Thursday morning, Geneviève's school day begins with a session of single-variable calculus that starts with the morning announcement from the principal.
"Good morning: today we finally bear good news from the extracurricular teams! The school won the state championship in the first round of the Square Root of the Answer math contest and is therefore automatically qualified for the national final! Congratulations to..." Glen, the principal, blares in the PA system.
"Éliane and Geneviève; the other two participants are in the other section but their contributions must still be acknowledged" Trent, the single-variable calculus instructor, highlights in front of the class.
"We might have won the state championship, our state had very little competition" Éliane points out.
"I will make a little sidebar on the Square Root of the Answer contest and briefly talk about the national final, which will be held in late April, and also of its content. The competition's final is about multi-variable calculus and, to qualify, there are two possibilities: either we win the state championship, or we reach a certain minimum score. In both cases, it's based on the top four on a team. If you want to participate next year, I strongly recommend reviewing the material of this course as well as study some multi-variable calculus notions" Trent continues taking.
"Finally, something positive coming from the extracurricular activities!" Lugh, another student, exclaims. "This year has been more than disappointing for sports..."
"If you are interested in participating next year, or simply to learn more about multi-variable differential and integral calculus, do not hesitate to come to the mathletics team's practice this Friday night!" Trent invites his students.
While most students in the group have barely started learning the concept of a definite integral, Éliane and Geneviève are both wondering how this concept translates to several variables, and everything they know about differential calculus as well. They do not suspect the content of the prize that comes with their victory at the Square Root of the Answer contest. No more than some students that aren't part of the team are now driven by a desire to obtain better grades in this course than they do. As if the desire to obtain advanced standing in college wasn't enough!
Speaking of getting advanced standing in college, the main instrument that allows students to do so depends on the obtention of an AP Exam of a given subject, in general 3 and higher, although certain institutions will demand a 4, and maybe even a 5. If this allows to lower the total tuition bill, so be it. As for differential and integral calculus, the BC version opens doors that the AB version will not open, for the only reason that the BC version will cover more material than the AB version, such as improper integrals and Taylor series.
As for the school's level of athletic under-performance, let's admit that the football season finished with a single win, and that all other sports teams struggle mightily. It does not make someone want to follow a sports team unless one is a friend of the athlete, or a relative.
"A definite integral corresponds to the area under the curve of a function between two bounds. If you calculate, for a given partition of a function f over an interval [a,b], a sum of f of x-i multiplied by their delta x-i, or the size of the partition, you will then calculate an approximation" Trent explains to his students.
"Sir, how do we get from this in one variable to several variables?" Éliane asks.
"Each variable will then have its own independent partition. If, in one variable, the definition of a definite integral is simply the limit of max delta x-i going to zero of the sum of f of x-i times delta x-i, in several variables we would then be talking about the limit of max delta x-i, max delta y-j, max delta z-k, all of which independently go to zero of the multiple sums of f of x-i, y-j, z-k times delta x-i, delta y-j, delta z-k"
For a multi-variable function, we will not be taking about indefinite integrals as we would in one variable; the integration bounds can also be functions (that do not contain the variable to integrate over) and the order in which we integrate the variables over may affect the formulation of the bounds.
"She did it, Éliane has just ruined everything" Shane grumbles. "As if Geneviève wasn't obsessed enough as it was... will that be on the test?"
"No but it could be useful if you intend to major in disciplines where multi-variable calculus becomes important. In addition, the Riemann sums form a basic method to calculate integrals numerically if we are dealing with a function that does not have a closed form integral"
"When we calculate Riemann sums, the choice of where we'll take our f of x-i, y-j and z-k will be important, but, for a number I(a,b) to exist, certain conditions must be met on our function or the limits on their Riemann sums. Other than it must be continuous over the domain, what has changed in several variables?" Geneviève asks in turn.
"Conceptually, not much: just that your partition is no longer just a line but a rectangle, or a parallelepiped. You may then, according to Fubini's theorem, calculate a multiple definite integral in any order, so long as your function is continuous"
"Multiple integral? We barely learned what does a definite integral represents in one variable!" Lugh asks.
"So long as the function is continuous, you integrate one variable at a time, and you use the result of the integral with respect to the first variable to calculate the second integral and you do so until you have integrated over all variables"
I feel that the rest of the year will be long in single-variable calculus; they just won't stop bugging us with multi-variable material! All that they will accomplish will be to befuddle us! Shane thinks while he realizes that Trent distributes special problems to people so that they can better understand the computation of multiple integrals, but without touching the Jacobians, at least for now.
Calculating multiple integrals is perfectly fine, but the final of the Square Root of the Answer contest contains derivatives of multi-variable functions, as is the case for every multi-variable calculus course I can imagine! For the first round, we needed to double down, more than double down, Éliane and I, after the AMC12 to learn the material of the second half of the year on time for the competition! Geneviève ponders, while she is questioning herself on her roadmap for the rest of the season. Which one to prioritize between the AIME and the final stage of the Square Root of the Answer? When we just got started learning definite integrals, I may as well ask myself how it will happen when calculating derivatives of multi-variable functions: last week, we began touching upon the basics of differential equations as an application of indefinite integrals, let alone doing this in several variables... On the one hand, I do not expect a whole lot out of the AIME, but on the other hand, I know it's much more widely recognized than the Square Root of the Answer. However, I believe it's realistic that we obtain a good result on the SRA that will make everyone proud here; If I must double down again, or triple down even, to make sure it happens... I know exactly what learning the entirety of a multi-variable calculus course in two months will look like, but I have no desire to get my feet wet until the end of the AIME next week! She loses herself in her thoughts while there are two students qualified for the AIME on the school's team: Éliane and Geneviève. The other two participants at the SRA narrowly missed their AIME qualification, having scored a total of 99 and 97.5 points on the AMC12 respectively.
"Éliane, how do you feel about the AIME next week?" Gen asks Éliane.
"At this point, what matters is simply to participate. Remember when the results from the AMC-twelve were released..." Éliane then answers.
"The administration didn't make such a big deal like they did on the SRA today. Especially not when the AMC-twelve results arrived in the middle of the mid-year exam season" Gen continues.
Historically the institution found a way to qualify a student or two on the AIME on any given year. If a freshman or sophomore competitor desires to qualify for this competition, which is the second round of a chain of competitions that leads to the International Mathematics Olympiad, he or she may enter the AMC10, everyone else must enter the AMC12. The qualification thresholds on the AIME coming from the AMC10 are the lowest score between the top 2.5% or 120 points. For the AMC12, one would then talk about the lower of the top 5% or 100 points. Please note that both competitions are tests comprising 25 multiple-choice questions, for which a correct answer is worth 6 points and a blank response is worth 1.5 point, therefore the maximum score on either competition is 150 points.
"One last thing that will make your life a lot easier when calculating definite integrals: the fundamental theorem of calculus! Let f be a continuous function over the interval ]a,b[..."
"Why ]a,b[? Do we need the function f to be continuous at the endpoints?" Lugh asks his teacher.
"In fact, no. The only discontinuities allowed are at the endpoints of an integration interval; thus, if we want to calculate an integral of a function that's piecewise continuous, we can simply calculate the integrals on each interval over which the function is continuous, and the global integral is the sum of the partial integrals. Let's return to the subject at hand and, if there exists a function g(x) such as f(x) = g'(x) over the interval ]a,b[, we then have"
Let's return to the subject at hand and, if there exists a function g(x) such as f(x) = g'(x) over the interval ]a,b[, we then have" [https://img.wattpad.com/57c7dc245649e2624be83ee91db04b796c96e5a3/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f6a717952626136652d54465a31413d3d2d313238383633383734392e313732396362653964383530346166383435303135363935303134352e706e67?s=fit&w=1280&h=1280]
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And then Trent alluded to Chasles' relationship without naming it... but we also have, following the definition of the definite integral, the property of linearity and, when the integration bounds are reversed, the integral is thus multiplied by -1.
-------------------------
When Friday afternoon comes, no one, not even the people who took part in the first round of the SRA, would suspect that there would be a crowd for the ultimate meeting of the team held prior to the AIME. Under normal circumstances, there is no one that wants to attend the team practices without being on the team at this point of the school year; there are way too many people that desire to meet with their champions for the usual room of the team.
"What is the meaning of this?" Trent asks, astonished by looking at the size of the crowd for the session.
"Geneviève, I owe you some apologies. I should have said that I didn't take pre-calculus at the same time as you. Can you explain to me how student loans involve the material of pre-calculus?" Dylan asks, while there are several others among the crowd that also have questions pertaining to the mathematics of bank loans.
"Do you want to be my valentine? If you accept, you must also accept to answer every mathematical question that I could be asking you on Valentine's Day!" Lugh asks his calculus companion.
"Valentine's Day is still in two weeks, I believe it's a little too early to go around begging for valentines! Even though the basic formula of bank loans actually comes from the calculation of annuities, it says that the principal is equal to the payment times a formula that contains an exponential in the numerator and the interest rate in the denominator. It's really the exponential term in the numerator that makes student loans a pre-calculus problem" Gen explains to Dylan.
"When I took pre-calculus never did we talk about student loans as an application of the material; I loved your article in the student newspaper about student loans as well as its follow-up on federal loan forgiveness programs; as a graduation gift, may I obtain a date with you?" A senior admirer asks her who, unlike the other admirers that came there in great numbers, seemed to prefer her chronicles about personal finance in the student newspaper.
"Because of high attendance, the math club meeting will be held in the auditorium" Trent announces to the crowd.
"Never would I have believed the school capable of winning a state championship even in a competition like the SRA! Seeing so many people here for their brand new darlings... even in my wildest dreams, I imagined that, as much that, to the campus' eyes, you are very special, in these math contests that you hold so dearly, everyone is at your level" Marcia points out, taken aback by the newfound admiration of the student body for its mathletics team.
The social situation has completely changed, and I am not sure that people will remain for the rest of the session. Or then they would be present not so much for the material but for us. I have a basic idea of what it means to calculate multiple integrals, but calculating derivatives of multivariate functions is beyond me, Gen thinks while she would do well to position herself as far away as possible from the other team members so that the newcomers, that, at most, she came across without knowing them, will not crowd out the others. Before this unexpected victory at the SRA, never was I at the center of social attention outside of the day grades are released, she keeps thinking, while Trent asks for her shoe size.
"Your shoe size please..."
"Why?" Gen asks, surprised to be asked about her shoe size.
"The newest member of the booster club decided to give new pairs of shoes to our champions. It's not the best prize in the world, except that it's better than nothing. A reward is well-deserved!" Trent answers his star student.
"All right, I wear ten" she answers.
After all members of the team have declared their shoe sizes, the child of the new booster in attendance simply sends out a text message to his father the pairs of shoes to give them at the end of the practice while past AIME problems are handed out to the people in the crowd. For instance, ((3!)!)!/3! = N times X! Calculate N+X.
"What are these stupidities?" Dylan asks, while ignoring everything about these mathletic contests that are responsible for this crowd.
"The answers must satisfy two constraints on the AIME: they must be integer numbers and be within zero and nine hundred ninety-nine. The easy part is the denominator, six. It's the numerator that will cause all our problems: three factorial yields six, six factorial yields... in short, the factorial of a natural number N, or N! corresponds to the product of all natural numbers comprised between one and N. Three times two times one equals six" Geneviève explains, while having visibly finished the problem long before everyone else around her.
"These competitions might be stupid, but if that's what gives us a little bit of pride this year..." Dylan sighs.
"Oh, Dylan, you haven't covered the factorial, the exponential, or the logarithm yet" Gen points out to him.
The numerator yields donne 720! But 720!/6 may also be written as 719! times 720/6 and therefore 120 times 719! For the current problem, N=120 and X=719, thus the answer to this question is 839.
"And now for the notion of a multi-variable limit! In several variables, one must take pathways into account, but since infinitely many paths exist to get to a point (a,b) in a two-dimensional space, it is, in practice, easier to prove that a multi-variable limit does not exist than to prove its existence. To do so, we substitute y=f(x) into the function g(x,y) while ensuring that we have f(a)=b. If we obtain two different results for two functions f, then the multi-variable limit does not exist" Trent lectures the attendees.
"But why?" Lugh asks without having the slightest idea that posing y=f(x) represents a pathway on the plane.
"You take two different pathways to get to the same point, but you get different results. Thus, a function cannot approach a certain value!" Trent answers him.
The segment on multi-variable limits ends and those that have any actual mathematical interest will love what is to follow: the PARTIAL DERIVATIVE! We apply the definition of the single-variable derivative to a specific variable, and we treat the other variables as if they were multiplicative constants in a total derivative, and the partial derivative is noted, ∂, or del.
"Dylan, that's enough! You brought me here to learn multi-variable calculus with your friend! Do you realize that we are stuck in pre-calculus and that it will take years for either you or me to be able to do these things?" Randy, the cornerback, complains while he is right next to Marcia, who is not in the same calculus BC section as Geneviève.
"Stop making a fool of yourself!" Dylan shouts. "You wouldn't want to disappoint her in the middle of the session! She may not want to help you for tests if you continue to act like this!"
"What to do when you want to calculate a rate of variation in a direction that partially contains one variable and partially contains another?" Marcia asks, while sandwiched between Geneviève and Randy.
"Marcia's question is interesting because it makes use of an object called a directional derivative. You calculate one by taking the scalar product of a directional unit vector with the vector whose components contains all the first-order partial derivatives at a point. This vector is called the gradient" Trent explains to her.
Now, several students in the room start blushing and shaking in their seats while they face the material that make them squirm, and some among them are there simply because they believed it was cool to meet with the darlings of the day, especially students that might wish to gain in popularity. One of these students, near the edge, is about to crack while he understands absolutely nothing about what's going on.
"I DON'T UNDERSTAND ANYTHING!" Cory howls, a student who, despite all his attempts to pay attention to the material in this session, has way too many questions on his mind: what the hell is a limit, what's a total derivative, what can partial derivatives be used for, and especially why come to a training session of the mathletics team after regular class hours.
In doing so, Cory grans Éliane's attention who, like Geneviève, must be wondering how they came from intimate training sessions to an immense lecture that would give someone the impression to be attending a multi-variable calculus lecture at LSU or at other universities of this type... especially when the majority of the students in the auditorium starts clamoring that they are lost in the material, that they understand nothing about partial derivatives.
"Now that you have a better idea of how a lecture session could look like in college, one might be wondering how many people here actually understand anything about it. Obviously, by taking into consideration that there are a lot of people in the room that I don't teach to in my calculus BC course, I expect that there are some students that effectively don't understand anything" clamors Trent among the vast amount of incomprehension howls. "Raise your hand if you aren't in my class"
What did I do to find ourselves in this position? Why, all of a sudden, everyone wants to come to the team's practices? Could that risk be hurting the successes of the team during the SRA final? Geneviève thinks while the strangeness of this situation makes her head spin, even though she doesn't struggle on the material itself at all. If it's my very own popularity, or that of the team, that's responsible for it... this is nothing more than a stroke of luck: the same victory in a more favorable athletic context would not have had the same impact. Despite this, to what extent is being capable of performing on the AP Calculus BC Exam after only half a year of preparation representing an accomplishment? To what lengths are students willing to go to attain popularity among the student body? Visibly, on the second point, that includes attending a lecture on multi-variable calculus while they are not ready to take such advanced math courses!
The session ends and she is very surprised to see the father of one of the students attending arrive with four shoe boxes with clearly identified shoe sizes! The other three come to collect their pairs of shoes before Gen does...
"Here's your pair of shoes!" Cory's father exclaims in front of his own son, alongside Gen, shoes in hand.
"Wow, Vans Skate Cloud! A little bit more and Vans would be sponsoring the Square Root of the Answer! Thanks a lot"
"My son does not apply himself in the classroom as much as he should, but since I live in this town, never did I see a school of the area win anything on a mathematical level!" Cory's father comments to Gen, after she took the shoebox.
"Your son has a lot of courage to openly admit that he's lost in multi-variable calculus material. A lot of people would have preferred to keep quiet. I never saw the academic teams attract the attention of anyone here except the parents of its members, or their friends" Gen comments in turn.
"Multi-variable... differential calculus?" Cory's father shouts, astonished by the content of the session. "What's my son's crazy idea? He has the same pre-calculus instructor as the tight end and the cornerback! He might admit that he's lost in the material, I believe it will inspire him to take pre-calculus seriously"
I must acknowledge that my popularity has exploded, or at least the team's. Sometimes the team can be popular even though the individual members may not be. Like a football team during its good seasons, its more popular players will often be the quarterback, the wide receiver on a more offensive team, the linebacker or the defensive tackle on a more defensive team. I feel people will only care about the SRA because the AIME will fly well above their heads, even though the material on the AIME is tractable on the level of pre-calculus, she thinks as the training session ends, still shaken by the size of the crowd present for the multi-variable calculus lecture.