February 13 arrives and that's the start of a whole new week that happens in class. Marcia finds Geneviève, who does not seem to be in the best of moods after having to comb through the various requests from other students that seek to become her Valentine. Lugh, for example, formulated the wish that Geneviève must answer every mathematical question that he asks on that day. What about Cory or Randy? She remembers the former because je openly declared what so many kept to themselves on that night, and the latter is simply one of Dylan's teammates that was wondering what he could have found interesting to attend a multi-variable calculus lecture and claimed to have done so as a favor to either Geneviève or Marcia. And other students that dream to capitalize on their popularity to raise their own.
"Before you ask once again about the AIME, I didn't receive anything yet, except that I have other problems. I can't seem to sort out who will be my Valentine for tomorrow. As tempting as it might be to have more than one, I am not sure it's a good idea. If I only have two with whom to do it, I will willingly celebrate it on two nights, with each of those people... Dylan seems to be very close to you and not just because he believes he's able to get the most out of you academically" Gen then tells Marcia.
"He just hasn't appreciated that Randy told him his four truths in front of me. But Randy, unlike Dylan, is not a senior" Marcia replies to Gen.
"Cory gave me this pair of shoes that I currently wear" Gen then points at her shoes.
"I don't know for Cory, however"
This Friday we shall see if the next multi-variable calculus session will be as crowded as the previous one; last time, there were more people than both AP Calculus BC sections combined. We shall see then whether my popularity is a flash in the pan or not, but I believe there won't be as much of a crowd this Friday especially since there are several students that don't understand anything right now, she thinks while the other potential valentines approaches her with offers as crazy as the last one. Regardless, several of these valentines attended this session, she recognizes some of them even though most of them kept quiet during the lecture.
"Can you be my valentine for tomorrow?" the potential valentines all take turns asking her.
"Before we begin, I have something to ask: did you attend this multi-variable differential calculus lecture?" Gen asks the potential valentines.
"Why?" a potential valentine asks her.
"Just to know from where you might have known me. Sometimes we don't have the same impression of someone depending on the context under which we have known a given person" Gen then answers that valentine.
"Your're the mathematical mascot here, you are doing nothing to hide it, especially not since this calculus lecture..." another potential valentine confides to her.
"Did you understand anything about it? Same thing for those who attended. And especially don't be afraid to admit you didn't understand anything" Gen asks the potential valentines.
"All that I understood is that, for a limit to exist in several variables, the function must take the same value irrespective of the pathway taken to get there" yet another potential valentine answers her.
"I must admit I didn't understand anything" a fourth potential valentine adds to this cacophony.
I know very well that lacking the necessary mathematical talent to perform multi-variable calculus does not make someone less of a person, she keeps thinking while she must juggle between several people. Like Cory, who comes to see her for the first time since that lecture; he just texted her.
"Hi Gen. My dad gave you these shoes; do you like them?"
"Hi, Cory; yes, I like these shoes"
"Do you want to be my Valentine tomorrow?" Cory asks her.
"But why is it that you want to be my Valentine? You might have the courage of your convictions but promise me that you will be honest regarding your mathematical needs! Your dad told me that you didn't make as much effort in class as you should..."
"Yes, we have just begun to cover exponentials and logarithms. You talked about student loans in the student newspaper, the homework contains loan payments and loan terms"
I wonder what he understands about exponentials and logarithms; just calculating the installment payment requires the exponential, while calculating a loan term with a known installment payment, typically maximum, requires a logarithm, Geneviève thinks while everything she knows about Cory can be summed up as staging an intervention during a multi-variable calculus lecture and that he has the same pre-calculus teacher as do Dylan and Randy! Speaking of Dylan, here he comes...
"Here's what you asked me for the French project" Dylan announces, while handing her his part in an envelope containing a wish card and another document relative to the daily life of a high school teacher.
"It's better than nothing, after some weeks of inaction on your part, thank you, Dylan. That said, do you want to do the pre-calculus homework together with Randy and Cory?" Gen asks Dylan.
"Maybe my own friends in the pre-calculus course will want to come, too, if you are here to help us to do is... may as well make everyone benefit from it!" Dylan responds to him.
"Thank you, Dylan, may we meet each other and the entire pre-calculus gang that has some questions relative to student loans and composite interest. We will then sort out who has valentines or not"
Thank God that the school does not rank students within a class, Dylan thought. That said, pre-calculus is more readily accessible, and there is a clear-cut objective, as well as a practical reason to do so. Not like the last lecture where people attending didn't quite know what the material would be used for. However, people who didn't have the homework on student loans in pre-calculus would not have had a vested interest in learning more about the mathematics of student loans (well, loans in general); everything that mattered to them on a conceptual level to the others was already covered in the student newspaper. There, Dylan, Cory and Randy invited all their available friends in pre-calculus in hopes of making their lives easier, and they had enough friends to fill an entire classroom after regular class hours.
"First question: suppose that we invested a thousand dollars in a bond bearing five percent interest per year for three years. How much money will we have at maturity?" Dylan asks, who wonders how such a calculation is made.
"From what I understand, Dylan, is that the first fifty dollars of interest earned during the first year will also bear interest the following year" Randy comments.
"Composite interest is such that, if we have an initial principal, an interest rate per period and a number of interest periods, we have:
Where S is the final sum, C, the initial principal, r, the interest rate per period expressed as a decimal number, and n the number of periods [https://img.wattpad.com/57718484020ef8a1bd29433ef0372b871024a451/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f536b54654f764b4e664a716448513d3d2d313238393433383138302e313732616234646133396236666130363834373030393939393932362e706e67?s=fit&w=1280&h=1280]
Where S is the final sum, C, the initial principal, r, the interest rate per period expressed as a decimal number, and n the number of periods. We then have C = $1000, r = 5% and n = 3, therefore we have:"
"Bravo, Cory, but if you want, for example, to know for how long to leave money in an investment to obtain a given sum, what are we doing then?" Geneviève asks, attentive to all questions of personal finance that the others could be asking [https://img.wattpad.com/096cba60631afb71aeae9aee75f6b2474ac284cc/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f677035397845646f47355a4e77413d3d2d313238393433383138302e313732616234646339323431343534623831363430363432333137352e706e67?s=fit&w=1280&h=1280]
"Bravo, Cory, but if you want, for example, to know for how long to leave money in an investment to obtain a given sum, what are we doing then?" Geneviève asks, attentive to all questions of personal finance that the others could be asking.
"Why are you talking about investments when the reason of Dylan's invitation was about loans?" Cory asks, annoyed.
"Loans and investments both use exponentials and logarithms" Gen answers him.
Did you know this text is from a different site? Read the official version to support the creator.
In general we would have, in the event we desire to calculate the duration to leave a certain amount of capital in an investment to obtain another sum at the end of it:
This is where one makes use of the logarithm, and the logarithm's basis shifts from 10 to 1+r, or from e to 1+r, and the answer to Cory's question then becomes, she thinks, with the intermediate steps on the board: [https://img.wattpad.com/a3ce17da00f6cef505fef130210e3ea5b8fd4c14/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f4e2d4b48535545737559694170413d3d2d313238393433383138302e313732616234653034303933646539313431363930333934393936392e706e67?s=fit&w=1280&h=1280]
This is where one makes use of the logarithm, and the logarithm's basis shifts from 10 to 1+r, or from e to 1+r, and the answer to Cory's question then becomes, she thinks, with the intermediate steps on the board:
"You said in the very first meeting for the AP French group project that, if our student loan balance at graduation was sixty thousand dollars, it would cost nearly seven hundred dollars per month just to repay the loan over a ten-year period" Dyl... [https://img.wattpad.com/ce57e3a10b62247f88f23a1003b415d9caea127f/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f5f63304350783659557842714a513d3d2d313238393433383138302e313732616234653363363836383834363435333538393034393339362e706e67?s=fit&w=1280&h=1280]
"You said in the very first meeting for the AP French group project that, if our student loan balance at graduation was sixty thousand dollars, it would cost nearly seven hundred dollars per month just to repay the loan over a ten-year period" Dylan comments on the group project.
"Dylan, I assume that you want to know how to calculate a loan installment payment? I see that the statement began with the present value of an annuity since, from the lender's viewpoint, a loan represents an annuity, assuming people repay their loans on schedule. In general, for loans, the present value of the annuity represents the loan principal" Gen responds to Dylan's comment.
"I may be wondering why we sought to grab Marcia's attention so much..." Randy comments.
"Marcia was straight to the point when giving answers. Unlike Geneviève..." Dylan sighs.
"It takes a couple of algebraic manipulations to calculate the monthly payments, but since the payments are monthly, and that the interest rates are usually yearly, you must then convert the yearly interest rate into monthly rates, while knowing that, for these calculations, you want the value t such that you get
"It takes a couple of algebraic manipulations to calculate the monthly payments, but since the payments are monthly, and that the interest rates are usually yearly, you must then convert the yearly interest rate into monthly rates, while knowing t... [https://img.wattpad.com/516cd2dc9b3a46fc0da36ba12ef3d7cc7d43cab8/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f49746139653873757a4a706462673d3d2d313238393433383138302e313732616234653864393764636162373834313433393334313332322e706e67?s=fit&w=1280&h=1280]
The annuity formula is given as follows:
And then we want to determine V knowing that C is worth $60,000, n = 120 months [https://img.wattpad.com/591fc9e2227c678cd051b09c4620c1f80f737acf/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f3263684e593364764c63677374773d3d2d313238393433383138302e313732616234656132663562353937363537373730313431373336362e706e67?s=fit&w=1280&h=1280]
And then we want to determine V knowing that C is worth $60,000, n = 120 months. If we take r = 0.0654 (i.e., the interest rate of a consolidated federal loan is the same as the interest rate of a direct graduate federal loan) in the monthly payment equation, then t = 0.00529. But with two algebraic manipulations one obtains:"
"My parents show me their credit card statements, I am willing to accept there is a minimum amount to repay, but there is a box I don't understand" Imélie, another student in pre-calculus, a female, asks, with her hand raised [https://img.wattpad.com/c35b7ac794a8cf7186e3bce28f4748cedabf3ac7/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f39494e35696f656d7a55702d51773d3d2d313238393433383138302e313732616234656435333964386661663836343634373636343638312e706e67?s=fit&w=1280&h=1280]
"My parents show me their credit card statements, I am willing to accept there is a minimum amount to repay, but there is a box I don't understand" Imélie, another student in pre-calculus, a female, asks, with her hand raised.
"What box are you talking about?" Gen asks Imélie.
"The minimum repayment time box"
"Not all credit card statements will have such a box. We will show how to calculate a required time to repay a credit card. To fixate the ideas, suppose that we have a credit card with a twenty percent yearly interest rate with minimum monthly payments to make, with a three-thousand-dollar balance and a minimum payment of two hundred fifty dollars"
Once again one needs to return to the basic annuity formula, but this time around we want to calculate n. She feels that the students are no longer reasoning in terms of pre-calculus but in terms of personal finance; the « eureka » moment would then happen as far as how the material is used in everyday life, or at least in the real world. Four algebraic manipulations later, another set of questions appear in the students' minds while the following equation appears on the board:
Four algebraic manipulations later, another set of questions appear in the students' minds while the following equation appears on the board: [https://img.wattpad.com/5be63a71bef8001c27ca7e5714800f33f5c19a7c/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f2d615249515f59436b57533133773d3d2d313238393433383138302e313732616234663063653961343938353431323331393837373534302e706e67?s=fit&w=1280&h=1280]
"There's something that bothers me; what happens if the left-hand side term is smaller than zero?" Imélie asks.
"The fraction contains three strictly positive variables. For the left-hand side term to be smaller than zero, Ct/V would be required to be greater than one and consequently that Ct must be greater than V, this would mean that the payment would not even be enough to cover the interest. This will be important for what is to follow because the logarithm of something comprised between zero and one will always be negative so long as the base is larger than one" Gen explains to Imélie.
"The minimum time would then be the n?" Randy asks while they are awaiting the answer to get the homework done.
"So are you telling me that a credit card represents a loan?" Cory asks, astonished at the mathematical realization that it represents.
"Yes. A short-term loan, but a loan, nonetheless. You have more freedom for any payment beyond the minimum payment" Gen then answers Cory's question.
By seeing the students perform the calculations, even the football players, they arrive at the conclusion that the minimum repayment is, express in months:
"Now that you have a better idea of how pre-calculus is used in real life, we have another question, unrelated to exponentials and logarithms: who here does not have a valentine for tomorrow? Raise your hands please [https://img.wattpad.com/208fa4cfa35ccbef322695eab6dd06b07ea1a524/68747470733a2f2f73332e616d617a6f6e6177732e636f6d2f776174747061642d6d656469612d736572766963652f53746f7279496d6167652f764745314c4f7677496b6d5142773d3d2d313238393433383138302e3137326162346636343434386662316533323438323636363132342e706e67?s=fit&w=1280&h=1280]
"Now that you have a better idea of how pre-calculus is used in real life, we have another question, unrelated to exponentials and logarithms: who here does not have a valentine for tomorrow? Raise your hands please..."
The last chance for Geneviève to help those who don't yet have a valentine to find one, herself included. Dylan appeared too fixated on Marcia to me, I believe he would be happier with Marcia than with me, but between Lugh, Randy and Cory, my heart can't decide. All three have their advantages. However, at the risk of regretting my decision, even though I know no one will get the same satisfaction out of it, I must take it as soon as possible and then I will be able to focus on my homework in other subjects, such as in chemistry, history, and government. The homework of these three subjects will take me hours so I'd better have the peace of mind, but I have the impression that Lugh just wants to take advantage of me, and I could say the same of Randy and Cory. However, since I have no course in common with them, I believe it would be a breath of fresh air to have a valentine from outside of my usual circle, she thinks while she reviews why one other another, especially while considering that these three boys were her only points of contacts in the current pre-calculus class. She is forced to acknowledge that Randy has some academic potential, while Cory already has a foot in the door. The others can rapidly find valentines among the people in the room, most of whom have kept quiet and are content with taking notes of the complete solution so they can then apply them with the problem's parameters.
"It might surprise you, but I would like you to be my valentine" Randy announces, after the others left the room, except for Cory and Randy.
"We must hurry up to resolve this, but you are the last two remaining" Gen then tells the two boys.
"I think I misjudged you during this multi-variable differential calculus lecture. After everything Dylan told me about you, and about your friend Marcia, you gave me the impression that you lived only for advanced material. But now I know that you're nicer than I initially believed" Randy comments on that situation.
"Gen, I am forced to admit that, unlike the other nerds that I might have known, you actually can link knowledge to more concrete situations" Cory tells her.
"That's a little weird for a confession but thank you. It will not engage you to anything beyond tomorrow, I must warn you immediately. I don't want you to hold false expectations towards me" Gen warns them.
"That's unlike you, Gen..." Randy sighs.
"I don't want to be mean, but there is only room for one" Geneviève announces them while she only wants to return home to start doing her three assignments for tomorrow. "Cory..." she announces as a whisper.
Once she returns home, Gen stops thinking to anything related to Valentine's Day, but if Cory seems to appreciate her because she thinks differently from the others... they might click between each other. She chains the three assignments, initially without complication, but it becomes increasingly longer to write the final assignment; when it comes to talk about legislative gridlock that happens when both houses of Congress are controlled by different parties, or the White House is controlled by a different party from Congress, she struggles to formulate coherent statements on the big question about the factors that contribute to gridlock, such as the legislature's partisanship, and the resulting desire not to show their opponents can legislate adequately. What's happening? It seems I'm unable to think clearly... why can't I... this is unlike me! I wish for a little bit of sleep! She thought while running out of juice to flesh out her answer and, by 1:30 AM, she decides to stop everything, exhausted and visibly needing sleep.