:u: not that i’m aware of. however i didn’t like the textbook my class used so i got principles of mathematical analysis by rudin and it was pretty good. you can find the international edition for pretty cheap. it’s not like a schaum’s outline but it did a much better job of explaining concepts than the textbook the class used.
that class is a bitch for everybody though. i’d have to say it’s by far the hardest math class. if you’re not comfortable with proofs maybe you can go back to your notes for your math 100 class or something. usually math majors take their first upper division course that’s some sort of introduction to problem solving and proofs. trying to jump into real analysis without a real solid understanding of the thought process required for proofs is like trying to learn footsies against john choi so it might be helpful to go redo easier proofs from past classes.
Linear Algebra low tier? No way… How can you say this when a lot of ODE’s depend ON Linear Algebra… If you do Markov Chains, you use matrices which… guess what, uses Linear Algebra. Yep, both continuous and discrete markov chains… The mathematics of hypothesis testing in statistics uses… wait for it… wait for it… Linear Algebra!
In short, Linear Algebra is upper mid tier at the very least…
If you are the mindlessly plugging equations type, then you should not take Real Analysis. You have the wrong mindset necessary for this class. In fact, you have the wrong mindset for mathematics period. It is an illusion that real life will just throw you cookie-cutter problems that you found in your Stewart’s edition of Calculus. What do you do when you don’t have a nice rule to solve a situation? Just mindlessly plugging in things into equations is something a computer can do, and it can do it better than you no less. That kind of math requires no thought at all. It is boring.
The Calc I,II,III you took is sort of an illusion in the sense that it is a diluted version of real analysis that is intended to be a sort of “catch-all” course for students whose majors require mathematics. These classes teach HOW to do something without really explaining WHY you are doing it (well that isn’t entirely true. Odds are the professor goes through the proofs of the theorems, but this is not the focus). Real Analysis is the WHY. It is in some sense the “real” calculus. Yes some problems can be modeled using elementary functions, but what happens when you come across a problem that doesn’t “fit” one of the examples in the book? This requires critical thinking and not mindless algorithms. Real analysis is one avenue you can use to deduce information about such things. Just blindly following what a professor or a book tells you without understanding WHY is insufficient. In fact, doing something without knowing why one is doing it is not understanding at all. If you don’t know WHY you can’t evaluate something like the limit as x goes to zero of |x|/x or sgn(x), that is an issue. Responding with something like “because my calculator/CAS/Wolfram says so” doesn’t demonstrate understanding, but rather a dependence on an inanimate object to do the thinking FOR you. In fact, this is the very ANTITHESIS. of. mathematics…
In other words, if you want to get better at mathematics, you cannot do so unless you discard your current mindset. Concurrently, if you want to get better at Real Analysis, you need to reevaluate your entire stance on mathematics and realize that the time for reliance on computers and algorithms is gone.
It’s because the derivative is the quantification OF change. Simply taking the difference in change is a crude estimation of change at best. Not only would you have to take infinitely many differences, it is only good at the points you would take the difference of. Why would you want to do that when the derivative of the function GIVES you the change that is occurring… at ANY point in time no less! Doing it your way, you would first have to take two points in time, say x1 and x2, then calculate f(x1) and f(x2), THEN you would have to take the difference of f(x1) and f(x2) and take the absolute value on TOP of all of that… EVERY TIME. Why would you want to do this when all you have to do to find the change is to simply plug in x1 into f’(x)? f’(x1) means: the amount of change at time x1 is the number I get when I plug in x1 into f’(x).
This is exactly what I was talking about when I replied to “Dances with Ninjas.” Simply plugging in a number into a formula is insufficient to understanding.
To respond to your frustration, the reason why is because of the theorem that is in your Calculus textbook: If f is differentiable at a, then f is continuous at a. In other words, if I can take the derivate of a function f, then this function f has the properties of a continuous function. This means that this function is a function that has a limit as you approach a… AND it is a special function in the sense that you can evaluate the limit of it simply by plugging in a into the function. You don’t need to take left and right hand limits because it has the property of being continuous which is guaranteed to us because the function is differentiable, which is what that theorem said to begin with. So now that we know the function f is continuous, we simply need to realize that being continuous means that there is no abrupt change, but only gradual change without interruption. Since the function is continuous at a, this means the following: 1) The limit of f(x) as x goes to a exists. So, if a is 3, and I take values close to 3, such as 2,9, 2.99, 2.999, 2.9999, as well as values on the other side of 3 such as 3.1, 3.01, 3.001, 3.0001, that the value of f(x) that I get is the same AND that it approaches a fixed number. You no doubt did this the first week of your calculus course and most certainly did this in a lab exercise. THIS is why. This is what they meant when they said “A small change in x results in a small change in y” THIS is why. 2) f(a) is defined. This means that if I plug a into f(x), the value is a number. 3) And this is the important part… the limit of f(x) as x goes to a is the SAME AS just plugging a into f(x). In other words, I don’t need to evaluate the limit with left and right hand limits. The fact that the function is continuous means that it is a function whose limit can be evaluated EASILY by simply plugging in the value a into f(x)
In sum, KNOWING a function has a derivative and that this derivative f’(x) can be evaluated at a point a TELLS YOU that the function is CONTINUOUS at a. Dances Wtih Ninjas… THIS is the kind of thought you must learn.
No, not essentially. It IS the area of a rectangle or box (assuming by box, you are referring to a two-dimensional object)
It will blow your mind even more then that this idea is used in probability to determine the probability that for instance a bus will arrive between 10 and 10:30. This kind of distribution is called the uniform distribution.
This is because you are assumed to have a solid grasp of precalculus. If you do not, either make a note of it and ask your instructor or go ask your tutor/help center about it.
Finally, as a general request, there already is a math thread on SRK, if we could keep it contained in one thread, this would make it easier to respond to questions and clarifications rather than searching through several threads. IOW, please respond to anything I or others will have said in the thread below:
Calc 2 is definitely the toughest of the Calc 1/2/3 series.
The trick to Calc 1 is as was mentioned, understand the principles of what is going on, what does it mean? If you know what it means, you can reverse engineer that back into how it is put together and then chain rule, quotient rule, all that stuff just sort of falls into place.
Actually, as someone who has to use Calculus everyday in teaching people who have never taken Calculus (boy do the liberal arts econ majors hate me… of course I wonder why you would major in Economics, the most mathematically intensive liberal art {Social Science} and not expect to be pushed mathematically but whatever) and I often find its helpful to actually have some mathematical literacy. People don’t know how to read math. The way we teach math is so flawed. We teach children from a young age how to do math, but we don’t teach them how to “read” math. We teach kids basic algebra but then don’t teach them what is actually happening, how to read the formula, what does the equation tell you… If you have decent math literacy, you can get your way through the first half of a math undergrad with a small amount of work and practice.
I have said it multiple times, Real Analysis… its top tier math my friends. Its useful, it has real (hence its name) applications, yeah… I hated taking Real Analysis but boy has it saved my rear as an economist.
Oh and before we start hyping up mathematics professions, economists aren’t top tier (math / physics) but hardcore mathematical economics is at least close to say engineering. Actually, most of the engineering profs and I have very similar math backgrounds… I.E. Masters degrees in Math or at the very least enough graduate level math to apply for a M.A.
In high school I did Calc 1/2/3 Diff Eq…no AP test for Calc 2/3 Diff Eq so I had to retake them. I knew what I was doing and still had to retake Calc 2 twice. One time was circumstances, the other time I busted my ass and still failed. Why? Trig integration. Fuck it. Nothing cute, just fuck trig integration. On some "predict the original version by memorizing the derivatives and work it backwards…GTFO.
Trig integration is just using trigonometry to convert integrals that you don’t have antiderivative rules for into ones that you do have antiderivative rules for. I’m guessing you weren’t up to par with your trig and so that is why you had such a hard time with it.
From experience, I find that the majority of people who have difficulty with calc do not have a solid grasp of precalculus… or even algebra for that matter. I honestly believe that you will not have a foundation for understanding calculus without it. I wish I had a dime for every time somebody got a problem wrong because they didn’t know basic logrithmic or trig properties on a homework assignment or an exam…
I must be in the minority but I found calc 1/2 the easiest and calc 3 the hardest in terms of understanding. I did fine in all of the classes, but in 3 I had trouble actually picturing what was going on and why I was doing things. As a result on the tests I just spit out formulas, got an A, and had no idea why I just did that vs where I use calc 1/2 things almost everyday.
had nothing to do with being up to snuff on trig and everything to do with having to memorize…my memorization is assed out, especially on things that I can’t apply to money (thats my solution, if I can’t sole something, I equate it to maoney and I get the correct answer…seriously)…its why Chemstry and Biology were a nightmare for me and physics was a cakewalk…IMO there are two types of subjects (may have said this before)
Biology & Physics.
In biology, the ‘body’ of knowledge (no pun) is based upon memorization.
In Physics - if you understand the principles enough, you can figure everything out. Physics 1 is nothing but f=ma…as you get further along its still the same principle but with advance math attached to make sense of the more complex variables (such as waves and particles)
Chemstry is down the middle and is the closests thing there is IMO to trig integration, it requires you to memorize a bunch of scenerios and at the same time figure them out…that is my weakness. If there was no memorization involved…pssh push-over. I crushed every other form of integration there was , I flew thru all my other math classes (when my grade wasn’t based on homework hehe)…but next to diff EQ, it has been the only other math where a large portion of it was based on memorization and recognition of things in problems just to back track enough to actually get a solvable problem.
And its worse because it meant nothing. Every other mathematical principle I’ve come across, had a true purpose…trig integration didn’t, wasn’t until Material & devices were we studied particle movement, that I ever saw it in school. I believe I used it a little in E&M, but nothing to the extremes that the class forced us to go thru. shudders
Fuc ktrig integration
Could you elaborate more on (as well as give examples of) how you personally would conceptualize bio, chem, and physics problems using your “framework”? If you were unable to conceptualize it in this manner, could you list attempts you made and why they did not work?
Regarding memorizing trig integration, do you mean trig substitution (iow you have a function in terms of x and switch it to theta, then integrate, then use trig to get it back in terms of x) or trig integration (where you have powers of something like csc(theta)cot(theta) and have to pull out a power of cot(theta) and use identities to get it in terms of mostly csc(theta), but with a cot^2(theta) in there somewhere?
Regarding trig identities, one thing that helps me is that rather than memorize all of them, one thing you can do is memorize a few and recall how to derive the rest. As an example, I start out with
sin^2(theta) + cos^2(theta) = 1. Divide both sides by sin^(theta) to get 1 + cot^2(theta) = csc^2(theta). Or I could divide both sides of sin^2(theta) + cos^2(theta) = 1 by cos^2(theta) to get tan^2(theta) + 1 = sec^2(theta). That’s 3 equations for the price of memorizing one. If you say that’s still too much, notice sin^2(theta) + cos^2(theta) = 1 is called the pythagorean identity. This is not by coincidence. I’m pretty sure you remember pythagoras’ theorem of c^2 = a^2 + b^2 right? (where c is the hypotenuse and a and b are the sides). Divide by c^2 and you get 1 = (a/c)^2 + (b/c)^2. But, what are a/c and b/c but the sine and cosine ratios…
Similarly, for other crazy identities, I start out by multiplying (cos(a) + isin(a))(cos(b) + isin(b)). Equating real and imaginary parts gives me the addition formulas for cosine and sine respectively. Then to get sin(2theta) I just let a = b in the sine addition formula, so sin(2a) = sin(a + a) = sin(a)cos(a) + sin(a)cos(a) = 2sin(a)cos(a). A similar argument shows that cos(2a) = cos(a + a) = cos(a)cos(a) - sin(a)sin(a) = cos^2(a) - sin^2(a). From here I can derive the rest of the identities. That is at least 4 identities I have displayed for the price of one.
You don’t necessarily have to memorize things if you understand how it all works…
Meh, I’m only in College Algebra/Business Applications in my junior college right now. We’re on Matrices right now, not really complicated, except I HATE Row Operations. I had taken Pre Cal my Senior year in HS. Had I not of had senioritis I would have aced that class easily, but I did get a high B so I didnt do too bad. Im not exactly sure what math classes I will be taking next semester, most likely Physics and maybe some Chemistry. Whatever is after COllege Algebra, I will be in that class.
Didn’t see this when it was posted, but here’s my answer, off the top of my head (not sure if I’m right.) Laplace transform is used instead of Fourier whenever you want to be able to use the unit step function, which you can’t use with Fourier transforms. So Laplace transforms can be used even for transient analysis, while Fourier transforms only give you steady-state answers.
Transfer functions are useful in general because they allow you to work in the frequency domain, and allow you to design for useful things like good settling time or phase margin. You might be interested to know that EE majors don’t always have the luxury of separating the feedback network from your amplifier, due to loading effects, which has resulted in some really exciting new ways for learning electrical network theory in order to design feedback networks. If you’re interested, the book to read (I’ve only read the first four chapters fully and skimmed the rest,) is
If you like geeky control systems stuff like this for ee applications, it’s exciting stuff. Some gruesome linear algebra is involved in proving the theorems, but the theorem results themselves are computationally efficient (or so I’ve read.) Got what I needed and moved on.
Edit: Hm… Okay, I’m reading here on wikipedia that ideal low pass filters (ie, rect functions) have Fourier transform pairs, which suggests that I’m wrong about step functions only having transform pairs for Laplace but not Fourier. But I’ll stick to my guns. I distinctly remember it all has something to do with Fourier transforms being taken by integrating from minus infinity to infinity, while Laplace transforms are integrated from 0 to infinity. Now, if I can only just remember what that has to do with step functions…
2nd edit: Okay I got it.
3rd edit: Okay, it looks like, technically, I’m wrong about there being no Fourier transform pair for the unit step function. wikipedia has the expression here
but it looks so unwieldy that I think I’m right in principle. Use the Laplace transform whenever you need the unit step function and its response.
Not only that, but you can’t rely on just memorization and do well past a certain point. If your method is to memorize how to add, subtract, multiply, and divide fractions and decimals, a formula for percent, the area of a rectangle, triangle, circle, difference of squares, FOIL, factoring, the quadratic formula, 45-45-90, 30-60-90 triangle ratios, definition of slope, slope intercept form, point slope form, polynomial long division, graphing parabolas and figuring out how the graph of f(x+h) is related to f(x), trig identities, etc. without knowing what you’re doing, then you already have no room left for calc.
On the other hand, somebody with a solid grasp of functions, basic geometry, and some algebra can learn the content of a yearlong high school trig course in a matter of weeks. This is still true at higher levels of math. Somebody who knows what she’s doing (yeah PC!) can learn differential geometry, even if it takes years. Somebody trying to get by on memorization will never, ever be able to learn it.
yeah you crazy, dawg. i gotta say based on how much bitching my classmates were doing and tutoring friends that calc II has to be the second hardest math class. certainly the most difficult lower division class. I say real analysis I is first then calc II then whatever class people take to learn how to do proofs. usually the first upper division class. those three classes really force you to step your thought process game up.
true dat. i got a saying that i tell people that ask me for help is to know what you know. people look at me like i’m crazy but i’ve encountered times on tests where i’ve forgotten lemmas or wasn’t sure if i could use them but because i was able to fully understand a few things I was able to derive like whole chapters worth of material and was able to solve the problems. if i just tried to memorize every single theorem and lemma without trying to understand them i would have been assed out.
agreeably, calc 1 was pretty easy. I took college calc1 in junior yr of HS through some chicago mayor program. compared to previous math courses, it actually took a lil while to get, but eventually got there. math courses were always my favorites/easiest. probably because, as mentioned here, it wasn’t about raw memorization but understanding really how the shit works/relates.
that was several yrs ago. all this calc 2 talk is making it sound very fun/interesting/a challenge tho. plus some other classes it sounds like I’d love to try. if I ever go back to school.
I probably should’ve taken calc2 already but with all the wonkiness of 3 different highschools/college within a few years, and not doing very well on the test that Highschools use to place you (I was totally placed too low for most of my classes), everything got wonky. got stuck in this easy ass algebra2 class or some shit in freshman year. only person in the class with perfect scores (and above perfect scores because everyone else in the class needed extra credit)…sad to say it was a bit boring and a waste of a year. then moved on to HS/college around soph/jr. year. totally skipped geometry though took trig in HS and precalc/calc1 in college. then had to do summer school for geometry last -_-…
problem is without practice I just forget everything : / while fun, not gonna spend time relearning and learning new stuff if it won’t benefit me. when I eventually want to move on from my current job that’s probably the area I’ll be looking in. gonna sub to that math thread. and google some of this stuff. more procrastination, yay.
It’s been over a decade since I finished calc, data analysis, partial diff equations, discrete, and all that other shit and even though I might say that I forgot everything it’s not entirely true. When I need to remember something for a project I just read up on it and remember it.
If you need to remember it, I’m pretty sure if you have a good book you can. For low level math like multi-variable calculus the internet is fine, but the higher you go, the more useless it becomes for attaining appropriate information directly. As you progress you’ll be relying more on the smarter professors in your area and others’ research materials.
yeah I’m looking at even simple shit from early HS and don’t remember a damn thing about what is supposed to happen in the equation. but like you said, it’s most likely not totally forgotten, just need a refresher to unlock the memories or something. I hope.
Calculus I (the one you are taking now) is going to always be hard compared to anything you have taken in the past. Simply put, it is introducing something TOTALLY new. Going from walking to a full marathon of long hours of studying things that are foreign to you is never easy. I can’t speak for anything past Calculus II (because I am taking that right now) but from what I am learning the pace is FASTER ( mainly because there is MORE to learn ), as for how hard it is…you will learn to cope with it and adjust to the hours of putting time into it.
Suggestions:
Find someone in your class that is not an asshole, but knows what the fuck they are doing. Make friends with them and study with them. It makes learning the subject a LOT EASIER. Do hw together, prepare for test together, and ask questions if you have them.
Go to your teacher after class and ask him/her questions about things you don’t understand. Sometimes it can be hard to ask during lecture because he/she might go fast and by the time you catch up…he/she is already 3 steps ahead. (mark with a * or circle something you have a question on and ask after class)
Don’t get mind fucked. It get’s tough, after putting a lot of time into studying and doing hw, then not doing so well on your test/quiz. A lot of people want to “act” like they know shit and are better in math. Sometimes you might wonder how other people are doing so well but you are trying your best and not achieving the same grades. TRUTH is: They had the class before, are putting in just as much time as you are but are studying something different that actually was on the test (unlucky circumstance), they have previous tests/exams from the class :woot: to study from.
Keep up the work. And remember don’t study hard, study SMART, it’s much more effective.
