, we need to multiply it by one-fifth in order to make an equivalent to what we were actually trying to find So that one got me pretty much justified but what Also came to my mind is This Remember we did inverse tangent integrand we were doing the table of integrals and at the end of 141 and We decided that anything in this form Was that okay, and most of you probably memorized that that's something that is used enough That was an a squared site So if that's something squared and that is the variable quantity squared we're integrating with respect to you it integrates to this Well, we can kind of knock out the plus c because we're going to have a Plus k that we decided on the inverse tangent problem was zero anyway So let's set that aside What if we had this might be even better than the way I worked it out? Yesterday What if we had that because that's kind of what we generated that one time how would that translate back to an integrand? Well, if that's one over a Inverse tangent of u over a that to me says that where I have an a I should really have a five Right and where we have a u squared. We should have an x squared So if that's what we really wanted to get for an answer it ought to look like this suppose I took the numerator up here and I'm going to try to get it in the form Here's the form. I'm kind of gearing toward right here If that's the form I'm headed toward and I've got a five squared there I'd better divide everything by what 25 So if I divide that by 25 25 down here by 25 and x squared by 25 So I haven't done anything that requires any compensation. It's numerator same thing. It's denominator So if I call that one and call this x over five, that's looking a whole lot like what we did And this I can call one again as long as I do what with that Bring the one over 25 out in front So is that Here we've got I don't I didn't like the fact that I went the other way this time But we've got a one-fifth and a one-twenty-fifth out in front there again We've got that factor of difference of one-fifth from what we generated so This is really all of this and what we have is in essence One-fifth of this is that okay? This isn't didn't I don't like it this way I did it actually another way I started with this and worked my way back to this I like that better, but it is this Thing that we started with This result that we have down here isn't it one-fifth of this Because here we've got a factor of one-fifth out in front here. We've got a factor of one-twenty-fifth It's a fifth of in essence what we found so by all three of these I think we can justify that the inverse tangent of X over five is The simplest way that we did it everywhere there was an X in the inverse tangent formula All we have to do is substitute an X over five that is the correct answer And what was that X over five? cubed over three X over five to the fifth over five and So that's the correct answer. I can tell that you're not quite as justified in your mind as I am in mind But I did basically work the problem three ways and Got the answer to be the same all three ways So we were missing a one-fifth in the other problem It's probably here's what we did We found this we integrated it It's it's missing a fifth Because in order for this to be an inverse tangent it has to have a one-fifth So we found this it's really one-fifth of the inverse tangent Not a test question, but I think that stuff that especially when we can connect it to a formula that we've had prior in this class is It's probably a pretty good thing to do All right, we ended yesterday with a Taylor series for e to the x actually it's a Taylor series instead of being centered at x equals a it's centered at x equals zero which technically makes it a Maclaurin series But you can call any of these Taylor series Taylor series are things that look like this We have higher order derivatives at a nth derivatives at a over n factorial x minus a to the n and If we let it start at zero and run forever, then it actually is Equal to we're going to talk about today a little bit about stopping it at a certain number How it approximates that function and any time it approximates a function then we have to talk a little bit about Error or at least the upper bound for the error if we're stopping it at the 13th derivative and 13 factorial and the binomial to the 13th power What kind of potential error do we have for e to the x or sine of x or cosine of x or inverse tangent of x or any other function Right now we're going to let it run to infinity So for this we let our a value be zero So we converted So x minus zero is just x So when we put in higher order derivatives the derivative of e to the x is itself So is it second derivative third derivative all the derivatives down the line? And when we evaluate all those derivatives at zero e to the x evaluated at x equals zero is e to the zero which is one so that doesn't change from one term to the other So this became one right all the higher order derivatives at zero. We're all one So if we have x to the n over n factorial from n equals zero to infinity We have a power series certain type of Taylor series For e to the x. I think that's probably about where we ended So the first term ought to be x to the zero over zero factorial, which is one x to the one over one factorial x to the two over two factorial X to the three over three factorial and so on so we didn't Kind of see if we think it's believable. I mean it seems to To work in this case where all the higher derivatives at zero or all one pretty simple version of this If this works just writing the same thing down again, so we have it on this sheet Let's see if it works for When x is one Well when x is one, so I'm plugging in a value For x everywhere. I see an x we should plug in a one We're supposed to let it go Forever, but let's just stop it at this term and see how well we're doing. We know e to the first is Approximately 2.718 So we've got a one plus a one there's a two There's one over two factorial, which is one over two which is one half There's one cubed over three factorial, which is one over Six there's one over 24 And there's one over five factorial. What is that 120? So we're supposed to let this go The terms are getting pretty small. The next one is what one over 720 So we're going to have some error, but let's see how well we're doing at this point We're supposed to get 2.718 roughly If we add these together, where are we at this point? Not bad So we're missing a whole lot of terms But we're still doing pretty well Basically truncating this that's called a Taylor polynomial as opposed to a Taylor series series means we're going to let it go To infinity a Taylor polynomial says we're going to stop it at a certain n value But we did okay in this case suppose we wanted Let me give you a reason for wanting it first. Let's suppose that We were Wanting a way to integrate that could we do that integration problem? By substitution a few of you are shaking your head that we could not Here's the attempt if we were going to use this by do this by substitution. It'd be e to the u You would be negative x squared Here's where I think the problem comes in. What's the derivative of you? We don't have an extra x floating around in the integrand. We can't create one ourselves because it's variable So this method is not going to work How could we do this problem? Well, we have a Taylor series a power series for e to any power How about let's get a Taylor series for e to the negative x squared And let's integrate that so if that's e to the x again, I'll just write this down We want something for e to the negative x squared. How do we do that? Same thing we did with the inverse tangent of x and Coming up with the inverse tangent of x over five everywhere. There was an x. We put in an x over five Everywhere there's an x We want to replace it with what? negative x squared so now we have this way of writing e to the negative x squared as this power series as a polynomial type thing polynomials are always a lot easier to deal with then Things that aren't some type of transcendental function polynomials will always be easier So we've got one minus x squared. Let's see what we've got here plus x to the fourth Over two what? negative X to the sixth over six Granted it's not all of them, but we've got the pattern going here. So if we want to Integrate this we should be able to integrate this site and accomplish the same thing So you can write something out that appears to not be integrable by methods that we've studied to this point in time patterned integration and We can integrate it in pieces in the series form So the integral of one would be x the integral of this x cubed over three Here we've got a one half and then we'll integrate that x to the fifth over five Here we've got a minus one sixth x to the seventh over seven So something that by normal means is not integrable convert it to a series It is integrable at that point. So we do need to keep it going in order for that equation to be maintained Probably shouldn't leave e to the x until we do this So we have e to the x written as a Taylor series McLauren series some type of power series we ended up trying to do a Problem the other day an inverse tangent problem We went outside of the in of the interval of convergence and it didn't work So we better see if that Possibly could happen with this particular power series. How are we going to find an interval of convergence? how did we there are Different ways that these types of series can converge they can converge for a single value Like that at x equals zero or at x equals three a single value of x only They could converge for all values negative infinity to positive infinity and they can Converge for some interval so how did we Figure out how did we come up with that interval of convergence? Ratio ratio test so let's take this thing right here and do a ratio test and see What is the nature of the convergence for this particular series? So we want the limit so we're doing ratio test to try to find when this particular power series converges for e to the x So we want to see what happens way out there to the right what's in the numerator I haven't done a ratio test for several days in plus first term So if the description of the term is x to the n over n factorial Then the n plus first term would be x to the n plus one over n plus one factorial Let's get rid of the complex fraction So we'll multiply by the reciprocal of the denominator Okay, tell me what things kind of knock out numerator and denominator. What's left where? Okay, and that happened because n plus one factorial is really in plus one Times in factorial is that right? So the n factorials reduce That's correct and in plus one in the denominator And how do they reduce? Leaving what where? x in the numerator so this Numerator has x to one degree larger than the denominator So you can reduce them, but it leaves an x to the first in the numerator So we don't know what x is let's keep the absolute value notation there in plus one in is Certainly positive How about that as in approaches infinity? Don't we want to know where this is? less than one right Isn't it always less than one As n gets infinitely large this denominator gets infinitely large x is Going to be eventually overtaken by this tremendously large denominator So this thing approaches zero which is less than one all the time So what's the interval of convergence for all values? So it's not a certain value of x or a certain Interval of values for x. It's for all x So the interval of convergence is negative infinity to positive infinity that means it works all the time We're not going to have that situation like we had the other day trying to find the inverse tangent of two Two was outside of the interval of convergence It kind of fell apart that it didn't converge So this can be used basically all the time convergent for all values of x We're not going to finish this today. I'm trying to think of what we can get to that Let's mention this and then we'll use it in our next two So we're changing the terminology slightly from Taylor series, which means we're going to let it run all the way to infinity To a Taylor polynomial So a Taylor polynomial means we're going to truncate it at some point in time We're going to talk about error when we're definitely not talking about that today We'll talk about that on Monday, but Taylor polynomials if we have let's say T sub n so it'd be T sub 3 T sub 4 So it'll be a certain in value that's subscripted that'll tell us not how many terms we want because Sometimes terms disappear because they have coefficients of zero. We're about to see that But it'll tell us the value for n For which we're going to stop that particular series So I'm going to use in there so I can't use it down here So I'm going to use another letter so from i equals 1 to n. I'm sorry i equals 0 to n This is our nth Taylor polynomial Now wherever we had ends before we're going to have to put i's other than that. It's the same thing So we had the nth order derivative now. It's an i in that position We had an n factorial now. It's an i factorial and we had an x minus a To the end now. It's x minus a to the i So this in is exactly the same as this one it tells us where we're going to stop So if we stop at n equals three, it's a third Taylor polynomial Stop at n equals seven. It's a seventh Taylor polynomial not equal to the series There'll be some error involved And we'll address that error After we get these let's let's today get these two. I think we can get them both Let's have a function sine of x and We want to develop a Taylor series. I guess a Maclaurin series for sine of x So x minus zero or just x to the n Let's start at n equals zero and let it go on forever. So this is a Taylor series We're going to talk about some of the Taylor polynomials associated with us, but for right now Let's generate the whole series We are going to need some higher derivatives We'll see a pattern here derivative of sine is Cosine excuse me derivative of cosine is and derivative of negative sine is Now we should start repeating right derivative of negative cosine is back to sine and This thing process It's going to happen happen in cycles of four Sine to cosine to negative sine to negative cosine and back to sine to cosine same thing cycles of four Well for n equals zero We should have the zero derivative. We decided the zero derivative although it sounds kind of stupid is really the what? the original function at zero over zero factorial x to the zero So what is the original function at zero? What was the original function? Sine of x right, so what's the sine of zero? Zero Doesn't really matter. What else I write so the first term there really isn't a first term That's actually going to be there to help us describe what the sine function is Now is this going to happen repeatedly? It's going to happen for this term Because the sine of zero zero not going to happen here It is going to happen here Not going to happen here. So every other term is going to be zero that sound right So for n equals one. We want the first derivative at zero over one factorial X to the one well What's the first derivative at zero first derivative is cosine? What's the cosine of zero? one over one factorial x to the one So that looks like x to me n equals two second derivative at zero over two factorial x to the two The second derivative was negative sine of x. What's negative sine of zero? Zero we already decided that every other term is going to drop out but the third derivative at zero three factorial X to the third The third derivative is negative cosine of x So negative cosine of zero is negative one So we have what negative x cubed over three factorial We don't need to generate the next term because it's actually going to be what? It's going to be zero. So every other terms going to be zero Are we going to alternate signs on the terms that remain? Well Here's the first term that's actually present is cosine of x the next one that's actually present is negative cosine We lose this one. What's the next one is? Cosine of x it appears that they're going to alternate in sign right s i g and sign So let's leave out the zeros. It's probably not enough for the pattern yet But if you had to wager a guess as to the next term that actually appears We know we're going to lose the x to the fourth term, right? The next term that appears is x to the fifth. Is it going to be positive or negative? It's going to be positive And I guess the only Leap we're making is what's the denominator? What would you guess based on the pattern that we have five factorial? And if we wanted to continue this is the pattern that look like a sine function It is Whether we think it looks like one or not it is this is how your calculator probably does Sign functions.  I don't know to what power 13th power You can probably check your manual out. That'd be something good to do over the weekend, right? You wake up at 513 in the morning Just get that calculator manual out and see how it's it you're giving me that look again How it actually is doing sine functions or inverse tangent functions, but my guess is that it's doing this Up to maybe x sub 13 over 13 factorial So it's not a Taylor series, but a Taylor polynomial This is probably going to be one of those that you're going to want to memorize I mean once you get the pattern going So let's give a couple of memory aids before we go on to the next one Is it the sine and odd function? Is that correct? Sine is an odd function What would have to be true about the sine function in order for it to be in classified as an odd function? Okay, there is no symmetry on either side of the y-axis that would be an even function Okay, which we're going to get to that in a minute But an odd function Symmetry to the Origin So we're going to have symmetry Okay, I like that Jacob. I think you said it symmetry to the origin And how do we test for symmetry to the origin the f of negative x is equal to the negative of the f of x? If that's true A couple little question marks there if that's true, then it's an odd function. Well the symmetry to the origin So for this point here, it's symmetric images right here on the other side of the origin For this point here, it's symmetric images there on the so we do have symmetry to the origin And this is the kind of the test that creates that kind of symmetry So is the sine of negative x is that the negative of The sine of x well just pick a value See if that makes sense sine of negative pi over six is That the negative of the sine of pi over six This is a fourth quadrant angle the sine of a fourth quadrant angle is negative So you would take the reference angle, which is pi over six take the sine of the same reference angle and just negate the value, right? negate it because this is in the fourth quadrant this is In the first quadrant same answer this one's negative So we better negate the first quadrant answer to make sure they're equal it is an odd function is that helpful in remembering that particular power series to know to remember that the sine is an odd function That helps me because That's odd That's odd That's odd that's odd everything in here is right The factorials are odd the powers of x are odd. It's an odd function That must be the sign We're not going to get the other one today, but I want to do a quick example here Since I brought up pi over six We have a series a polynomial type series for the sine function And it's x minus x cubed over three factorial well everywhere. There's an x. Let's replace it with a pi over six and see if it works So it should be x which is pi over six, you know, I'm feeling kind of lazy today, too I don't really want to write any more terms out Let's just see how well we did okay, so the sine of x is Is x and I know there's a whole lot of other terms to this But let's just stop right there. Let's just say sine of x is x How well did we do you be surprised we actually did pretty well What's the sign of pi over six? I get that trig look every time we do a trade problem that No, that's third equivalent of 30 degrees. What's the sign of 30 degrees one half Okay Now what is the number pi? Which is what this side is the number pi divided by six? That's a little more than three divided by six 0.5 to Three six, that's not bad We didn't have to go very far in that Taylor series. It's pretty good approximation already now. This is too much, right? So what's going to happen next? We're going to subtract some away. Let's just see by putting in the next term how much better it is So the sine of pi over six on the left side is one half So on the right side, we're going to use now. We're going to gather in the next term I mean, you know what's going to happen. We're going to subtract some but guess what we're going to subtract too much Therefore we have to add some in with the next term and that's how those things proceed But what's pi over six that quantity cubed? over Six or three factorial Subtracted from this thing right here pretty good, right? That's pretty darn close to the sign of pi over six and we only use the first two terms of the Taylor series So as you progress and gather in this term and this one X to the ninth X to the eleventh it's going to be very very accurate That is more than likely how your calculator is doing a sign problem Whatever number you enter in if you enter it in in radians Which this is in radians. It's going to use that value If you enter a value in in degrees, it's going to convert that to radians so that it can use This particular power series Now we do have time for this we don't have time to develop the cosine I'm just going to write down what we have at this point thus far For the sign isn't the cosine the derivative of the sign That correct so you've got some things that are going to kind of connect these that you can make sure that you've memorized the right thing We're not developing the cosine yet using the Taylor series But let's see what we're going to get once we do that. What's the derivative of X? It's one What's the derivative of? X cubed over three factorial well the three factorial is really one over three factorial, right? That's a coefficient and what's the derivative of X cubed and we'll simplify that in a minute The coefficient here is one over five factorial Derivative of X to the fifth coefficient here is one over seven factorial Derivative of X to the seventh and that process is going to continue the same way. There we go That three knocks our three factorial Which is three times two times one the three's knockout so we have two factorial that process should continue Five factorial the five and the five knockout so you're left with four factorial That's sounding kind of even to me The seven knocks out with this seven so you're left with six factorial Here we've got an X squared guess that's looking better all the time There's what the Taylor series or the McLauren series since we're centering it at zero is going to look like for cosine Wouldn't it be nice if cosine were an even function that would help you remember that it's cosine and even function it is Cosine is an even function Now what might throw you off a little bit. I dotted my you there's that okay if I dot my use The first term might throw you off in this this whole even function thing, but if you remember X squared over two factorial X to the fourth over four factorial You could really kind of back up to from this one this technically is what X to the? zero over zero Factorial is that right and it's still zero is even right So it is even cosine is an even function the test for something To see if it's even It doesn't matter whether you put in an X or a negative X you get the same thing is The cosine of negative pi over six Is that the same thing as the cosine of pi over six somebody justify that and we're done for the day We're not gonna I mean a song won't do it. We can't sing a song and it's justified Where are we in this cosine of a fourth quadrant angle? This is cosine of a first quadrant angle Isn't the cosine positive in the fourth quadrant and also positive in the first quadrant If the reference angle is the same it doesn't matter if your first quadrant or fourth quadrant The cosine is the same in both if this had been second quadrant and third quadrant is the cosine the same in both of those Yes, so this is a true statement which in all cases. It's true cosine of X is an even function Have a great long week