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Calculus-based Rage.
I figure a forum of people devoted to various geekeries have the best chance of sympathizing with my rage at the situation I am faced with.
Today in calculus class, we were discussing derivatives, and my teacher put a simple function on the board:
f(x) = ax +b
...and asked us to find the derivative of said function. I raised my hand for the first time in a long time (I am not so great at calculus, and had recently obtained some help from a friend in linear algebra), and explained what I'd learned: that according to the constant multiple rule and the rule of derivatives of constant fuctions, the derivative is "a." He paused for a moment and then joked that none of that was correct, before correcting himself and proceeding to obtain the same answer with the difference quotient. Prior to seeking some help from my friend, I found the difference quotient to be tedious, drawn-out, confusing, and not at all conducive to how I learn, with a pile of opportunities to make errors on top of that (depending, of course, on the complexity of the derivative). So, for the past 2 weeks, I have been doing all of my homework with the quotient-derived rules for differentiation. I found that calculus suddenly became much easier to understand and do, as well as noticed the inefficiency of finding derivatives using the difference quotient. As my friend said, "Its like using binomial expansion when you know Pascal's Triangle."
This is about the point at which the rage kicked in. My teacher turned to me and said, "[WindUpBird], for the purposes of tomorrow's quiz, you will not be allowed to use any of these rules to find derivatives on tomorrow's quiz."
Me: *blink* "Wut?"
Him: "You need to know how to derive these rules for them to be of any use to you."
Needless to say, I was pretty upset. Not only did half an hour of tutoring from a friend teach me more than 2 months in this man's class, but isn't it beside the point how I find said answer, provided I show proof of how I came to my answer? According to my book, said rules are BEYOND what the rest of the class should know currently, so haven't I in fact surpassed expectations by learning this other method and applying it? Finally, why should I be forced to use a method that I have found to be directly contrary to the manner in which I learn?
As far as the ability to derive the rules goes, I really see no point. I achieve the goal, so why trouble myself with a method in which more opportunities for mistakes present themselves? I feel the same way about this as I do about screwdrivers: there is no reason why one should need to learn how to forge a screwdriver before one can USE a screwdriver.
So: can anyone tell me if this makes sense or not? Am I just being spited by a man who is upset that his quiz will be a piece of cake for me; or is he right, and I should force myself to attempt to learn in a fashion that confuses me and destroys my grades?
Today in calculus class, we were discussing derivatives, and my teacher put a simple function on the board:
f(x) = ax +b
...and asked us to find the derivative of said function. I raised my hand for the first time in a long time (I am not so great at calculus, and had recently obtained some help from a friend in linear algebra), and explained what I'd learned: that according to the constant multiple rule and the rule of derivatives of constant fuctions, the derivative is "a." He paused for a moment and then joked that none of that was correct, before correcting himself and proceeding to obtain the same answer with the difference quotient. Prior to seeking some help from my friend, I found the difference quotient to be tedious, drawn-out, confusing, and not at all conducive to how I learn, with a pile of opportunities to make errors on top of that (depending, of course, on the complexity of the derivative). So, for the past 2 weeks, I have been doing all of my homework with the quotient-derived rules for differentiation. I found that calculus suddenly became much easier to understand and do, as well as noticed the inefficiency of finding derivatives using the difference quotient. As my friend said, "Its like using binomial expansion when you know Pascal's Triangle."
This is about the point at which the rage kicked in. My teacher turned to me and said, "[WindUpBird], for the purposes of tomorrow's quiz, you will not be allowed to use any of these rules to find derivatives on tomorrow's quiz."
Me: *blink* "Wut?"
Him: "You need to know how to derive these rules for them to be of any use to you."
Needless to say, I was pretty upset. Not only did half an hour of tutoring from a friend teach me more than 2 months in this man's class, but isn't it beside the point how I find said answer, provided I show proof of how I came to my answer? According to my book, said rules are BEYOND what the rest of the class should know currently, so haven't I in fact surpassed expectations by learning this other method and applying it? Finally, why should I be forced to use a method that I have found to be directly contrary to the manner in which I learn?
As far as the ability to derive the rules goes, I really see no point. I achieve the goal, so why trouble myself with a method in which more opportunities for mistakes present themselves? I feel the same way about this as I do about screwdrivers: there is no reason why one should need to learn how to forge a screwdriver before one can USE a screwdriver.
So: can anyone tell me if this makes sense or not? Am I just being spited by a man who is upset that his quiz will be a piece of cake for me; or is he right, and I should force myself to attempt to learn in a fashion that confuses me and destroys my grades?
Comments
Find your answer first with your nice method, but put it on the bottom of the page. Then go up to the top of the page and do it his way. If the answers are the same, you're good. If not, find out where you went wrong in the top, and if you can't, just change some of the numbers so the end result is the same as your way. Your teacher won't be able to discern the difference if you do your work in a non coherent, winding manner.
If I understood what subject you're learning, I assume he wants you to use simple algebra to subtract two infinitely close numbers in order to find the derivative of the function.
You, on the other hand, want to say something like " f(x)=x" => f'(x)=1".
You're right, but the theory behind this is too important to be neglected.
If all you care about is learning what rules to apply with no desire to understand why these rules work or where they come from, you may not be suited for math.
I suppose my problem is more that I feel like I can do something in a more straightforward manner, but I'm being prevented from doing so because the rest of the class hasn't gotten that far yet. Granted, said rules probably might be useless for higher derivatives, but in any event I already have the background to learn that, I've just discovered it differently.
I've already started deriving all of the rules I know so that I can do it with ease tomorrow; however, I still feel like I should be able to apply what I know.
Also, even something that looks as simple as the power rule can be tricky. There are different ways to prove it when the exponent is a positive integer, negative integer, rational number, or a real number in general.
If you would like to prove the power rule for the case you wish to apply it on the top of your test, your teacher would certainly be unreasonable to not give you credit for knowing the material. However, unless you have been given a long string of repetitve problems, this is unlikely to save you time.
EDIT: fuck it, I'll try.
f(x)= sin(2x3)
f'(x)= [sin(6x2)][cos(2x3)]
f"(x)= I'm not doing it right now, I get enough calc homework as it is. It's going to be long and drawn out, but I assure you that you don't need to touch any (delta)x nonsense in order to do it.
If that's wrong (and it probably is, it doesn't seem right), then whatever. I said I suck at doing derivatives of trig functions, and we haven't touched derivatives of trig functions in about a month.
Like...for the first 2-3 we were doing derivatives, my teacher wanted us to use the difference quotient, but after that he didn't want to see it again, because he agreed that it is confusing. For the quizzes where he wanted us to use the difference quotient, I just did this:
edit: I fucked up and realized my fuck up. Going to edit it now.
Oh yeah, that delta(x) nonsense? Try doing any real physics without it.
f'(x)=f'(x)g(x)+f(x)g'(x)
STFU Translation, I SUCK BALLS AT DERIVATIVES OF TRIG FUNCTIONS. God damn.
I'll do the problem properly in a few hours once I'm done with my real homework.
See, you little bastards don't seem to realize, either through laziness or stupidity, that the difference quotient is important because it can tell you an instantaneous rate of change. This will be important to you if you have any plans to understand university physics. If you approach physics thinking that the derivative is just a set of rules for making quick calculations, you'll never understand what's going on.
Further, if you know how to use the difference quotient, you won't have to be worried about forgetting the hundreds of different rules you're going to see for obtaining derivatives. You'll be able to derive the rule from scratch.
But NOOOOOOOO! You're a teenager! You obviously know more than your broken down old teacher. I mean he just has a college degree, and you haven't finished high school . . . Oh, wait.
Anyways....
f(x)= sin(2x3)
f'(x)= [cos(2x3)][6x2] (I wasn't sure if it was chain rule or not...after consulting my book, guess it was the chain rule)
f"(x)= [cos(2x3)][12x] + [-sin(2x3)][6x2][6x2]
......= 12x cos(2x3) - 36x4 sin(2x3)
Also, that's not the chain rule.
12x cos(2x3) - 36x4sin(2x3)
12x [cos(2x3) - 3x3sin(2x3)]
Also, you still haven't factored completely.
I just did the fucking problem, which you said I couldn't do...and yet you still call me a shit talker? If I had a scanner, I'd scan the page in my book that says "Chain rule" at the top and then proceeds to give an example problem that is almost identical to the one you posted (except the one in the book didn't want a second derivative). I don't have a scanner, so I can't do that, though. You'll just have to take my word for it (and what does my word mean, I'm just a shit talker, right?) 12x [cos(2x3) - (3x3)sin(2x3)]
And finally, if you take a careful look at my first post in this thread. I never actually said that I could do it. I just simply said that I knew it could be done. The fact that I JUST did it is icing on the cake for me.
After you went back and edited about five different times after I told you that you were wrong. That's some bitter icing.
Your pride in this is just an indicator of how little you know. No, that wasn't the challenge. The challenge was to do it with the Power Rule, which is what Mr. Bird was trying to use.
Do you think anyone wants to use the difference quotient all the time? Of course there are simpe rules that derive from the difference quotient, but you need to understand the difference quotient first. You obviously don't. That's an indicator of maturity. Yes, I'm sure you're smarter than a lot of your teachers. That's why they have college degrees and you don't.
Also, saying that a college degree = intelligence is about as naive as saying that having a high gpa = intelligence. When I did it the first time before looking at my book, I knew I was wrong (like when I had it with all the addition signs and shit). I even said, right after doing the problem, that it was wrong. You didn't tell me shit. When I noticed that I put cosine instead of sine at the end of the problem, I caught that mistake by myself. As for not factoring, I really don't give a shit. Factoring != completeness. That's like saying 2x+2y is less correct than 2(x+y). It's not. It's just a different way of writing that expression. You never specified if it needed to be factored until well into the "challenge". You also just said "use your rule to find the derivative of sin(2x3)". You never said "use your rule blah blah and get the answer in one try and factor it completely". Yes, it matters. He never said Power Rule. He said "derivative of a constant function" and "constant multiple rule".
I go to a private school, to get an A you need to be getting between 95-97%. That grade was from a few weeks ago when midquarter grades came out. Currently I'm getting a 92, which is a B+ at my school, would be an A if I was going to public school.
If I'm really as dumb as you say I am, then grades certainly don't = intelligence.
As for the next 3 quarters of Calculus, we'll see.
However, trying to use them without proving them first, as many high school students tend to do, turns calculus into mostly pattern recognition. This may make things easier in the short term, but means that you will go into university-level classes lacking some important tools, especially if you plan on using the AP exam to avoid retaking the class.
An example of a mistake you might make if you try to avoid difference quotients is trying to find the limit as x goes to 0 of (sin x)/x. If you did not use the sum angle formula and the difference quotient to take the derivative of sin x, you may be tempted to use L'hopital's rule without realizing the circularity of what you have just done.