Discussion in 'Math' started by cicada207, Dec 27, 2014.
can i help me step by step this exercise , thanks very much!
I don't know, can you help you?
This looks like homework. You need to show YOUR best attempt to solve YOUR homework. That will give us an idea of what you are doing right, what you are doing wrong, and what suggestions and hints might move you in the right direction. We will NOT just do your homework for you.
first do partial function......
As the other readers have suggested, you should try this yourself so we know what way you would like to do it.
Did you learn how to do a partial fraction expansion yet?
I will start by giving one hint though, and that is when you have a sum in the numerator you can break the problem down into two smaller problems by separating the two sums to start with. That will give you a new form like:
were N1 is one numerator and N2 is the other, and D is the same denominator for both.
We dont know what way you want to approach this yet so it's hard to say what you really need right now.
It worries me how a bit of basic algebra and decomposition can boggle peoples minds when they reach the Laplace/Fourier milestone.
I'm speculating quite a bit here, but my guess is that you would find the people that struggle with these kinds of things tend to be highly field-dependent, which basically means that how they perceive things is primarily within the context of other things and that if you change the context they have trouble grasping that it is still fundamentally the same thing. Engineering and most technical disciplines lend themselves to people that are highly field-independent, meaning that they see things as being composed of smaller things that can, in turn, be viewed independent of the context. People that are highly field-independent have problems keeping things straight in situations in which context is the dominant factor, which often is the case with things like project management. I saw one study that estimated that about 60% of Americans are at least moderately field-dependent while 90% of engineers are at least moderately field-independent and many are highly field-independent. Conversely, a very significant fraction of marketing, sales, and business executives are highly field-dependent.
WBahn: Very interesting view you have there.
Maybe you also heard the somewhat funny 'proverb' which i'll try to remember here which is a twist on another quotation...
"Engineers are people that learn more and more about less and less until they know everything about nothing, while Managers are people that learn less and less about more and more until they know absolutely nothing about everything and thus can pass themselves off as Engineers."
I have ran into several people on the web who are attempting to solve complex circuit problems but dont understand simultaneous algebraic equations yet.
I have found some solved examples with Laplace transformation. I my opinion, they are very helpful. Try to analyse these examples and then solve your exercise.
At that point some engineers can pass themselves off as managers, too!
I've seen that many, many times, both here and in the classroom. I think there are a few factors at play and that the blame lies in a few different places. In the classroom it is sadly all too possible for people to advance from one course to another without learning the fundamentals. Sometimes that's because the course doesn't emphasize or even teach the fundamentals, but rather just a bunch of memorized formulas and recipes for how to do a handful of very specific things. Other times the fundamentals are covered and perhaps even stressed, but the requirement that students actually demonstrate comprehension of them in order to pass the course is weak. To be sure, that's easier said than done and I've been guilty of it myself -- no instructor (well, damn few) likes failing a student. On top of that, faculty are held hostage to the all-important course survey that students fill out at the end of the course and that, most places, has a huge impact on whether the instructor will be retained or, even more so, whether they will receive tenure. So faculty all too often make decisions based on what will improve their course surveys rather than what will truly serve their students in the long run.
On the student's end, all too many students don't care about learning the fundamentals or even how to apply the formulas and recipes they've been given in a solid manner -- a course is just a requirement that they have to get checked off in order to get a piece of paper that they believe will result in some company throwing lots of money at them. The easier the course is the more they like it and the harder the course is the more they hate it -- and that is reflected in their course surveys.
Even the students that do care about truly learning things are caught in a bind because, at the end of the day, they only have so much time and working the problems that affect their grade takes higher priority than really learning the material. So unless the problems are tightly aligned with learning the fundamentals, that part is going to slip.
All of this feeds into a vicious cycle because if a student enters a course ill-prepared for it, then they will have a strong tendency to be satisfied with just getting by in the next course (because just doing that is going to be a struggle). If the teacher is faced with a significant portion of the class that is ill-prepared they often end up watering down the course and not getting the fundamental material across like they should, which then results in an even larger portion of that class leaving the course ill-prepared for the next course and so on. But what happens is that the student eventually hits a brick wall and, to proceed, they have to take a huge step back and learn all of the things that they didn't learn over the prior two or three years and that is so discouraging for many that they continue to struggle on getting deeper and deeper in the hole (and in debt).
For engineering disciplines, the problems start in what is being turned out by the high schools. Students enter technical college programs without adequate math skills and very often lack proficiency in junior high level algebra. So they start out in a huge hole and the colleges face a catch-22 in trying to deal with it since they can't realistically bring them up to speed and still complete a four-year degree in four years (and that is a big driving factor these days given the high cost of tuition and student loan debt that these kids end up carrying) but they can't reject such a high number of applicants based on not being adequately prepared (which is what SHOULD happen, but isn't going to).
For the on-line situation there is yet another dynamic and that is that people routinely try to take on something starting at Level 5 when they should be starting out at Level 2 at best. There's nothing to stop them from doing so and they are usually completely unqualified to gauge where they should be starting out in the first place. I suspect most of us have been guilty of that from time to time -- I know I have.
I don't know if the OP is going to come back or not since it should be apparent that we aren't going to just work their homework for them. But hopefully they will make some effort at a start so that we can help out.
"Even the students that do care about truly learning things are caught in a bind because, at the end of the day, they only have so much time and working the problems that affect their grade takes higher priority than really learning the material. So unless the problems are tightly aligned with learning the fundamentals, that part is going to slip."
All too true again, and it makes me kind of sad to see this kind of thing. I often found myself wanting to get into some deeper exploration of the subject matter with some students but they almost always complain that they simply do not have the time. I believe that steals from them some of the beauty and joy of theory and even application.
It's always, "Big test coming up on Monday, have to study.", but never, "Big test coming up on Monday, have to learn and understand."
My remarks in post #5 was primarily aimed at students who have been examined countless times on the fundamentals associated with maths where that of which would achieve entry level merit so as to progress on to university and study an engineering discipline. Yes, of course, the beauty of Laplace transforms is to reduce the problem to an easily digestible algebraic one. But there is no excuse to be experiencing any difficulty with high-school/college algebra (no matter what your discipline) when you're learning advanced techniques; it's futile.
As an engineering student, I've had the opposite experience. Instructors are at the mercy of the curriculum, which usually forces a tight lecture schedule, making completion of the required material the overriding priority. There simply is no time for depth. Generally, even with motivated students, our ability to absorb and explore the material is limited by the rate at which new material is presented. In particularly dense courses, neither instructor nor student has any "free time", so to speak. For example, Calculus II -- a class in which I was genuinely interested -- felt like a whirlwind of topics and techniques; there was no possible way we could have explored the significance or beauty of even one of the topics, let alone the subject as a whole.
It's frustrating, and I blame the academic system. It's a peculiar notion that all undergrad degrees should take roughly the same amount of time to complete. To this end, departments devise roadway-like curricula with the courses serving as sign posts, and the destination being a graduate (and not necessarily an engineer). Professors are hamstrung by this approach because their primary responsibility is not to educate, but to get the students to the next sign post. Students, in turn, are trained to view the process as a means to the destination. Worse still, the universal criterion by which our journey is judged is a letter grade at each sign post, which further encourages study techniques that tend to short-circuit the learning process.
For better or worse, students adapt to the conditions of their situation. If our algebra skills suck entering college, it is because our algebra education sucked in high school. If, upon hiring us after university, you find that we don't have the foggiest idea of what we're doing, blame the system. Some time ago I was having a conversation with a friend who is a working engineer and ABD in a PhD EE program. I asked him at what point in his academic career did he start getting "real" courses, where the coursework involved real depth and learning. He sighed and said that there was no such thing, that as an engineering student the burden is squarely on me to go deeper. My experience certainly agrees with this assessment. And that's fine; good engineers work hard to become good engineers. But the realization, from a student's perspective, puts the entire academic rigmarole in quite a different light.
While I agree with much of what you've said, there is plenty of blame to go around and the student has to shoulder their share of it. When I teach I always have Help Sessions weekly and my policy is simple -- I will stay until the last student is ready to leave. On occasion this has meant leaving at 4am. I even had this policy when I was running mentoring sessions for Physics II as an undergrad. At least then I was getting paid for the scheduled two hours whereas my Help Sessions now are completely on my own time and dime. The result is completely predictable each semester -- tons of students come to the first one but as soon as they discover that I won't just show them how to work the homework problems (though I often work similar problems as examples) and that want them to put forth the effort and that I will work with them, even sitting right next to them guiding them from step to step, they don't come back. Each semester there will usually be two or three students that come every week and work on their stuff asking questions as needed and those students often stay well past the official end time. But that's two or three students out of a total enrollment in my courses of typically 120 or more. So even when in instructor is willing to make the time available, only a tiny fraction of students are willing to take advantage of it.
I also disagree with your friend, at least based on personal experience. I have had LOTS of courses that involved depth and real learning both at the graduate and undergraduate level. Deep enough and with enough learning that I was able to design simple electronic systems on spec for a few customers and was able to design and build two research grade test systems for NIST as an undergraduate while a co-op student and also designed and built a support system for them as an independent contractor while a graduate student. But I do believe that I was lucky in both high school and college in that, both times, I went to quality schools that had quality faculty that held students to reasonably high expectations. In all of my high school and undergraduate education I think I could only say that I had two or three instructors that I found it very difficult to learn from. But I worked with students (from other schools when I was their peers and more generally later on) and so I recognize that this may be the exception rather than the rule. I really saw it at a for-profit university that I taught at for one semester -- their students were horribly unprepared for the course I was supposed to be teaching them and so I set about bring them up to speed. The graduate course in particular was responding well and getting quite involved when someone from the undergraduate course complained that I was expecting too much from them and so I was ordered to abandon my efforts and stick to the junk curriculum. I concluded that they didn't care about the students actually learning anything, they merely wanted to string them along for as long as possible paying nearly twice the tuition as at a real university. I couldn't get out of there quick enough.
(I love hearing about dedicated teachers; kudos for that!)
I'm certainly not arguing that students are exempt from any blame. It may have been a while since you were on the other side of the desk, so I offer a friendly reminder: as frustrating as bad students can be for teachers, they are also frustrating for their fellow students. Fortunately, demanding coursework usually weeds out the laziest, so that by the third year of, say, an engineering undergrad degree, it's fair to say that those remaining are willing, motivated students. And at this point, if a professor's generous Help Sessions are lightly attended, I would not suspect laziness as the cause. In my experience, Help Sessions are useful for students that are struggling with a specific technique or concept from the lecture. The expectation is that one attends such things to get help working through a few relevant problems, which is truly valuable if a) the student really needs the help and, b) if the student has the time. By this point in their academic careers, students are swamped with difficult coursework and must employ academic triage to best manage their time. The dynamics of the situation make it self-selecting that only a few will attend: The top-tier students won't need the extra help; they do it on their own. Those in the middle-tier could be A students in your class if they took advantage of these extra work sessions, but they may need to allocate their time to another class (in which they're bottom-tier). That leaves only the students who are genuinely lost, which should be precious few.
In any case, I obviously cannot speak for your experience, which may well be the more prevalent scenario.
You are very fortunate to have had such an excellent education! Was it luck, or an institutional thing -- you found a rare academic culture that truly fosters education -- or perhaps it was a generational thing? While I've had good teachers (bless them each and every one!), most of my educational experience seems to be the opposite of yours. I would like to think that my friend and I are the exception, that great education is the norm, but it is difficult to believe. All I have to do is think about how we teach math (at least in the U.S.) and I become depressed: from primary school through undergrad, it's all techniques and zero mathematics.
The dynamics are certainly complex, there's no doubt about that and no simplistic answer can truly sum it up. What I've noticed time and again is that the students that come every week tend to be the among the top tier students. They almost certainly would have done just fine had they not come, but they wanted to learn as much as they could. The rest were certainly a mixed bag but the ones that really, really should have been coming every single week tended to only come late in the semester (other than the initial rush in the first week or two) just before a big project was due and after they had already pretty much sealed their fate.
I think it was a combination of all three. I graduated from high school in 1983 and got to observe some very interesting dynamics in my school district first hand (and also in retrospect). At the time that I was in elementary school there was a fad that had started a decade or so earlier called "open concept" which was a mish-mash menagerie of muddle-brained policies. The basic idea, in practice, seemed to center around letting students explore on their own and learn at their own pace and that rigid settings only inhibited the learning process. So schools were built without walls between classroom (just things like bookshelves or other furniture to divide class areas) and students were free to come and go as they wished between their class area and the library and the playground. Not surprisingly, a LOT of kinds spent all their time on the playground. But this was considered just fine since obviously that was how these kids were most productively exploring their world. The result was that when I was in kindergarten, which was nothing but playtime, and my sister was in fifth grade she could barely read and the expectation for me (my dad and I ran across the documents a couple decades later in stuff my mom had kept) was that I be able to recite the entire alphabet by the end of second grade! My parents saw what that was doing to my sister so we moved specifically to get us out of that district in the middle of my kindergarten year. The school I ended up in was almost no playtime and these kids were reading and writing so I had to play catch-up big time. In the middle of my first grade year they built a new elementary school a couple blocks from my house and I was supposed to go their the following year, but it was open-concept and my dad flat out told the school board that that wasn't going to happen and that we would move again if we had to. The school board gave us a waiver (school boundaries were quite rigid back then) so I stayed at my traditional (closed-concept) school. When I went to junior high there were about six or eight elementary schools feeding my junior high and mine was the only one that was closed concept. My high school was physically open-concept but by then the fad was fading and so the way in which classes were taught was closed-concept. We had about five junior highs feeding us but the only path that a student could go from K-12 and be closed concept the entire way was from my elementary school. When I graduated something like 2/3 of the honor students were from my elementary school. The notable exception was our valedictorian who went to my junior high but an open-concept elementary. I was talking to her dad one night and we got to discussing that and I asked how she (and her younger brother) was able to get such a good education at that school and he said that every night when they came home he and his wife took it upon themselves to teach them the things they should have learned in school that day.
So I was fortunate (with a big dose of intent on my father's part) to be in -- and be kept in, thanks to my dad -- a good school system. I was fortunate that my high school had a big emphasis on academics, unlike the other high school in my district, so I was able to get a very solid education that emphasized fundamentals. In college I was fortunate (but not by chance) to go to a well-respected school (the Colorado School of Mines) and, even more so, fortunate that I chose to major in Engineering Physics. The culture in the physics department very much stressed fundamentals and problem solving and all of the students were very serious and competitive and motivated to learn, not just to get a degree. It prepared me very well to take engineering courses and excel in them.
Because I was four years older than most of my peers (thanks to a stint in the military and time spent as a co-op student at NBS/NIST) I got to observe firsthand the effects of the calculator fad when virtually none of the "traditional" students could write a program to do multi-byte multiplication and division while every one of the handful of older students had no problems at all. The reason turned out to be simple -- almost none of the students four years behind me had ever been taught how to do multiplication and division! They were in the age range that four function calculators had just gotten cheap enough for them to be required to use them from first grade on. I was at the tail end of the "outdated" ways of doing things so I had to learn how to use trig and log tables (had I been just two years older I would have still used slide rules in the science classes, but as it was scientific calculators had just gotten cheap enough at the time I entered high school). The end result is that I think I was lucky and caught a very good time in the transition. But for the kids a few years behind me, since this was the new fad of "putting technology" in the classroom, it was pretty much universal around the country. Since then it has been a swinging pendulum at different rates in different places, so today it is very hit and miss. I've encountered many "honors math" high school students who literally could not tell me what 9*6 was and were actually proud of it because they saw no point in being forced to memorize useless facts when they had calculators. Of course, it crippled them when it came to learning much of anything, such as factoring polynomials. And, of course, the people teaching things like factoring polynomials have to "dumb down" their content to match the students they have to work with, thus feeding the vicious cycle.
Thanks for the detailed reply. It's striking (and more than a bit worrying) how seemingly arbitrary factors, such as where and when one lives, can have such profound effects on education. It can be life-changing! But it reinforces how incredibly important it is for parents to take an active role in their kids' schooling, something they are likely to do only if they themselves received a good education.
There's certainly a correlation, but there is a lot of cultural influence as well. Different cultures view the value of education very differently (and this changes over time). Historically, for many of the immigrant surges into the U.S., the people coming here were predominantly very poor and very poorly educated. But they saw education as the key to their children succeeding in this country, even when they felt that it was "too late" for they themselves. In many of the countries they came from it was all but impossible for a poor person to get an education and, in many places, even if they did the class system was so strong that it was unlikely to make much of a difference. But, sadly, there are other cultures that have come to view education as a waste of time and effort -- frankly it's an attitude that I can't get my head around, even having seen it numerous times firsthand plus countless documentaries and the like chronicling it. I can't help but conclude that a number of factors may have conspired to create that same sense of futility in those cultures, primarily that getting a good education wouldn't actually make any difference and that their children are stuck in the same life that the parents are in.