The Script of My High School Calculus Class Final Project

The following is exactly what the chapter title says it is. I actually had to voice this and edit it together, and I had voiced both characters myself, chopping it together in audacity. I pulled an all-nighter and did all the night right before it was due too. I had the video in a USB that I used to have the video shown in class, but I think the USB got sent to the dry cleaners in my suit that day... I'll upload the video onto my YouTube if I ever find it, but for now, here's the script:

Carrie: Hello class, so yesterday, we went over how we can apply integrals to calculate displacement and distance traveled, and when you begin to think of them as objects moving at different rates, that’s very important, rates over time, it begins to make a lot of sense how and why we can use integrals this way. Now today, we’re going to dive a bit deeper into that graphically.

Cordon: Oh hey, Carrie, the guys were wondering if you’re free to go street racing again tonight.

Carrie: Oh hey, Cordon. Sorry, but I’m busy recording some video lectures for my class. Would you like to join us?

Cordon: Oh, nah, I’m good. I was gonna go join them actua-

Carrie: Here, why don’t you come try this problem?

Cordon: Uh, sigh, alright.

Carrie: Here, try to find the area of this shape. (y=-½x^4+x^2+3 and y=0) [11.021]

Cordon: Well, uh, I don’t know. I never learned the formula for this shape. I mean, those are kind of semicircles? So… area of a circle… half it, wait no, there’s two.

And then times two and add this rectangle maybe? Height is 3… times 4… well, more like 3.5… And when I add it all together, that’d come out to… approximately 11?

Carrie: Haha, you’re so silly, you imbecile. You absolute buffoon of a troglodyte. That’s not how you find the area of this shape! That wasn’t even close to a Riemann approximation! My brother everyone, isn’t he funny?

Cordon: Ugh, c-come on. You know I never took calculus.

Carrie: And you never will with that attitude. Try to be a little more enthusiastic!

Cordon: Ugh, fine. Well, how do you find the area of this shape, Carrie?

Carrie: I’m glad you asked, Cordon! Let’s break this down a bit and look at some simpler shapes. Here, can you tell me the area of this square?

Cordon: That’s easy. That’s just base times height.

Carrie: Right! Or more precisely, you take a side and square it, but I didn’t expect you to have the cephalic capacity to synthesize that formula when given the information that the shape is a square as that would require you to know some basic geometry. Your answer isn’t wrong.

Cordon: I was better at math than you in middle school! I tutored you in geometry!

Carrie: And yet I still had a higher grade than you because you were lazy, disorganized, and had horrible learning habits.

Cordon: Oh by Juno’s divine wrath, I swear I’m going to punt you accelerating so fast through the stratosphere one of these days, they’re gonna discover a new hole in the ozone layer, you pedantic little possum.

Carrie: Clears throat. We will get to acceleration very fast, but please stick with me here. Now, while we do understand rectangles such as squares as a base multiplied by height…

Cordon: Bruh.

Carrie: Let’s try to find the area of this square through an integral.

Cordon: I don’t, I don’t know what that-

Carrie: Do you know what an integral is, Cordon?

Cordon: I just- um, sigh, is it the “area under the curve?” That’s what I always hear.

Carrie: Good job! So you’re not completely a hopeless moron!

Cordon: You know, I don’t have to take this. I can walk away at any-

Carrie: So in the case of a square, the “curve” would be the line y=5, and because we are only looking at the area under y=5 from 0 to 5 as it is a square, after all, we will set up our integral as the integral of 5 from 0 to 5. Take the antiderivative and plug in the numbers… 5x becomes 25. 0x is 0. Subtract and we have 25, the area of the square which we would get from multiplying 5 by 5.

Cordon: Oh, I think I get it! So by taking the integral of something, we are essentially turning the “curve” into the height and multiplying it by the length of the base, which is the difference between the upper and lower bounds.

Carrie: Not exactly! This is what you get for making stupid assumptions, you stupid stupid excuse of a human being. Let’s dissect our example a bit more shall we? The word Integral more so means sum than it means product. We are adding together small segmented areas together, in a way, like with Riemann’s sum.

Cordon: That’s what I tried to do earlier!

Carrie: Haha, not even close. Let’s take our function earlier, y=5, and turn it into f(x)=5. Earlier, when we took the antiderivative of f(x), we got, let’s call it g(x), g(x)=5x+C, though because we are integrating, the C cancels out from the subtraction, so let’s ignore it for now. Now tell me, Cordon, what does g(5)-g(0) mean in the context of f(x)?

Cordon: Hmm, well, if g(x) is the antiderivative of f(x), then that means that f(x) is the derivative of g(x). In that sense, g(5)-g(0) gives us the difference, or in other words, change in the y, or in other words, change in height of the function g(x), over our bounded domain of 0 to 5, or in other words, our base. Now to find the area under the curve of f(x), we need to know the different heights of the curve at every point across the length of our base, so that when we do base times height, we are multiplying the base by all the different heights. If we consider that every point in f(x) corresponds to a change in y over the change in x of g(x), by then finding g(5)-g(0), we are essentially finding the sum of all the various heights as they change from the length of our base of 0 to 5, giving us the area under the curve.

Carrie: Uh, erm, wow

Cordon: Disorganized and lazy. Your words.

Carrie: Well, it seems you’ve got a great understanding of integrals now, Cordon. Here, why don’t we go back to our original question then? (y=-½x^4+x^2+3 and y=0) [11.021] {Area under the curve}

Cordon: I, uh

Carrie: You get it now, right?

Cordon: We’re not doing this.

Carrie: Doing what?

Cordon: You know I can’t solve this. What’s the point? Just to humiliate me?

Carrie: Cordon, I would never! You just seemed like you could use a challenge! I suppose I’ll solve it for you then, so watch closely.

Cordon: I literally absorbed none of that. You know I don’t have the foundations to! You can’t just mouth sounds and pretend that it’s a sentence, Carrie!

Carrie: You know what? Why don’t we move on to how integrals apply to velocity, acceleration, and distance traveled and try some different problems that you might find a bit more applicable in your own life, Cordon? Suppose that, um, Gordon is driving…

Cordon: Really?

Carrie: Yep, he’s uh, he’s driving his usual route, starting from his house to the other end of the city. His velocity in miles per hour can be described by the function v(t)=1/2x^2-5x+10. Tell me his displacement, how far he is from his starting point, at time equals 5.

Cordon: Hm, okay. So, do I just… plug in 5?

Carrie: Of course not, you sack of scum! That would just give you his velocity at time equals 5.

Cordon: Okay…? Care to explain this one then?

Carrie: Gladly! Think about it this way. Time is our domain, so we can think of it as our base in our base times height. In this case, our base is 5 again, so now what are our different heights?

Cordon: Ah, that would be our various velocities! The different rates of change of position at different times. So integrate v(t) from time equals 0 to times equals 5… 8.33!

Carrie: There you go! Doesn’t this feel good?

Cordon: Eh, street racing feels better still.

Carrie: We’ll see about that. Why don’t you try to find the total distance traveled in that time then?

Cordon: The… The total distance traveled? I-is that not the same?

Carrie: Of course, you wouldn’t know the difference! Our velocity isn’t always positive! That means, at times, Gordon is driving back closer to his starting point, so when you found displacement, it accounted for that. I’m asking you to find how much distance in total he traveled.

Cordon: Uhhh, how do I know at what point he started driving backward and how far he went?

Carrie: Think about it this way, Cordon. When Gordon starts moving backward, the position curve would reach a peak. It would stop increasing and start decreasing, causing a maximum. So just find where there is a maximum from time equals 0 to 5.

Cordon: Yeah, I see, uh, one problem… how do I find that?

Carrie: Laughs You’re so cute when you admit how dumb and uneducated you are. If velocity describes the rate at which the position changes, then the moment at which the velocity changes from positive to negative describes the moment the position stops increasing and begins decreasing. Watch this, let’s find the zeroes of our velocity.

Now that we have 2.764 and 7.236, we can discard 7.236 considering it’s outside of our domain. Now, we’ll check both sides of 2.764 at time equals 2 and time equals 3. We get 2 and -0.5, so our value changes from positive to negative. That’s no good! So we’re going to draw a frowny face.

Cordon: Oh, a frowny face, great, what, so am I like a toddler now?

Carrie: Because a frowny face resembles a maximum, we know that there’s a maximum there. Now Cordon, explain back to me how we know there’s a maximum at 2.764.

Cordon: Well, our velocity changes to negative at that point, indicating a change from increasing position to-

Carrie: Frowny face.

Cordon: Uh, do I have to? I can’t just visualize a position graph going up and then-

Carrie: Use. The. Frowny. Face.

Cordon: Going from positive to negative is bad. Because negatives are inherently perceived as bad within our society. So that makes a frowny face, which looks like a maximum.

Carrie: Good! Now that we know where our velocity is negative, we can find our total distance traveled. We split up the integral from 0 to 2.764 and from 2.764 to 5, find them separately, and then add them together. That gives us 15.787. Are you getting the hang of this yet?

Cordon: Uh, sure. Yeah, sure, whatever. I’m getting pretty sick of this… Can I go-

Carrie: Gordon is driving again. He’s driving under the influence and going at a velocity of 97 miles an hour in a school zone on Wednesday, March 11th of 2015 at 8:17 a.m.

Cordon: Hold on…

Carrie: In Brookfield Wisconsin by Fairview South Elementary on Bermuda Boulevard where elementary kids cross the road.

Cordon: You just doxxed us. Why are you bringing this up?

Carrie: Suppose that t equals the seconds after 8:17 am.

Cordon: IT WAS AN ACCIDENT

Carrie: WAS DAD ALSO JUST AN “ACCIDENT” THEN?

Silence.

Carrie: It was about 13 seconds after 8:17 when the “accident” happened. Gordon had been accelerating at a rate of 0.0021 miles per second squared for those 13 seconds. Tell me, Cordon, what’s Gordon’s displacement at the 13th second?

Cordon: Displacement, right?

Carrie: Mhm

Cordon: Well, since Gordon only went in one direction for those 13 seconds, it’s pretty much the same as the total distance traveled in those 13 seconds, which is just the area under the curve… which we’ll have to derive from our acceleration. We know the initial velocity is 97 miles per hour which we’ll have to convert to seconds, and with an acceleration of 0.0021 miles per second, take the antiderivative, we’re looking at a velocity curve of v(t)=0.0021t+(97/3600). Then we take the integral of that from 0 to 13, which should come out to 0.528, the varying heights being the different velocity rates over the length of our base, second 0 to second 13.

Carrie: Good. Bonus question, Gordon, I mean, Cordon, and this should be easy. At exactly what acceleration did you drive that 1.76-ton Nissan 300ZX into those children?

Cordon: Oh. Yeah? Yeah, you want the acceleration at impact? Sure… sure! You want me to find the force too? Do a little bit of physics for you, tell you the force I hit those kids with too?

Carrie: Your words.

Cordon: Deadly, okay? It was with deadly acceleration and force. Is that what you wanted to hear? I didn’t need a good STEM education to tell you that.

Carrie: It was a bloodbath…

Cordon: And yet you didn’t stop me…

Carrie: What!? How was that on me?

Cordon: You were in the car as well… You could’ve stopped me at any point, and yet, you were on board with it the whole time

Carrie: Don’t act like it was my responsibility to save those kids from your poor decision-making skills! I’m still traumatized. I’m the victim here!

Cordon: Yeah? Is that so, Ms. Victim, all high and mighty on your noble pedestal? Picking up the pieces of your life, turning it all around. Too good for us now, huh? No… you don’t actually hold any disdain for me… you hate what I remind you of. An old you that you can only delude yourself into shaking away…

Carrie: What? No, you literally drove into an assembly line of children while I was in the car, Cordon. I despise you in particular.

Cordon: Now you’re just nitpicking at the irrelevant details. The point is, you’re not yourself anymore.

Carrie: No! I’m not the me that you want me to be anymore. I like who I am now!

Cordon: Do you? Do you actually enjoy math to the extent that it’s become your entire life now? It’s all you ever talk about! Do you like who you are now, or do you just like having structure? Some sort of framework to stick yourself into. You did always crave identity and validation… but you know, you used to hate everything related to academics. Remember that? Remember when we would drive by college towns on rainy days and through puddles, soaking all those nerds and their nerd books in dirty rainwater? Do you think we haven’t noticed? You… sigh… You’re just who Dad wants you to be now!

Carrie: W-what!? And, so what? So what if I am? He would be proud! Of me! Don’t tell me who I am or what I used to be. Life makes sense for me now! I’m concerned for you, Cordon! You can’t just keep racing through life like you are now!

Cordon: No, no! Who cares if he would be proud of you now or not?! In what world should that mean anything to you?

Carrie: In this world, Cordon, and it should mean a lot to you too! None of that would have happened had we just listened to him a bit sooner. He just wanted the best for us.

Cordon: Oh my Nascar, listen to yourself! It’s like you can’t come to terms with reality, and you’re trying so hard to make sense of it all in your head, clinging on to the idea that maybe you could… I don’t even know, keep him alive in spirit? That if you just started doing all this, new, weird, being a responsible-and-productive-member-of- society stuff you’ve got going on now that then you could rationalize all this trauma or something. This isn’t you! This whole, finishing your education thing. It’s absolutely insane, and I’m tired of it. We’re all tired of it. You need to let go. It’s like, I don’t even recognize you anymore.

Carrie: I- I don’t…But, no… I can’t… I have to do this.

Cordon: Do you? Do you have to do this? His expectations aren’t your obligations. No one. No one is expecting anything from you. Anything other than, maybe, I don’t know, than to live… You don’t even have to do that if you really don’t want to, but… just… live for a while, damn it.

Carrie: I… I am living, Cordon. Life is going.

Cordon: Really? Like this? Just going through the motions. You don’t actually enjoy this, do you?

Carrie: It’s not about that. It’s not about whether or not I like or enjoy this. I never have been. I need this. I- I don’t know what I’d be without this. What I’d do without this. How I’d live without this.

Cordon: I don’t know… maybe like how you’d always lived? Like how we’ve always lived? Before all this crazy stuff happened, this whole, starting your career thing. You never needed this before. You don’t need it now.

Carrie: C-can I really go back to that?

Cordon: Of course, you can… We could start with doing laps in the Costco parking lot.

Carrie: I’d like that.

Cordon: Let’s go be a menace to society. The guys are still waiting for us. We’ll always wait.