The Calculus Connection - 1. Chapter 1

Wednesday Afternoon

Week Two, Fall Semester

It’s the week after the beginning of the first semester of my second year in college. The first week was hectic, with getting to know where my classes were, the perfunctory schedule changes and administrative requirements like getting a parking permit, paying lab fees and the like. Finally able to devote most of my time at school to my studies, I enter the library and head to my usual place along the wall where I can quietly concentrate and not be disturbed.

My name is Sean O’Donnell, I’m twenty-one years-old and pursuing a bachelor of science degree in structural engineering, which means having to take, among other things, a lot of math. I just opened my Calculus book when I hear, “Hey, I never expected to see you here.”

I turn to see that it’s one of the boys—a very cute boy—who had bagged my groceries a year or so ago at Smith’s Grocery Store. Quickly scanning his body, I see that he had changed little from the last time I’d seen him; maybe five-foot-ten-inches tall and somewhere between 130 and 140 pounds. “Hey,” l smile. “Tyler. Right?”

“Yeah, Tyler Stuart” he smiles. “It’s funny, we talked a lot when I worked at Smith’s but I never got your name.”

I smile, realizing that I finally have his last name; I’d only known his first name until now. Tyler had piqued my interest when he worked at Smith’s. He had so many nice qualities that I liked just about everything about him. “It’s Sean,” l offer, extending my hand to shake his hand. “Sean O’Donnell.

“So, what happened to you, Tyler? All of a sudden you were just gone. You never mentioned that you were quitting your job at Smith’s.”

“I got a job working at Subway just up the road,” he replies. “The pay was better and I got more hours.”

“Well, you can’t argue with more money,” I smile.

He glances to the table at my Calculus book. “So, you’re taking Calculus too?”

“Actually, it’s third semester Calculus,” I correct.

He laughs. “No, I meant that you’re taking Calculus also.”

“Oh, sorry,” I apologize. “I’m majoring in structural engineering, which means lots of math.”

“Really? Well, I want to own my own business, so I’m majoring in Business. They have me taking one semester of Calculus, but I’m not so good with math. It’s really difficult for me, and I’m just hanging on by a thread.”

Seeing this as my opportunity to get to know Tyler better, I’m quick to take advantage of it. “Well, maybe I could help you, explain the concepts and make it easier for you. You’re probably studying limits right now, which was difficult for me too.”

“Yeah, limits,” he smiles. “I just don’t get any of it, so I’m kind of doomed at the very beginning. I’m thinking about dropping it.”

“No, don’t do that, Tyler. If you want my help, then give me until mid-semester. If you’re still struggling, then you can drop it. You have until mid-semester to drop a class without having to take a grade.”

I see what looks like a smile of relief. “I could sure use the help,” he says. “But are you sure, with your studies and all?”

“Yeah,” I smile. “I’m sure.”

“We’d have to work around my job,” he says. “I now waiter at a restaurant twenty hours a week.”

“We can do that, Tyler. I have a job too, but I’m sure that we can find a time that will work for both of us.”

His face lights up into a smile. “Wow! Thanks, Sean! You don’t know what this means to me, because I’d hate to have to drop a class. Why don’t you give me your cell number and I’ll give you mine and we can arrange a time to hook up.”

Hook up? I sure like the sound of that, although it’s unlikely that he intended it that way.

Thursday Evening

I’m sitting on my couch, studying when my cell phone chimes. “Hello.”

“Hi, Sean. It’s Tyler Stuart.”

“Hey, Tyler. What’s up?” I smile, thinking about the double entendre of what I’d just said. Of course, he probably wouldn’t get it.

“I’m calling to arrange a time for when we can hook up… to help me with Calculus, you know?” There are the words ‘hook up’ again. And I think he realized it too when he hastily added the last part. “I’m free tomorrow evening after 7:00,” he continues. “That’s when I get off work. I realize it’s Friday, and if you already have plans, I’ll totally understand.”

Plans? Hey, I can’t think of anything that I’d rather do than spend a Friday evening with you, Tyler. “No plans,” I reply. “Except helping you with Calculus. Seven o’clock tomorrow evening is fine. Where do you want to meet?”

“Well, I guess the library would be okay since I have two roommates who will probably just be in the way if we do it here.” Do it… I smile when I hear those words. I’d love to ‘do it’ with you, Tyler. “They don’t take studying as seriously as I do,” he continues, pulling me out of my reverie. “Don’t get me wrong, they’re okay guys; it’s just… well, you know…”

I chuckle. “No, I wouldn’t know given that I live alone.”

“Lucky you!” he laughs.

“So, how about we do it here at my place where it’s quiet?” I suggest. “And my couch is much more comfortable than those hard chairs in the library.”

“You convinced me,” he laughs. “Where do you live?”

I tell him the name of my apartment complex, which is within walking distance of campus. He tells me that it isn’t far from where he lives and to expect him about 7:30 pm.

Friday Evening

It’s a little past 7:30 pm when I hear the expected knock on my door. Dressed in a pair of dark blue Nike running pants, a blue and yellow striped rugby shirt and white ankle socks, I answer the door to see a very smart- and preppy-looking Tyler Stuart wearing tan chinos, a pastel blue dress shirt with a medium gray tie and gray Sperry Top-Siders—perfection personified. After he toes off his shoes at the door, I usher him to the couch.

“I brought dinner,” he says, shrugging off his backpack. When he bends over to set it down on the coffee table in front of the couch, I focus my eyes on his butt, noticing how nicely his chinos accentuate every seductive curve, particularly the cleavage. Hm… the same incredibly sexy butt that attracted me the day we first met. Obviously, the last year or so has been quite kind to you, Tyler. Then he opens it and withdraws a thermal sack containing four large, generously stuffed tacos.

“Wow, that was really thoughtful, Tyler,” I smile, quickly raising my gaze to his eyes as he turns his head to look at me.

“Hey, it’s the least I can do, considering that you’re giving up your Friday evening to spend it with me, helping me with Calculus,” he replies. Hey, I’d love to spend a lot more than a Friday evening with you, Tyler, and in bed would be the perfect place to begin. “Besides, it didn’t cost me anything; the cook likes me and was happy to do it.”

It’s no wonder. Everything about you is sexy and pleasing to the eyes, Tyler. You’re everything that a guy like me could possibly hope for. “I have Coke and Sprite,” I offer. “Or if you’re staying for a while, I have a specialty seasonal beer: Samuel Adams Octoberfest. I assume that you drove here, so I don’t want you to leave with a buzz and get stopped, or worse.”

“Yeah, I drove, and yes, I’ll be here for a while, so I’ll have the beer,” he smiles.

When I return from the kitchen with two bottles of Octoberfest in one hand and two drinking steins in the other—how else do you serve a German-style ale—he had moved to the dining room and had the tacos laid out on the table on their wrappers. I set the beers and steins on the table and go back to the kitchen for a couple of plates. With the tacos properly on plates, we sit down at the table and pour our beers into our respective steins. “Um… good,” he smiles after sipping his beer. “I don’t drink much, since I’m underage, but I really like this…” He looks at the bottle. “Octoberfest.”

“I don’t drink much either, Tyler, which is why I consider this beer a treat. It’s more expensive than the run-of-the-mill beers like Budweiser.

“So, anyway,” I say, picking up one of my tacos, “I had planned to order pizza for tonight, assuming that you hadn’t eaten, but tacos are a heck of a lot more tasty—especially these tacos. Thanks, Tyler. Again, it was very thoughtful of you.”

“No problem,” he smiles before biting into his taco.

“So, you’re majoring in Business,” I say midway through our meal.

“Yeah,” he replies. “Like I said, I want to own my own business.”

“Have you decided what kind of business it’ll be?”

“Yeah, since my jobs have mostly involved food service, I’ve been thinking about buying a franchise, like Subway. Like I told you in the library the other day, I worked there for about a year after leaving my job at Smith’s. I just started waitering a couple of weeks ago and the tips are good. But anyway, if a franchise works out for me, I might buy another one, or I might consider getting into the real restaurant business.”

“So it sounds like you have your future mapped out pretty clearly,” I smile. “I’m impressed, Tyler. Not only are you ambitious, but you seem to have a good head on your shoulders; a young man with a solid future.”

“Yeah,” he smiles. “So, what does a structural engineer do?”

“He designs competent structures that won’t fail or collapse. It can be buildings, bridges, dams… Things like that. Basically, he works with the architect to assure that the design is both feasible and competent. After college, I’d like to get a job with mid-sized architectural firm.”

“Sounds like a lot of math,” he says.

“And physics, spec reading and computer-aided design,” I add. “Just to name a few.”

“Just the math alone would eliminate me,” he says before taking another bite of his taco, followed by another sip of beer.

+     +     +

“So, where are you right now?” I ask as Tyler pulls his Calculus text book out of his backpack. We’ve finished our meals and are now seated side-by-side on the couch.

“Here,” he replies opening the book to a dog-eared page and pointing to the chapter on limits.

“Okay, Tyler, limits can be confusing. A limit is a subtle concept but the very foundation of Calculus and what makes it work. It’s elusive because it’s not a common idea in everyday life. The closest analogy to a limit that I can think of is the Riddle of the Wall. You stand facing a wall. If you walk halfway to the wall, and then you walk half the remaining distance, and then you walk half of that, and on and on, you’ll get closer and closer to the wall but you’ll never actually reach it. Here the wall takes on the role of the limit. Essentially, when you get to derivatives, which is the instantaneous rate of change of, say distance over time which is velocity, you make the rate of change smaller and smaller until it’s as arbitrarily close to zero as you want. But it can never be zero because the function doesn’t exist at zero.” I draw a crude sketch of a parabola, then draw a secant line between two points on the curve, then mark a point of interest on the curve between them. “Okay, the slope of the secant line is the average rate of change between two points on this parabola.” Then I draw consecutive secants closer and closer to the point of interest on the curve until I reach the point and draw a line tangent to the curve there. “This is the line tangent to the curve at this point, which is the slope of the parabola at this point. The secants can approach as close as we want to the tangent, but they can never actually become the tangent because the change in y divided by the change in x of the secant would be zero divided by zero which is indeterminate and undefined. Here the tangent is the limit of the secants as their lengths approach zero, the actual slope of the curve at this point and the limit of taking the length of the secant closer and closer to zero. So you can get as close to the actual slope at this point on the curve, but you can never actually reach it with the secant line. You decide how close you want the secant to be to the tangent until you’re willing to just call it the tangent. Remember, the tangent is the limit, the value that you’re looking for.”

I give several examples of equations where the limit approaches a numerical value or approaches zero or approaches infinity. Then I give examples where the limit has different values when approaching a point on a curve from the left as opposed to approaching it from the right, which makes it a discontinuous function at the point and nondifferentiable.

He smiles. “Okay, I think I’m starting to get it, Sean.”

“So, what I did was take the secant line closer and closer to the curve, making it smaller and smaller until it’s length vanished at the tangent at this point on the curve, which is the line that touches the curve only once, the limit of what we’re looking for. A value doesn’t have to actually exist at that point; it just has to approach the same limit from both sides in the vicinity of the point. That’s how limits work. It’ll become clearer to you when you study the method of increments, the process used to find the derivative, which in this case is the slope of an equation, like this parabola, at any point you choose.

“Differential Calculus is the mathematics of rates of change, Tyler. A rate is a fraction, and the denominator of a fraction can never be zero. It can be as close to zero as we choose it to be, but it can never actually be zero.”

After that, we thumb through the pages of the chapter and I try to clarify the concepts with which he’s having difficulties. Then I venture into the method of increments for finding the derivative of a function and how the limit works as we take a small difference of the change in y divided by the change in x and then shrink it towards zero. “Wow. You make it so simple, Sean. You’d make a really awesome teacher,” he says when we finish. “I think I really have a better understanding of limits now and why we’re studying them. The instructor hasn’t done a good job explaining it like you have.”

“That’s the problem with too any math instructors, Tyler; they assume that you should already know what they’re teaching you. So why even take the class? Either that, or they have no connection with their students and why they’re taking the course. Not everyone taking math has the mind of a mathematician.”

“Yeah, right. Well, I’ve had about as much math as I can take tonight,” he says, sitting back on the couch with his hands on his knees.

“I hope it doesn’t mean that you’re going to leave now, Tyler. We could watch a movie. I have streaming Netflix on TV.”

“The problem is I have to be at work at ten in the morning, Sean. We open at eleven for lunch. We could do it tomorrow evening, though, if you don’t have other plans.”

Other plans? Absolutely not when they conflict with time spent with you, Tyler! “I don’t,” I smile. “So, then it’s a date?” Since he’d used the term ‘hook up’ twice with me, I feel comfortable referring to tomorrow evening as a date.

“It is,” he smiles. “I get off at 6:00 pm, and since I don’t have to work Sunday, I can sleep in on Sunday morning if it gets too late.”

“Well, if it gets to be that late, I would expect you to stay here rather than have to drive home when you’re tired, Tyler.” There, I put it out there. Who knows? He might actually take me up on it; though, I wouldn’t expect us to end up sleeping together if he does. Still…

And to make it sound even more like a date, I add, “Oh, and don’t bother eating before you come, because I’m planning to have supper ready by the time you get here. So, I’ll expect you anytime after 6:00 pm?”

“It’ll probably be closer to 7:00, but you don’t have to do that, Sean.”

“Oh, but I do,” I smile, getting up from the couch. “I pride myself on being a good host. So 7:00 o’clock tomorrow evening it is.” Tyler stuffs his Calculus textbook into his backpack, zips it closed and then gets up too. “Oh, by the way,” I add, deciding to risk it as I walk him to the door. In for a penny, in for a pound. “You look very dashing tonight, Tyler. Dressed to impress. I look sloppy compared to you.”

He opens the door and smiles. “It’s how I have to dress for work. Usually I dress down after work, but since you’re so impressed by the look, I might just come dressed like this tomorrow too.”

“You should,” I smile. “It makes you look dashing and smart.” Cute is the word that I really want to say, but that would be revealing way too much so early in our friendship.