Introduction to Calculus and Lines

Well, welcome to the official start of calculus. Congratulations, you made it in the class. Today we're going to talk about a review, a review of a lot of the math 2 concepts and some basic algebra concepts that you really need to have down in order for you to be successful in calculus. The first place we're going to start is section 0.1, chapter 0. We're going to talk about lines, just basic lines. We're going to go through some families of curves. We'll get into some trig functions, your favorite, right? And then we'll continue calculus. That's a weird line. There. What's special about a line? What do you know about lines? They have two points. Well, they have infinite points, but specifically you need at least two, right? What else do you know about lines? Do they curve at all? Do they curve lines? Do they end? What do you need to know about a line in order to graph it? You need two points. You need two points or you need one point and specifically one other thing. Slope. You've got to have slope. Lines have slope. They're straight. They don't end. They have slope. In fact, the slope is what we're going to talk about for the first part of today. The slope is pretty much how a line rises or falls. Now you might have been walked through this a long time ago. We're going to go really quickly through how to find the slope of a line. And we're going to invent the formula ourselves. So let's take a generic line and we'll pick two random points on it. We're not going to be specific on it because in order In order to create a formula, you can't really say something specific like any particular point. We need to work for all points. Can you tell me if I've got two points, how can I make a distinction between these? What does every point have? Coordinates. What are the coordinates? Someone over here, what are the coordinates generally for a point? X1. X1. Okay. So we know that any point that we draw. It's going to have the coordinates x, y. This one will and this one will. But the problem is we need some way to tell a difference between those two points. How are we going to do it? What do you think? With some numbers. What are those? What numbers? One and two. Are they going to go on top of our x or below our x? Below. In place. Where now? Now if I put them up here, we're talking about x to the first power. So we're going to put an x to the, an x1 and an x2. A y1 and a y2. Saying that this is our first point and our second point, whatever those points may be. We can't use real numbers because then it wouldn't work in general for any two points. Can you tell me along the x axis how far is this? One? If this is the point, how far is that? Yeah, it's for sure. How far is this point? Yeah, are you guys with me on this? Nudge your head if you're alright so far. If you're not, you go like this. We okay or no? Alright. If this was like the point, to get to, you'd go over 3 and up 5, right? So then this would be 3. This is not 3 times 5. It's x1, y1. So we're going over x1. How far are we going up? Good. And here? Now when we talk about Typically, a long time ago when you were first introduced to slope, the teacher probably said, yeah, it's how your line rises or falls. But then they also said, slope is defined as what over what? Let's go ahead and let's try to identify what our rise is and what our run is. What would you say would be our rise? This way? This way. So if we find the difference between those two numbers over there, we're going to find the rise for our line. What is the difference between those two numbers over there? How do you find the difference, this distance here? Sure, yeah, if this was 10 and this was 3, the distance between them would be what? Yeah, you'd subtract them, right? You'd do 10 minus 3. So here we're going to go, well, it's not 10 and 3. It's y2 and y1. And so our rise we're going to call y2 minus y1. Can we do the same thing with the run? How far is our run? What's that distance represented as? x2 minus x1. I thought you were going to answer, instead you sneezed. I was like, she's off. And then, pew, darn it. Let's hope for a good one. Yeah, we got x2 minus x1 for sure. Does that look familiar? Yeah, if we use the letter instead of the word slope, what letter am I talking about here? If we use M instead of slope, we got our formula. Which is kind of, you've seen it before right? You've probably seen it, have you seen it invented before like this? If you haven't well this is something new for you. If you have well you've seen it again. This is how you invent the slope formula. The reason why we couldn't use specific points is because we wanted to be able to plug in any two points. I give you, right? So using that, if you call your points x1, y1, x2, y2, you can find a slope for anything. Now, the one reason why I invented this for you is I want to show you that we can actually create an equation for a line from that slope formula. So let's talk just a bit about equations. What we're going to do is we're going to manipulate that formula by fixing one point. We're going to be able to get the formula for a line. So here's what we're going to work with. We're going to start with M equals Y2 minus Y1 over X2 minus X1. What I'm going to do is I'm going to fix this point. Say this is any specific one point and let the other one float. What that does is it changes this formula into this formula. So, y2 minus y1, there's no more y2. You see, here's the thing. A fixed point is a point with a little number under it, a subscript. Like this would be a specific fixed point, x1, y1. This is a specific fixed point, x2, y2. What I'm doing by fixing one point, only one point, I'm saying that's the one point I'm fixing, x1, y1. I'm gonna let the other one be floating. So, I'm gonna erase that y2 and the x2. That means I'm gonna have y minus y1 over x2. for x minus x1. That says one point's fixed. This point is going to be formulaic. It means you can plug in an x and get out a y. You ever seen an equation for a line? Nod your head. You've all seen it because you're all here, right? It has places for you to plug in x and get out y, doesn't it? So we need that inherently. Otherwise, we don't have the equation for a line. We have a specific two points. We don't want that. We want the equation for a line. Now, here's the cool thing. Is there a way that you can solve for y minus y1? How would you get rid of... of this denominator. Do what now? Someone said I just have bad ears. See? Multiply it. On one side or both sides? So if I multiply this by, oh that worked, cool. X minus X one, and over here, X minus X one. Is this gone? You guys are so quiet. Is this gone? Yeah, they can't do that. For sure. For sure. On the right-hand side, tell me what I'm left with. On the left hand side, I'm going to reorganize this. I'm not going to have x minus x1 times m. I'm going to have m times x minus x1. You still OK with that so far? Multiplications, commutative, doesn't matter what I have first, what I have second, no problem. Maybe you're not familiar with it written this way, but I'll bet you've seen this before. Yeah, what is that called? No, no, it's not slope intercept. Point slope. Why is it called point slope? Mathematicians are very unoriginal. It's called point slope because it's named after what you need to complete it. You need a point and a slope. That's why it's called point slope. So by manipulating our slope formula, interesting, isn't it? We use this. We fixed only one point, let the other one float. Worked around a little bit. We now have point slope. Pretty useful stuff. Shall we do an example? Would you like to see something that we can use this stuff with? I think you've seen it before. It should be reviewed for you. But let's go ahead and let's see if we can first get the cobwebs out of your head because I know you weren't doing math over Christmas bread, were you? or a holiday break, whatever you're doing. Were you doing math? I was, I was redoing this class to make it extra super special for you. You should feel honored. But let's go ahead and try to find the equation of the line that passes through these two points. We'll get the cobwebs over our head, we'll try to use the slope formula, and then use point slope. Or what points? So find the equation. Through these two points. Whenever your teacher taught you how to find the equation of a line, they taught you you need, you absolutely have to have two things. You have to have one, what? Point. If you have one point, very good. And somebody else, you also have to know the? Get into the slope. We'll be able to find the slope. Firstly, do we have a point? Yes. Actually, we have two of them. We're set, right? Do we have a slope? Is it given to you right now? No. Can you find it? Yes. Go ahead and find it. By the way, I usually do this, I'll walk around the room. If you need help at this point, let me know because now would be a good time for me to help you. What I want you to do right now is find the slope. If you don't remember how to find the slope, you assign x1, y1 to one point, x2, y2 to one point, and plug it in that formula. Go ahead and try that right now. Let's see what you all did. Just want the slope for now, we'll work on the equation in just a second. So as far as the slope goes, we need to pick one of these points to be x1, y1, another point to be x2, y2. You just got to make sure it goes xy and it goes 11 and 22, right? Those numbers got to be together. in alphabetical order, you can't go YX. Which is gonna be your X1? Do you all pick negative two for your X1? It really doesn't matter, does it? I could pick this one as X1, but then that would have to be Y1. So once you pick. one letter, the rest of them have to fall into place. Now typically people just want to make it easy and they put this one, x1, y1, x2, and y2, which I'm guessing most of you did that one. The other way, you're going to get the same exact slope. It's just your, if you had a negative or a sign, it's going to be in the opposite spot, not a big deal. Let's go ahead and plug this in. We have y2 minus y1. What am I going to write down if I'm supposed to do y2 minus y1 right now? Two. Two, like that? Minus, okay, good. So we're subtracting, but that also has a negative sign. I'm going to be real careful about that. I'm going to put it in parentheses to show that I'm subtracting a negative right there. What is subtracting a negative? What's that become? Great. Okay, well yeah, it ends up being addition. And then for X2 minus X1, we're going to have our 8 minus negative 2. Same idea. What is our 2 minus negative 3? How much do we get? Over? Can you reduce slopes? Absolutely. What is it? Quick show of hands, how many people were able to find the slope? Good, that's starting to remember, that's fine. If you're not, work on that later. Revisit this, try to follow through this example, see if you can do it on your own and then come up with that one half. Are we done? We're about halfway there. We have the slope. Do we have a point still? If we've got a point and we've got the slope, we should be able to fill out point-slope. It's actually the same exact formula as slope, we just have now fixed one point and we have the slope. So we know we're going to be filling out y minus y1. Check it out. If you already have your y1, you can leave it. If you want to make it easier on yourself and not deal with those negatives, you could use that point. That'd be fine as well. It doesn't matter what point you have after you've already identified your slope. Now I'm going to stick with the same one to keep it kind of continuous for us. So we have y, what am I going to write? One. One minus one. Negative three. Perfect, okay. Again, I'm going to use some parentheses saying that's a negative number in there. Equals, I'm supposed to have my m. What's my m? One. Uh-huh. And x? Minus negative three. Perfect. Thank you. Let's clean this up just a little bit. We're going to have y plus 3 equals 1 half x plus 2. If it's asking for point slope, you know what? You're done. That's it. But this isn't exactly the easiest way to graph a line, is it? Like that. Is there a way we could make it easier? What would you do? Sure, we want to get rid of those parentheses. We'll distribute that. If we distribute the right side, I'm still going to have this y plus 3, but I'm going to get 1 half x plus what? One. Good. One half times 2, that's going to give us 1. And lastly, the last step is what? Y. Yeah, if I do that, I'm going to get y equals 1 half x. That's something we're real familiar with. We know how to graph that pretty well on the xy axis. What is that called, by the way? Yeah, whenever we have y equals mx plus b, when we have some number times x plus or minus some constant, we know that that's going to be called, that is slope intercept. And it's pretty easy to graph. It gives you what you need to graph a line very quickly. And again, the reason why it's called slope intercept is that's what you have. That m, well, that's our slope. What's the b stand for? Y intercept. Yeah, good. Y intercept. Can you graph it? What we just graphed. How would you graph something with 1 half x minus 2? Can you tell me what is my y intercept in this case? So when we're graphing slope intercept, it says if we have a negative 2, that means we're going down or left, which was. What's that mean? So we know our y is down two, and it says we're gonna put a point right there. That's where our minus two's coming from. Next we use our slope from the point that we just graphed, not the origin, but the point that we just plotted to find our next point. Our slope is, what was it? Up or down? Up. Up how many? One. And over to the? Right. Over to the right how many? Two. Yeah, the positive or negative tells you whether you're going up or down. You always go to the right. So if we had negative one half, that would say go down one, but you're still going to the right. You're always going to the right. The plus negative tells you up or down. You with me on this so far? So we're going to go up one, you all said to the right two. Now that we have two points, we know that the two points delineates a specific line, unique line. We graph it, we make sure we label this, and we're done. Show of hands, how many of you feel okay with graphing these lines using slope formula and the point slope? Good deal, alright. Now, there's a couple more things we've got to talk about about lines before we talk about parallel and perpendicular, which we're going to do in just a second. If I told you that... This was a line, y equals to some number c, where c is a constant. What type of line is that? They're all straight lines, right? They're all straight lines. That's a straight line. What I think you might mean is, is this a vertical line or is that a horizontal line? Horizontal line. Horizontal line. You know the way you can tell? What variable do you have up there? Y. That means it's going to cross the y-axis. Whatever variable you have says that's the intercept that you have. So there's no way to have this line crossing the y-axis. Does that make sense to you? There's no way to do that. The only way you can do that and have a constant line is like this. That's a horizontal line. So if you have a y equals, that's saying you're going to have a y intercept at that number, and it's going to be horizontal. So when y equals a constant, you're talking about a horizontal line. What's the way I can make a vertical line, just straight up? Yep. If x equals a constant. We have an x-intercept. That's a vertical line. The way you think about that, if you have y equals, if you try to fit it into this formula, the slope intercept, the point slope, you've got a slope of zero, right? There's no slope there. Slope of zero is going to give you a horizontal line. If you don't have a y, though, that means you're going to have an undefined slope. That means you're going to be a vertical. Oh, yeah. Let's manipulate one equation. We'll see if we can put it into slope-intercept form. We'll talk about some parallel perpendicular lines, and then we'll go on to some trigonometry. I can tell you're excited. You excited? Yeah. Some of you are giving me death bricks right now. You know that? Right here we have an equation that's kind of a standard form of a line. It'd be standard form if we added the three to one side. And then we'd have the standard form. Is there a way you can put this into slope intercept? Go ahead and try that for me. Just go ahead and solve for y and make that slope intercept. We want to make sure you can do it. What would you do first if you're trying to solve that thing for y? Yeah, you're trying to isolate it. What's your first step in doing that here? You could add 3, then you could subtract 4x, and then you could divide by 2. Another way to do it, you could subtract 2y, right? It's probably going to eliminate a step for you. If you subtract 2y, you're going to end with... 4x-3 equals negative 2y. Now typically we don't like to do that because we don't want to have a negative coefficient from a y, but we just have to divide by a negative. If you're good with signs, you can do that pretty easily. How do we get that y by itself again? Just make sure if you divide, you've got to do it everywhere. And you have to have the same exact thing and you have to be good with your signs. So 4X over negative 2, what are you going to get out of that? Negative 2. How much? Negative 2. 2? X. And then plus or minus? Plus. How much? 3X. Equals? 3. Sure, maybe you write that a little bit different and skip that in the form that you want it. Y equals negative 2x plus 3 halves. Could you still graph that? As a matter of fact, sometimes it's nice to keep it in standard. I asked you to do it in slope-intercept form, but I also want to show you this. If you had this in standard form, which would be that. Have you ever learned the cover-up method for graphing from standard form? You ever seen that before? If you want to find the x-intercept, cover up your y-intercept. Y, divide by 4, you know this is going to cross the x-axis at 4 thirds. It's easier than graphing a fraction and going up and over, down and over. If you want to find the y-intercept, you cover up the x, divide by this number, positive 2. That's going to cross the y-axis at positive 3 halves. You can graph a line like that as well. So just a little refresher on this. Hello show hands, how are you people doing? Okay, so far on our lines. You still alright? You awake still? This should be review. This is review for you. I know some of you are thinking, where's the calculus? Just wait for it. The calculus is going to come, whether you want to or not. Just hang on for a second. Enjoy the nice slow stuff, but really absorb this if you're a little rusty on it. What do you know about parallel lines? That's actually the definition. Did you hear them over there? We have the same exact slope. That means we have parallel lines. It's kind of like climbing a mountain. climbing stairs, right? The way stairs work is they're parallel. That means you're going up and over the same rate. Otherwise, on these stairs over here, if your stairs didn't go up and over, you're going to be like this. You're going to be like, oh, these are nice. If they were different slopes, they intersect somewhere. We don't want that to happen for stairs. We don't want that to happen for parallel lines. So when we talk about parallel lines, what we're talking about are lines that have the same exact slope. Do perpendicular lines have the same slope? No, no they don't. Actually, perpendicular lines meet at a very specific angle. What angle do perpendicular lines meet at? So if one line's like this, the other line's got to be like that, right? It means if one slope's positive, the other slope's negative, so we know that's going to come into play. Also, there's a... It's not just a negative slope. That's going to be something like this, right? That's not going to cross at 90 degrees. We want to make it actually just a little kicked over. How do I make it so it meets exactly 90 degrees? It's not only negative but also a? Negative reciprocal. Very good. A couple of you said reciprocal. So perpendicular lines are lines which have, are lines where the slopes are negative reciprocals of each other. I was begs the question, if I give you an equation, can you find a line that's parallel and or perpendicular or perpendicular to a given equation? Can you guys do that? Let's try that out real quick. That'll be our last little timid review problem. Hopefully these have been timid for you. We haven't even made it 30 minutes in. We're going to get to some tricky just a bit. Yeah. So let me say that I want to find the equation of the line that passes through this point and parallel to this. Thank Find an equation passing through a set point in parallel to a given equation. What's the two things you need for sure in order to make the equation of a line? Slope and what now? Do I have a point? Cool. Do I have a slope? Do I have a slope right now? Not yet. I've got to work on it, but could you find that slope? Let's find the slope. Go ahead and do that on your own real quick. Give me about five seconds. You guys should be pretty good at this. Is the slope negative 2? No, what am I missing? That's the reason we talk about slope intercept, right? It's to find that slope very easily. Your slope's negative 2 thirds. Now what I'm asking for is the equation of a line that's parallel to that line, but now goes through that point. So what slope am I going to use if I want to find the parallel line to this? Three halves. I'm going to use three halves? But what the parallel to this? Same. Oh, parallel lines have the same slope. So what we're going to write down, we're going to find our slope. We want the parallel slope. So we're going to write down the parallel slope. If it's parallel, it's going to be exactly the same. We know that our m is going to be negative 2 thirds. Now what point do we use? What point do we use? Speak up. What point do we use? Does this 4 have anything to do with this problem, actually? All we cared about was finding the slope. Once we have the slope and we've already identified a point, Wow, we can just plug that into our point slope formula to find the equation for our line. So we'll do y minus 7 equals negative 2 thirds x minus 6. Quick show of hands, how many people feel okay solving for slope? Keep your head down if you understand that the slope we're supposed to use is still negative two-thirds because we want to find a parallel. Do you see where the six and the seven are coming from? Good deal. Can you work that out and make that slope intercept form? Okay, let's do that together. I know that y minus seven, that's gonna stay there. On the right hand side, tell me what I'm gonna get. And then plus or minus, what do you think? Plus. Plus how much? Four. Good. Final step, if we add set four, because we got 2 3rds times six. 2 3rds times six, you can simplify the fractions, or we get 12 3rds, that's four. Add our seven to both sides. And we're done. What if I change the problem, and instead of having parallel, I ask you to do perpendicular? Could you still do it? Let's talk about the only changes that would occur up here, okay? I'm not going to change the whole problem, I just wanted to walk you through it here. If I'm talking about perpendicular, would this process change? Would this change? What's that going to become? Okay, so this would become 3 now, so... Here now we're talking about perpendicular. With the 7 and the 6 change. No, we're just talking about slope. What we're going to talk about this is parallel perpendicular only has to do with the slopes. This would change to our 3 halves. This would change to our 3 halves x. It would be minus how much? Minus 9. And if we added 7 to both sides here. We get 3 halves x minus 2. So we find both the parallel and the perpendicular slope. Do you feel okay about our basic lines so far? Would you like to learn a little bit about angles of inclination and use a little bit of trigonometry here? And you're like, no, not really. Too bad. We can do it anyway. Let's talk about angles of inclination. We're going to use it at some point. Are there any questions before I erase any of this stuff? Are you sure? Are we having fun yet? Wouldn't you rather be in here than out there in the rain? Some of you are like, no, I'd rather be in the rain. Honestly, right now, I prefer that. Just lie to me. Just lie and say, yes, this is awesome, Leonard. I love this class. You're the best. Thanks, guys. Appreciate it. If you did that, I would just watch this video over and over again and have you go, Thanks Leonard, you're the best. Yeah. Thanks Leonard, you're the best. Yeah. Thanks Leonard, you're the best. Yeah. See, I just have a running move. I do it with my workout. Thanks Leonard, you're the best. Bam. Thanks. Bam. So cool. I know, I'm a dork. Oh my, angle of inclination. Let's talk about how we can use angle of inclination and relate it to a slope. We're going to start with some line. Random life. When I talk about the angle of inclination, what we talk about is the angle that any line makes with the x-axis. So the angle of inclination, these actually would be the same angle. If you've taken geometry, you know that those are the same. We're really just talking about this one now. We want to find some way to represent this line as having that angle. If we think about the x and the y-axis, notice that this we can represent as a change in x. You guys have seen that terminology before, the delta x change in x? And this would be, well, the change in y. Now think back to your trig days. This is just basic trigonometry. Is there a trig function that relates this angle and these two specific sides? Remember this would be a 90 degree angle. What does that? This is a triangle, which it is. What's this side called? Hypotenuse. That's hypotenuse. What's this one? According to this angle. That's the... Good. And this one is the... Which one relates adjacent and opposite? This one. Not sine, sine would be opposite over hypotenuse. Tangent does. Or cotangent, we don't deal with cotangent. Cotangent's adjacent over opposite. We want to deal with probably the easy one, tangent. If we talk about tangent, the tangent of that angle... Is equal to the opposite over the adjacent. Nod your head if you're okay with the tan, opposite over adjacent. If you're not, you're definitely going to want to review your trigonometry before attempting this class. We deal with a lot of trigonometry in here. But here's the cool thing. What do we already define as change in y over change in x? Or rise over run? What do we already define that as? So then we have this relationship. We know that tan theta, well, that's delta y over delta x. But this is also the same thing as slope or m. If you bring all this together, Slope's equal to tan theta. Do you see the relationship between your slope and your angle of inclination? It's the same as the tangent of that angle of inclination. Because tan is defined as opposite over adjacent and so is slope, we can make that jump. What's kind of cool is it says that if you know the angle, well you can find the slope. Getcha? If you know the angle, you can do that in a calculator, you'll be able to find the slope. If you know the slope, you can find the angle. Those things are intertwined. They're equal to each other. Along that tangent. So let's, shall we try one? Would you like to see an example of how this is done? Would you? Yeah. You're in the most mellow class I've ever had. Would you like to see one? Bring it on. Let's say that your angle is 30 degrees. Someone quickly, 30 degrees as radians is what? 5 or 6. Good. The same thing as 5 or 6. Yeah. I want you to find the slope of the line that has an angle of inclination of 30 degrees or pi over 6. Here's how you do it. We know for a fact that m equals tan theta. Don't forget that. That's your equation. That's what you do now. You know that. that the slope is equal to the tangent of that angle. What's our angle? What's our angle? 30 degrees. 30 degrees. So if I plug in my 30 degrees, or I plug in my pi over six at the same angle, okay, if I plug that in there, then I know the slope I'm looking for is equal to the tangent of pi over six or 30 degrees, wherever you want to work in. I like pi over six. Now, all those with your unit circle tattooed on your right arm, okay, look down there. Can you tell me what tangent of pi over 6 is? Can you tell me what tangent? Remember, tangent, you have to define a sine over cosine, right? So you define sine of 30 degrees or sine of pi over 6. Put it over cosine, or have it memorized. Put it over cosine of pi over 6 or 30 degrees. Look at you in a circle if you have one. You should have one. You need one. Honestly, you should tattoo it on your forehead backwards. That way you look in the mirror and you can memorize it. It's a good idea. I haven't done it personally, but I'm waiting for someone to actually do that. You're pretty classy. Tangent of pi over 6 is sine over cosine. The sine of pi over 6, 1 half. Cosine of pi over 6, root 3 over 2. Correct me if I'm wrong, but I think those are right. I'm doing this. on the spot. Good, alright. Can you simplify that a little bit? Yeah, those twos are actually going to cross out. You know, divide those fractions, complex fractions, you're going to flip that, multiply. The twos are going to be gone. What you're going to end up with is 1 over the square root of 3, or if you rationalize the denominator, you know how to rationalize denominators, right? Multiply by root 3 over root 3, you're going to get root 3 over 3. Strangely enough, that's your slope right there. Your slope is root 3 over 3. Alright, let me recap. Did you get that? That was funny. Let me recap. Come on. Stand up next time. Maybe you'll get it. I honestly will recap though. Here's what you do to find the slope if you have the angle of inclination. You take your angle you plug it in and figure out the tangent of that angle. That's honestly it. I mean, if you can find the tangent of pi over 6, you have your slope. That is your slope, all right? Now, can you go backwards? That's going to be another question for us. What if I said I have now, I have a slope of negative 1. Can I find the angle of inclination? We only have one equation. The only thing that we know is that the slope equals tan theta. Over here we knew the theta, right? We were looking for the slope, the m. Over here we know which one? We know the M. So notice we're using the same exact equation here. We knew our angle. You could have put 30 degrees here very easily and done the same exact thing. Here we know our M. We know negative 1 equals tan theta. How do we find theta? You can do tan inverse, sure, of both sides. Take it to the left. We do tan inverse of negative 1 equals theta. If you do tan inverse of both sides, it would look like this. You'd have tan inverse. You have tan inverse. Tan inverse of tangent, gone. You have theta. Tan inverse of negative 1, that's what we have right here. This is what the question asks you, ok? Here's this in plain English. It says, I want you to tell me the angle so that when I take tangent of it, it's going to give me the angle. Negative 1. That's what tan inverse says. It's kind of a backwards way of looking at it. It's saying find me the angle. That way when I take the tangent of it, it's going to give me the value of negative 1. Do you understand the question there? It basically goes down to look at the unit circle. Find out where sine and cosine are the same but have different signs because you know tangent is sine over cosine, right? So you look at your unit circle. Find out where it's, what do we have? It gets 2 over 1. I don't remember the exact one. Find out where you have sine and cosine exactly the same, off by sine. That will give you negative 1. Have you found it? Do you have your unit circle out? Yeah. 3 pi over 4. That's exactly right. Oh yeah, I have it right here. So this happens where your sine and your cosine are the same, but off by a sine. That's root 2 over 2 over root 2 over 2. That's where that happens on your unit circle. So check that out. later if you don't have your union circle handy, you're going to have that same exact value only this one's going to be negative. That happens twice actually. If you're confused, it's, well, wait a second, don't I get two of these same values? Yeah, you do, but think about what you're actually doing. You're trying to find the slope of a line. It goes like that, right? It's gonna cross two quadrants. It's gonna have a slope here and there. One's gonna be positive, one's gonna be a negative way of looking at an angle. They're all the same, though. Three pi, what'd you say, three pi over four? Yeah. Three pi over four, or the negative version of that, used as a reference angle. So here we go. We got that negative one, so we know that this happened when theta was equal to three pi over four. Or if you want to translate that, that's 135 degrees. So either way we look at this, we can find slopes from angles, we can find angles from slopes. I'd say that this one's probably a little bit more tricksome for you because you're going to actually have to do a little bit of work on it. This you could probably just plug in a calculator if you memorize like the.877. thing, you know, that's like 3 over 2, yay. Or whatever you, does anyone want to be, I memorized those, I don't want to look them up. But if you do this on a calculator, it'll actually work out for you. It probably won't give you the square root of 3, but you can figure that out. Here, You're going to have to use a unit circle. If they give you a slope, you're going to have to find the angle to which sine over cosine makes that value. It's going to be something, you might have two of them. It's not going to matter, that's the same exact value of the angle. Choose either one of them, it'll work out the same. But find the angle to which you're getting that value. Do you guys see the process here? I know I went very quickly through this. Do you see the process? Yeah or no? How many of you feel okay with this? Does it take a little bit of work to get handy on that? Yeah, probably. Probably a little bit. bit more work to do. Let's see one more thing I want to go over. Let's talk about the distance formula real quick. It'll wrap up our very first section here. Are you sure there's no questions on the angle of inclination anymore? We've got time. So if I give you this on a test and I say I want you to find the slope if the angle is pi over 3, you do it? Could you do it if you had a unit circle? Yes. Oh, okay. That's a good question. If I say the slope is 1 half, can you find tangent where the angle would give you 1 half? Could you do it with a unit circle? Yeah. Okay. Practice that stuff. That's what I'm looking for. I don't care. Sure. To be honest with you, this is very much review stuff. What I'm trying to get you to get back familiar with, the tangent, sine, cosine, secant, cosecant, cotangent ideas, because we're going to move way past this. We're going to be using this within some problems, all right? So this isn't going to be a problem. This is going to be within some problems. It's kind of like factoring isn't everything in algebra. You just use factoring in everything in algebra. You get the analogy? So that's kind of what we're doing here. We're going to use a lot of trigonometry in this class. We're going to be doing things called derivatives and integrals, and they're all going to involve trig functions. So you've got to know the trig pretty well to be successful. People say that you go to calculus to finally fail algebra and trigonometry. You made it this far, but it's not the calculus that's going to hold you back. I promise. It's going to be your algebra, and it's going to be your trigonometry. I guarantee it. The calculus is actually quite easy. It's those concepts put together with calculus that makes it kind of hard. So if you're good at algebra and trig, you can. Absolutely fine. Stick with this class. Okay, shall we? Distance formula in about a minute and a half, then we'll call it a day. Let's do distance formula. We're going to do it the same way that we did our slope formula, which is we're going to pick two random points, X1, Y1, and X2, Y2. Only this time we're going to find the distance between them. If we have the x1, y1, x2, y2, well, we for sure know that's x1 and that's x2, and this is y1 and that's, well, that's y2. What we want to do now, though, is use something that relates this side. This side, this side, and that side, and that's a right triangle. What relates to that? Pythagorean theorem, absolutely, you're right. If we call this our distance, here's what we can say. We know that this length is x2 minus x1 by the same stuff that we... that we just did with slope formula. We know that this distance is y2 minus y1 by the same stuff we just used on our slope formula. This is the length, this is the length from those corresponding points. We wanna find the distance. If we do Pythagorean theorem, we've got d squared equals, what's Pythagorean theorem say? Sine one squared plus sine two squared. Sure, yeah, a squared plus b squared equals c squared, some of you know that, or I prefer a leg squared plus a leg squared equals the hypotenuse squared. because that tells you what you're doing, right? So if we take a leg squared, that's this, and a leg squared, that equals the hypotenuse squared. I already have that. Here's my first leg. That's the distance squared, plus the second leg. That's the distance squared. Can you guys see the Pythagorean Theorem at work here? Can you guys see the leg squared? Yeah? This is one leg, right? I'm just squaring it. Here's another leg. I'm just squaring it. And that has to be, by Pythagorean Theorem, equal to the hypotenuse. squared. So we got that. The only thing we need to do now, get rid of the square. How do I get rid of the square? Yeah, that just means the whole thing. Now we are going to omit the plus minus because, well, we can't have a negative distance, it doesn't make any sense. So we're just going to have the square root of this entire thing, d equals. By the way, oh, here's a good question for you. See where you're at. Will this square root get rid of this square and that square? What do you think? Does it work that way across addition? This multiplication? Sure. Addition? No way. No way no. Can't do that. And that's our distance formula. You know what, I'm not going to do an example for you because it works really, really, really similarly to our slope formula. Would you be able to, if I gave you two points, would you be able to find me an x1 and a y1? And an x2 and a y2? And plug them in. and not mess the signs up, right? Yeah. You just square this value, you square that value, add them, of course it's going to be positive, right, because you're squaring something, squaring something and adding it. Just don't forget to take the square root. You can either leave it in terms of square root or approximate it and give me a decimal answer. How many people feel pretty good about what we talked about today? All right, that's good.

We'll get into some trig functions, your favorite, right? And then we'll continue calculus. That's a weird line. There.

What's special about a line? What do you know about lines? They have two points.

Well, they have infinite points, but specifically you need at least two, right? What else do you know about lines? Do they curve at all?

Do they curve lines? Do they end? What do you need to know about a line in order to graph it?

You need two points. You need two points or you need one point and specifically one other thing. Slope.

You've got to have slope. Lines have slope. They're straight. They don't end. They have slope.

In fact, the slope is what we're going to talk about for the first part of today. The slope is pretty much how a line rises or falls. Now you might have been walked through this a long time ago. We're going to go really quickly through how to find the slope of a line. And we're going to invent the formula ourselves.

So let's take a generic line and we'll pick two random points on it. We're not going to be specific on it because in order In order to create a formula, you can't really say something specific like any particular point. We need to work for all points.

Can you tell me if I've got two points, how can I make a distinction between these? What does every point have? Coordinates. What are the coordinates? Someone over here, what are the coordinates generally for a point?

X1. X1. Okay.

So we know that any point that we draw. It's going to have the coordinates x, y. This one will and this one will. But the problem is we need some way to tell a difference between those two points. How are we going to do it?

What do you think? With some numbers. What are those?

What numbers? One and two. Are they going to go on top of our x or below our x?

Below. In place. Where now?

Now if I put them up here, we're talking about x to the first power. So we're going to put an x to the, an x1 and an x2. A y1 and a y2. Saying that this is our first point and our second point, whatever those points may be. We can't use real numbers because then it wouldn't work in general for any two points.

Can you tell me along the x axis how far is this? One? If this is the point, how far is that?

Yeah, it's for sure. How far is this point? Yeah, are you guys with me on this? Nudge your head if you're alright so far.

If you're not, you go like this. We okay or no? Alright.

If this was like the point, to get to, you'd go over 3 and up 5, right? So then this would be 3. This is not 3 times 5. It's x1, y1. So we're going over x1.

How far are we going up? Good. And here? Now when we talk about Typically, a long time ago when you were first introduced to slope, the teacher probably said, yeah, it's how your line rises or falls.

But then they also said, slope is defined as what over what? Let's go ahead and let's try to identify what our rise is and what our run is. What would you say would be our rise?

This way? This way. So if we find the difference between those two numbers over there, we're going to find the rise for our line.

What is the difference between those two numbers over there? How do you find the difference, this distance here? Sure, yeah, if this was 10 and this was 3, the distance between them would be what?

Yeah, you'd subtract them, right? You'd do 10 minus 3. So here we're going to go, well, it's not 10 and 3. It's y2 and y1. And so our rise we're going to call y2 minus y1. Can we do the same thing with the run? How far is our run?

What's that distance represented as? x2 minus x1. I thought you were going to answer, instead you sneezed. I was like, she's off. And then, pew, darn it.

Let's hope for a good one. Yeah, we got x2 minus x1 for sure. Does that look familiar? Yeah, if we use the letter instead of the word slope, what letter am I talking about here?

If we use M instead of slope, we got our formula. Which is kind of, you've seen it before right? You've probably seen it, have you seen it invented before like this?

If you haven't well this is something new for you. If you have well you've seen it again. This is how you invent the slope formula. The reason why we couldn't use specific points is because we wanted to be able to plug in any two points.

I give you, right? So using that, if you call your points x1, y1, x2, y2, you can find a slope for anything. Now, the one reason why I invented this for you is I want to show you that we can actually create an equation for a line from that slope formula. So let's talk just a bit about equations.

What we're going to do is we're going to manipulate that formula by fixing one point. We're going to be able to get the formula for a line. So here's what we're going to work with. We're going to start with M equals Y2 minus Y1 over X2 minus X1. What I'm going to do is I'm going to fix this point.

Say this is any specific one point and let the other one float. What that does is it changes this formula into this formula. So, y2 minus y1, there's no more y2. You see, here's the thing. A fixed point is a point with a little number under it, a subscript.

Like this would be a specific fixed point, x1, y1. This is a specific fixed point, x2, y2. What I'm doing by fixing one point, only one point, I'm saying that's the one point I'm fixing, x1, y1. I'm gonna let the other one be floating.

So, I'm gonna erase that y2 and the x2. That means I'm gonna have y minus y1 over x2. for x minus x1.

That says one point's fixed. This point is going to be formulaic. It means you can plug in an x and get out a y. You ever seen an equation for a line? Nod your head.

You've all seen it because you're all here, right? It has places for you to plug in x and get out y, doesn't it? So we need that inherently.

Otherwise, we don't have the equation for a line. We have a specific two points. We don't want that.

We want the equation for a line. Now, here's the cool thing. Is there a way that you can solve for y minus y1?

How would you get rid of... of this denominator. Do what now? Someone said I just have bad ears.

See? Multiply it. On one side or both sides?

So if I multiply this by, oh that worked, cool. X minus X one, and over here, X minus X one. Is this gone?

You guys are so quiet. Is this gone? Yeah, they can't do that.

For sure. For sure. On the right-hand side, tell me what I'm left with. On the left hand side, I'm going to reorganize this.

I'm not going to have x minus x1 times m. I'm going to have m times x minus x1. You still OK with that so far?

Multiplications, commutative, doesn't matter what I have first, what I have second, no problem. Maybe you're not familiar with it written this way, but I'll bet you've seen this before. Yeah, what is that called?

No, no, it's not slope intercept. Point slope. Why is it called point slope? Mathematicians are very unoriginal. It's called point slope because it's named after what you need to complete it.

You need a point and a slope. That's why it's called point slope. So by manipulating our slope formula, interesting, isn't it?

We use this. We fixed only one point, let the other one float. Worked around a little bit.

We now have point slope. Pretty useful stuff. Shall we do an example?

Would you like to see something that we can use this stuff with? I think you've seen it before. It should be reviewed for you. But let's go ahead and let's see if we can first get the cobwebs out of your head because I know you weren't doing math over Christmas bread, were you?

or a holiday break, whatever you're doing. Were you doing math? I was, I was redoing this class to make it extra super special for you.

You should feel honored. But let's go ahead and try to find the equation of the line that passes through these two points. We'll get the cobwebs over our head, we'll try to use the slope formula, and then use point slope.

Or what points? So find the equation. Through these two points.

Whenever your teacher taught you how to find the equation of a line, they taught you you need, you absolutely have to have two things. You have to have one, what? Point.

If you have one point, very good. And somebody else, you also have to know the? Get into the slope.

We'll be able to find the slope. Firstly, do we have a point? Yes. Actually, we have two of them. We're set, right?

Do we have a slope? Is it given to you right now? No. Can you find it? Yes.

Go ahead and find it. By the way, I usually do this, I'll walk around the room. If you need help at this point, let me know because now would be a good time for me to help you. What I want you to do right now is find the slope. If you don't remember how to find the slope, you assign x1, y1 to one point, x2, y2 to one point, and plug it in that formula.

Go ahead and try that right now. Let's see what you all did. Just want the slope for now, we'll work on the equation in just a second. So as far as the slope goes, we need to pick one of these points to be x1, y1, another point to be x2, y2. You just got to make sure it goes xy and it goes 11 and 22, right?

Those numbers got to be together. in alphabetical order, you can't go YX. Which is gonna be your X1? Do you all pick negative two for your X1?

It really doesn't matter, does it? I could pick this one as X1, but then that would have to be Y1. So once you pick.

one letter, the rest of them have to fall into place. Now typically people just want to make it easy and they put this one, x1, y1, x2, and y2, which I'm guessing most of you did that one. The other way, you're going to get the same exact slope.

It's just your, if you had a negative or a sign, it's going to be in the opposite spot, not a big deal. Let's go ahead and plug this in. We have y2 minus y1. What am I going to write down if I'm supposed to do y2 minus y1 right now? Two.

Two, like that? Minus, okay, good. So we're subtracting, but that also has a negative sign.

I'm going to be real careful about that. I'm going to put it in parentheses to show that I'm subtracting a negative right there. What is subtracting a negative? What's that become? Great.

Okay, well yeah, it ends up being addition. And then for X2 minus X1, we're going to have our 8 minus negative 2. Same idea. What is our 2 minus negative 3? How much do we get? Over?

Can you reduce slopes? Absolutely. What is it?

Quick show of hands, how many people were able to find the slope? Good, that's starting to remember, that's fine. If you're not, work on that later. Revisit this, try to follow through this example, see if you can do it on your own and then come up with that one half. Are we done?

We're about halfway there. We have the slope. Do we have a point still? If we've got a point and we've got the slope, we should be able to fill out point-slope.

It's actually the same exact formula as slope, we just have now fixed one point and we have the slope. So we know we're going to be filling out y minus y1. Check it out. If you already have your y1, you can leave it. If you want to make it easier on yourself and not deal with those negatives, you could use that point.

That'd be fine as well. It doesn't matter what point you have after you've already identified your slope. Now I'm going to stick with the same one to keep it kind of continuous for us. So we have y, what am I going to write? One.

One minus one. Negative three. Perfect, okay.

Again, I'm going to use some parentheses saying that's a negative number in there. Equals, I'm supposed to have my m. What's my m?

One. Uh-huh. And x?

Minus negative three. Perfect. Thank you.

Let's clean this up just a little bit. We're going to have y plus 3 equals 1 half x plus 2. If it's asking for point slope, you know what? You're done. That's it.

But this isn't exactly the easiest way to graph a line, is it? Like that. Is there a way we could make it easier?

What would you do? Sure, we want to get rid of those parentheses. We'll distribute that.

If we distribute the right side, I'm still going to have this y plus 3, but I'm going to get 1 half x plus what? One. Good.

One half times 2, that's going to give us 1. And lastly, the last step is what? Y. Yeah, if I do that, I'm going to get y equals 1 half x.

That's something we're real familiar with. We know how to graph that pretty well on the xy axis. What is that called, by the way?

Yeah, whenever we have y equals mx plus b, when we have some number times x plus or minus some constant, we know that that's going to be called, that is slope intercept. And it's pretty easy to graph. It gives you what you need to graph a line very quickly. And again, the reason why it's called slope intercept is that's what you have. That m, well, that's our slope.

What's the b stand for? Y intercept. Yeah, good. Y intercept.

Can you graph it? What we just graphed. How would you graph something with 1 half x minus 2? Can you tell me what is my y intercept in this case? So when we're graphing slope intercept, it says if we have a negative 2, that means we're going down or left, which was.

What's that mean? So we know our y is down two, and it says we're gonna put a point right there. That's where our minus two's coming from. Next we use our slope from the point that we just graphed, not the origin, but the point that we just plotted to find our next point. Our slope is, what was it?

Up or down? Up. Up how many? One.

And over to the? Right. Over to the right how many? Two.

Yeah, the positive or negative tells you whether you're going up or down. You always go to the right. So if we had negative one half, that would say go down one, but you're still going to the right. You're always going to the right. The plus negative tells you up or down.

You with me on this so far? So we're going to go up one, you all said to the right two. Now that we have two points, we know that the two points delineates a specific line, unique line. We graph it, we make sure we label this, and we're done. Show of hands, how many of you feel okay with graphing these lines using slope formula and the point slope?

Good deal, alright. Now, there's a couple more things we've got to talk about about lines before we talk about parallel and perpendicular, which we're going to do in just a second. If I told you that... This was a line, y equals to some number c, where c is a constant. What type of line is that?

They're all straight lines, right? They're all straight lines. That's a straight line. What I think you might mean is, is this a vertical line or is that a horizontal line?

Horizontal line. Horizontal line. You know the way you can tell? What variable do you have up there?

Y. That means it's going to cross the y-axis. Whatever variable you have says that's the intercept that you have.

So there's no way to have this line crossing the y-axis. Does that make sense to you? There's no way to do that.

The only way you can do that and have a constant line is like this. That's a horizontal line. So if you have a y equals, that's saying you're going to have a y intercept at that number, and it's going to be horizontal. So when y equals a constant, you're talking about a horizontal line.

What's the way I can make a vertical line, just straight up? Yep. If x equals a constant. We have an x-intercept. That's a vertical line.

The way you think about that, if you have y equals, if you try to fit it into this formula, the slope intercept, the point slope, you've got a slope of zero, right? There's no slope there. Slope of zero is going to give you a horizontal line.

If you don't have a y, though, that means you're going to have an undefined slope. That means you're going to be a vertical. Oh, yeah.

Let's manipulate one equation. We'll see if we can put it into slope-intercept form. We'll talk about some parallel perpendicular lines, and then we'll go on to some trigonometry.

I can tell you're excited. You excited? Yeah. Some of you are giving me death bricks right now.

You know that? Right here we have an equation that's kind of a standard form of a line. It'd be standard form if we added the three to one side. And then we'd have the standard form.

Is there a way you can put this into slope intercept? Go ahead and try that for me. Just go ahead and solve for y and make that slope intercept.

We want to make sure you can do it. What would you do first if you're trying to solve that thing for y? Yeah, you're trying to isolate it.

What's your first step in doing that here? You could add 3, then you could subtract 4x, and then you could divide by 2. Another way to do it, you could subtract 2y, right? It's probably going to eliminate a step for you.

If you subtract 2y, you're going to end with... 4x-3 equals negative 2y. Now typically we don't like to do that because we don't want to have a negative coefficient from a y, but we just have to divide by a negative. If you're good with signs, you can do that pretty easily.

How do we get that y by itself again? Just make sure if you divide, you've got to do it everywhere. And you have to have the same exact thing and you have to be good with your signs.

So 4X over negative 2, what are you going to get out of that? Negative 2. How much? Negative 2. 2?

X. And then plus or minus? Plus. How much? 3X.

Equals? 3. Sure, maybe you write that a little bit different and skip that in the form that you want it. Y equals negative 2x plus 3 halves.

Could you still graph that? As a matter of fact, sometimes it's nice to keep it in standard. I asked you to do it in slope-intercept form, but I also want to show you this. If you had this in standard form, which would be that.

Have you ever learned the cover-up method for graphing from standard form? You ever seen that before? If you want to find the x-intercept, cover up your y-intercept.

Y, divide by 4, you know this is going to cross the x-axis at 4 thirds. It's easier than graphing a fraction and going up and over, down and over. If you want to find the y-intercept, you cover up the x, divide by this number, positive 2. That's going to cross the y-axis at positive 3 halves. You can graph a line like that as well. So just a little refresher on this.

Hello show hands, how are you people doing? Okay, so far on our lines. You still alright? You awake still?

This should be review. This is review for you. I know some of you are thinking, where's the calculus?

Just wait for it. The calculus is going to come, whether you want to or not. Just hang on for a second. Enjoy the nice slow stuff, but really absorb this if you're a little rusty on it. What do you know about parallel lines?

That's actually the definition. Did you hear them over there? We have the same exact slope. That means we have parallel lines. It's kind of like climbing a mountain.

climbing stairs, right? The way stairs work is they're parallel. That means you're going up and over the same rate. Otherwise, on these stairs over here, if your stairs didn't go up and over, you're going to be like this. You're going to be like, oh, these are nice.

If they were different slopes, they intersect somewhere. We don't want that to happen for stairs. We don't want that to happen for parallel lines. So when we talk about parallel lines, what we're talking about are lines that have the same exact slope. Do perpendicular lines have the same slope?

No, no they don't. Actually, perpendicular lines meet at a very specific angle. What angle do perpendicular lines meet at? So if one line's like this, the other line's got to be like that, right?

It means if one slope's positive, the other slope's negative, so we know that's going to come into play. Also, there's a... It's not just a negative slope. That's going to be something like this, right?

That's not going to cross at 90 degrees. We want to make it actually just a little kicked over. How do I make it so it meets exactly 90 degrees? It's not only negative but also a? Negative reciprocal.

Very good. A couple of you said reciprocal. So perpendicular lines are lines which have, are lines where the slopes are negative reciprocals of each other. I was begs the question, if I give you an equation, can you find a line that's parallel and or perpendicular or perpendicular to a given equation? Can you guys do that?

Let's try that out real quick. That'll be our last little timid review problem. Hopefully these have been timid for you. We haven't even made it 30 minutes in. We're going to get to some tricky just a bit.

Yeah. So let me say that I want to find the equation of the line that passes through this point and parallel to this. Thank Find an equation passing through a set point in parallel to a given equation. What's the two things you need for sure in order to make the equation of a line?

Slope and what now? Do I have a point? Cool. Do I have a slope?

Do I have a slope right now? Not yet. I've got to work on it, but could you find that slope? Let's find the slope.

Go ahead and do that on your own real quick. Give me about five seconds. You guys should be pretty good at this.

Is the slope negative 2? No, what am I missing? That's the reason we talk about slope intercept, right? It's to find that slope very easily. Your slope's negative 2 thirds.

Now what I'm asking for is the equation of a line that's parallel to that line, but now goes through that point. So what slope am I going to use if I want to find the parallel line to this? Three halves. I'm going to use three halves?

But what the parallel to this? Same. Oh, parallel lines have the same slope. So what we're going to write down, we're going to find our slope.

We want the parallel slope. So we're going to write down the parallel slope. If it's parallel, it's going to be exactly the same. We know that our m is going to be negative 2 thirds. Now what point do we use?

What point do we use? Speak up. What point do we use? Does this 4 have anything to do with this problem, actually?

All we cared about was finding the slope. Once we have the slope and we've already identified a point, Wow, we can just plug that into our point slope formula to find the equation for our line. So we'll do y minus 7 equals negative 2 thirds x minus 6. Quick show of hands, how many people feel okay solving for slope? Keep your head down if you understand that the slope we're supposed to use is still negative two-thirds because we want to find a parallel. Do you see where the six and the seven are coming from?

Good deal. Can you work that out and make that slope intercept form? Okay, let's do that together.

I know that y minus seven, that's gonna stay there. On the right hand side, tell me what I'm gonna get. And then plus or minus, what do you think?

Plus. Plus how much? Four.

Good. Final step, if we add set four, because we got 2 3rds times six. 2 3rds times six, you can simplify the fractions, or we get 12 3rds, that's four. Add our seven to both sides. And we're done.

What if I change the problem, and instead of having parallel, I ask you to do perpendicular? Could you still do it? Let's talk about the only changes that would occur up here, okay?

I'm not going to change the whole problem, I just wanted to walk you through it here. If I'm talking about perpendicular, would this process change? Would this change? What's that going to become? Okay, so this would become 3 now, so...

Here now we're talking about perpendicular. With the 7 and the 6 change. No, we're just talking about slope.

What we're going to talk about this is parallel perpendicular only has to do with the slopes. This would change to our 3 halves. This would change to our 3 halves x.

It would be minus how much? Minus 9. And if we added 7 to both sides here. We get 3 halves x minus 2. So we find both the parallel and the perpendicular slope.

Do you feel okay about our basic lines so far? Would you like to learn a little bit about angles of inclination and use a little bit of trigonometry here? And you're like, no, not really.

Too bad. We can do it anyway. Let's talk about angles of inclination.

We're going to use it at some point. Are there any questions before I erase any of this stuff? Are you sure?

Are we having fun yet? Wouldn't you rather be in here than out there in the rain? Some of you are like, no, I'd rather be in the rain. Honestly, right now, I prefer that.

Just lie to me. Just lie and say, yes, this is awesome, Leonard. I love this class. You're the best.

Thanks, guys. Appreciate it. If you did that, I would just watch this video over and over again and have you go, Thanks Leonard, you're the best. Yeah. Thanks Leonard, you're the best.

Yeah. Thanks Leonard, you're the best. Yeah. See, I just have a running move. I do it with my workout.

Thanks Leonard, you're the best. Bam. Thanks.

Bam. So cool. I know, I'm a dork. Oh my, angle of inclination. Let's talk about how we can use angle of inclination and relate it to a slope.

We're going to start with some line. Random life. When I talk about the angle of inclination, what we talk about is the angle that any line makes with the x-axis.

So the angle of inclination, these actually would be the same angle. If you've taken geometry, you know that those are the same. We're really just talking about this one now.

We want to find some way to represent this line as having that angle. If we think about the x and the y-axis, notice that this we can represent as a change in x. You guys have seen that terminology before, the delta x change in x?

And this would be, well, the change in y. Now think back to your trig days. This is just basic trigonometry. Is there a trig function that relates this angle and these two specific sides?

Remember this would be a 90 degree angle. What does that? This is a triangle, which it is.

What's this side called? Hypotenuse. That's hypotenuse.

What's this one? According to this angle. That's the... Good. And this one is the...

Which one relates adjacent and opposite? This one. Not sine, sine would be opposite over hypotenuse. Tangent does.

Or cotangent, we don't deal with cotangent. Cotangent's adjacent over opposite. We want to deal with probably the easy one, tangent. If we talk about tangent, the tangent of that angle... Is equal to the opposite over the adjacent.

Nod your head if you're okay with the tan, opposite over adjacent. If you're not, you're definitely going to want to review your trigonometry before attempting this class. We deal with a lot of trigonometry in here. But here's the cool thing.

What do we already define as change in y over change in x? Or rise over run? What do we already define that as? So then we have this relationship. We know that tan theta, well, that's delta y over delta x.

But this is also the same thing as slope or m. If you bring all this together, Slope's equal to tan theta. Do you see the relationship between your slope and your angle of inclination? It's the same as the tangent of that angle of inclination.

Because tan is defined as opposite over adjacent and so is slope, we can make that jump. What's kind of cool is it says that if you know the angle, well you can find the slope. Getcha?

If you know the angle, you can do that in a calculator, you'll be able to find the slope. If you know the slope, you can find the angle. Those things are intertwined. They're equal to each other. Along that tangent.

So let's, shall we try one? Would you like to see an example of how this is done? Would you? Yeah.

You're in the most mellow class I've ever had. Would you like to see one? Bring it on.

Let's say that your angle is 30 degrees. Someone quickly, 30 degrees as radians is what? 5 or 6. Good.

The same thing as 5 or 6. Yeah. I want you to find the slope of the line that has an angle of inclination of 30 degrees or pi over 6. Here's how you do it. We know for a fact that m equals tan theta.

Don't forget that. That's your equation. That's what you do now. You know that.

that the slope is equal to the tangent of that angle. What's our angle? What's our angle? 30 degrees.

30 degrees. So if I plug in my 30 degrees, or I plug in my pi over six at the same angle, okay, if I plug that in there, then I know the slope I'm looking for is equal to the tangent of pi over six or 30 degrees, wherever you want to work in. I like pi over six. Now, all those with your unit circle tattooed on your right arm, okay, look down there. Can you tell me what tangent of pi over 6 is?

Can you tell me what tangent? Remember, tangent, you have to define a sine over cosine, right? So you define sine of 30 degrees or sine of pi over 6. Put it over cosine, or have it memorized.

Put it over cosine of pi over 6 or 30 degrees. Look at you in a circle if you have one. You should have one.

You need one. Honestly, you should tattoo it on your forehead backwards. That way you look in the mirror and you can memorize it. It's a good idea.

I haven't done it personally, but I'm waiting for someone to actually do that. You're pretty classy. Tangent of pi over 6 is sine over cosine. The sine of pi over 6, 1 half.

Cosine of pi over 6, root 3 over 2. Correct me if I'm wrong, but I think those are right. I'm doing this. on the spot. Good, alright. Can you simplify that a little bit?

Yeah, those twos are actually going to cross out. You know, divide those fractions, complex fractions, you're going to flip that, multiply. The twos are going to be gone.

What you're going to end up with is 1 over the square root of 3, or if you rationalize the denominator, you know how to rationalize denominators, right? Multiply by root 3 over root 3, you're going to get root 3 over 3. Strangely enough, that's your slope right there. Your slope is root 3 over 3. Alright, let me recap. Did you get that? That was funny.

Let me recap. Come on. Stand up next time.

Maybe you'll get it. I honestly will recap though. Here's what you do to find the slope if you have the angle of inclination.

You take your angle you plug it in and figure out the tangent of that angle. That's honestly it. I mean, if you can find the tangent of pi over 6, you have your slope.

That is your slope, all right? Now, can you go backwards? That's going to be another question for us.

What if I said I have now, I have a slope of negative 1. Can I find the angle of inclination? We only have one equation. The only thing that we know is that the slope equals tan theta. Over here we knew the theta, right?

We were looking for the slope, the m. Over here we know which one? We know the M.

So notice we're using the same exact equation here. We knew our angle. You could have put 30 degrees here very easily and done the same exact thing. Here we know our M. We know negative 1 equals tan theta.

How do we find theta? You can do tan inverse, sure, of both sides. Take it to the left.

We do tan inverse of negative 1 equals theta. If you do tan inverse of both sides, it would look like this. You'd have tan inverse. You have tan inverse.

Tan inverse of tangent, gone. You have theta. Tan inverse of negative 1, that's what we have right here.

This is what the question asks you, ok? Here's this in plain English. It says, I want you to tell me the angle so that when I take tangent of it, it's going to give me the angle.

Negative 1. That's what tan inverse says. It's kind of a backwards way of looking at it. It's saying find me the angle.

That way when I take the tangent of it, it's going to give me the value of negative 1. Do you understand the question there? It basically goes down to look at the unit circle. Find out where sine and cosine are the same but have different signs because you know tangent is sine over cosine, right?

So you look at your unit circle. Find out where it's, what do we have? It gets 2 over 1. I don't remember the exact one.

Find out where you have sine and cosine exactly the same, off by sine. That will give you negative 1. Have you found it? Do you have your unit circle out? Yeah. 3 pi over 4. That's exactly right.

Oh yeah, I have it right here. So this happens where your sine and your cosine are the same, but off by a sine. That's root 2 over 2 over root 2 over 2. That's where that happens on your unit circle. So check that out.

later if you don't have your union circle handy, you're going to have that same exact value only this one's going to be negative. That happens twice actually. If you're confused, it's, well, wait a second, don't I get two of these same values?

Yeah, you do, but think about what you're actually doing. You're trying to find the slope of a line. It goes like that, right? It's gonna cross two quadrants. It's gonna have a slope here and there.

One's gonna be positive, one's gonna be a negative way of looking at an angle. They're all the same, though. Three pi, what'd you say, three pi over four?

Yeah. Three pi over four, or the negative version of that, used as a reference angle. So here we go. We got that negative one, so we know that this happened when theta was equal to three pi over four.

Or if you want to translate that, that's 135 degrees. So either way we look at this, we can find slopes from angles, we can find angles from slopes. I'd say that this one's probably a little bit more tricksome for you because you're going to actually have to do a little bit of work on it.

This you could probably just plug in a calculator if you memorize like the.877. thing, you know, that's like 3 over 2, yay. Or whatever you, does anyone want to be, I memorized those, I don't want to look them up.

But if you do this on a calculator, it'll actually work out for you. It probably won't give you the square root of 3, but you can figure that out. Here, You're going to have to use a unit circle.

If they give you a slope, you're going to have to find the angle to which sine over cosine makes that value. It's going to be something, you might have two of them. It's not going to matter, that's the same exact value of the angle. Choose either one of them, it'll work out the same. But find the angle to which you're getting that value.

Do you guys see the process here? I know I went very quickly through this. Do you see the process?

Yeah or no? How many of you feel okay with this? Does it take a little bit of work to get handy on that?

Yeah, probably. Probably a little bit. bit more work to do. Let's see one more thing I want to go over. Let's talk about the distance formula real quick.

It'll wrap up our very first section here. Are you sure there's no questions on the angle of inclination anymore? We've got time. So if I give you this on a test and I say I want you to find the slope if the angle is pi over 3, you do it? Could you do it if you had a unit circle?

Yes. Oh, okay. That's a good question. If I say the slope is 1 half, can you find tangent where the angle would give you 1 half? Could you do it with a unit circle?

Yeah. Okay. Practice that stuff.

That's what I'm looking for. I don't care. Sure. To be honest with you, this is very much review stuff. What I'm trying to get you to get back familiar with, the tangent, sine, cosine, secant, cosecant, cotangent ideas, because we're going to move way past this.

We're going to be using this within some problems, all right? So this isn't going to be a problem. This is going to be within some problems. It's kind of like factoring isn't everything in algebra. You just use factoring in everything in algebra.

You get the analogy? So that's kind of what we're doing here. We're going to use a lot of trigonometry in this class. We're going to be doing things called derivatives and integrals, and they're all going to involve trig functions. So you've got to know the trig pretty well to be successful.

People say that you go to calculus to finally fail algebra and trigonometry. You made it this far, but it's not the calculus that's going to hold you back. I promise. It's going to be your algebra, and it's going to be your trigonometry.

I guarantee it. The calculus is actually quite easy. It's those concepts put together with calculus that makes it kind of hard. So if you're good at algebra and trig, you can. Absolutely fine.

Stick with this class. Okay, shall we? Distance formula in about a minute and a half, then we'll call it a day. Let's do distance formula. We're going to do it the same way that we did our slope formula, which is we're going to pick two random points, X1, Y1, and X2, Y2.

Only this time we're going to find the distance between them. If we have the x1, y1, x2, y2, well, we for sure know that's x1 and that's x2, and this is y1 and that's, well, that's y2. What we want to do now, though, is use something that relates this side.

This side, this side, and that side, and that's a right triangle. What relates to that? Pythagorean theorem, absolutely, you're right.

If we call this our distance, here's what we can say. We know that this length is x2 minus x1 by the same stuff that we... that we just did with slope formula. We know that this distance is y2 minus y1 by the same stuff we just used on our slope formula.

This is the length, this is the length from those corresponding points. We wanna find the distance. If we do Pythagorean theorem, we've got d squared equals, what's Pythagorean theorem say? Sine one squared plus sine two squared. Sure, yeah, a squared plus b squared equals c squared, some of you know that, or I prefer a leg squared plus a leg squared equals the hypotenuse squared.

because that tells you what you're doing, right? So if we take a leg squared, that's this, and a leg squared, that equals the hypotenuse squared. I already have that.

Here's my first leg. That's the distance squared, plus the second leg. That's the distance squared. Can you guys see the Pythagorean Theorem at work here?

Can you guys see the leg squared? Yeah? This is one leg, right? I'm just squaring it. Here's another leg.

I'm just squaring it. And that has to be, by Pythagorean Theorem, equal to the hypotenuse. squared.

So we got that. The only thing we need to do now, get rid of the square. How do I get rid of the square? Yeah, that just means the whole thing.

Now we are going to omit the plus minus because, well, we can't have a negative distance, it doesn't make any sense. So we're just going to have the square root of this entire thing, d equals. By the way, oh, here's a good question for you.

See where you're at. Will this square root get rid of this square and that square? What do you think?

Does it work that way across addition? This multiplication? Sure.

Addition? No way. No way no.

Can't do that. And that's our distance formula. You know what, I'm not going to do an example for you because it works really, really, really similarly to our slope formula.

Would you be able to, if I gave you two points, would you be able to find me an x1 and a y1? And an x2 and a y2? And plug them in. and not mess the signs up, right? Yeah.

You just square this value, you square that value, add them, of course it's going to be positive, right, because you're squaring something, squaring something and adding it. Just don't forget to take the square root. You can either leave it in terms of square root or approximate it and give me a decimal answer. How many people feel pretty good about what we talked about today? All right, that's good.

Transcript for:Introduction to Calculus and Lines

Transcript for:
Introduction to Calculus and Lines