As an undergraduate, I took five semesters of calculus. The last semester on partial differential equations supposedly gave me a head start for grad school in my chosen field of chemistry.

But, unlike my friends in engineering programs, I had little need for calculus except in a few physical-chemistry labs and classes. I can't remember using calculus in grad school, and since then I can't remember using it more than a few times to solve electronics problems. Instead of having to take all those calculus classes, two semesters would have provided sufficient knowledge. Classes in scientific data analysis, instrumentation techniques, and statistics would have made more sense, but at the time either the courses didn't exist, or no one suggested them.

I looked at the math requirements for an engineering school, and found the following requirements for a BS degree:

Calculus I and II

Ordinary Differential Equations

Discrete Mathematics

Statistics

That seems like a better series of course requirements (one semester each) for engineers than the "exposure" I got in college. My biggest impediment to learning calculus centered on the lack of a relation between it and practical problems. After the math professors got past a few simplistic examples, it was all x-this, y-that, and everything between plus and minus infinity. Practical examples of why nascent scientists and engineers might need a triple integral might have helped us understand calculus better. Or perhaps I'm just more of a practical than theoretical mind.

We now have many computer programs that can help analyze data, and that provide sophisticated mathematical tools. And using such tools takes the drudgery out of tedious mathematical analyses and equation evaluations. So, does it still make sense to load engineering and science students with more than an introduction to calculus? If so, what level of calculus should they reach? (Of course, math majors and students interested in calculus could still take more courses.)

During college, I had a dull-as-dirt calculus text I was glad to throw out. Since then, I discovered The Calculus Tutoring Book by Carol and Robert Ash. The IEEE Press published this book in the mid 1980s, and although now out of print, AbeBooks.com lists many copies at reasonable prices. I recommend this book highly to students who struggle with calculus. One summer I went through about half the book on my own, and worked many of the problems. Finally, I found calculus interesting and clear. It only took an extra 20 years to "get it."

What level of calculus should engineering students reach? Discuss in the comments section below.

Jon, I completely agree. I took only 2 semesters in college, but that was plenty.Where you were able to recall at least a few examples where you applied Calculus during your career, I look back and cannot recall even one instance.While the subject matter was valuable to broaden my overall foundation for engineering, the practical application for me was lost.My vision of applied calculus was for NASA engineers who designed missions to Mars; a far cry from my day-to-day in electronics products.

I remember watching Tom Hanks playing Jim Lovell in Apollo-13 and recall the scene where he was busily scratching out equations that may well have saved his life – I thank God that my life never depended on my calculus skills, or I wouldn't be writing this today!

When I was working as an engineer, one of the engineers in our group needed to calculate the area under a curve. He asked for help, and pretty soon there were six of us around his desk, arguing about how best to calculate the area. The point is, calculating the area under a curve is one of the most basic calculus tasks, and most of the engineers in my group (who were all out of college for a few years) couldn't remember exactly how to do it -- at least for this particular problem. I took three semesters of calculus, one semester of differential equations and one semester of vector calculus, and never used it, at least not directly. I know that it's important for engineers to understand how equations are derived, and calculus is often part of that understanding, but it really does't have much bearing on how good an engineer is.

I remember a physics lab in which my partner and I created a strip-chart recording that plotted a variable of some sort. Then we had to find the area under the curve. Because this was experimental data and not a function we could describe with an equation, we carefully cut out the "area" under the curve and weighed it on an analytical-type scale. Then we weighed a 10-by-10-cm piece of the same chart paper. After that, the "integration" was simple math.

Ebay has planimeters for sale from time to time, so one of those would do an integration of experimental data, too, but I don't know the accuracy of the results. Wikipedia has a short discussion about planimeters. Of course it uses integrals and partial-differential equations, so the underlying calculus is never far away.

I remember a physics lab in which my partner and I created a strip-chart recording that plotted a variable of some sort. Then we had to find the area under the curve. Because this was experimental data and not a function we could describe with an equation, we carefully cut out the "area" under the curve and weighed it on an analytical-type scale. Then we weighed a 10-by-10-cm piece of the same chart paper. After that, the "integration" was simple math.

Well, you could have taken your raw data points and done a Newton-Raphson summation, instead. Just assume the area is an array of rectangles of width equal to the spacing between samples of the independent variable and height equal to the data point.

I did the same thing in a biology class ages ago, where part of the assignment was to measure the area of a leaf. Weigh the leaf, weigh a precisely cut area ( or better, various sample square areas) from the leaf, do the math. Surprisingly, the instructor was astounded. He expected people to lay graph paper over the leaf and count squares.
There is a similar story of Edison showing one of his mathematician employees how to measure the volume of a light bulb. Instead of approximations and integrals, he filled an empty bulb with water.
The last industrial use I know of a planimeter was in a leather shop, measuring the before and after area of hides after various processing steps. The more calculus oriented versions were ball and disk integrators used in some mechanical analog computers.

That being said, learning the Calculus is a great thing for engineering students--even if it is never used, even if it is hard. How far to go is anyone's guess. But exactly how far up the mathematics ladder you choose to go determines your future possibilities. Period.

I'll bet many engineers haven't used much Trigonometry since they found CAD. And I know I haven't even done long division or extracted a single nth-root either. But that's not a reason to avoid learning it.

My high school sophmore son just asked me this question. His older brother, who is a freshman in Aerospace Engineering, took lots. That was in high school. He took AP Calculus B/C and then, as a senior, took multivariate calculus. They call this Calc 3 in our schools. Actually, it is taught at the University of Illinois at Urbana-Champaign. So, he has a college transcript before getting into college. He could have started with differential equations, but decided to take the equivalent of Calc 3 at his university.

Is this all really necessary? I have used it many times in my career. I am not using it now, but I probably use it in some projects in the near future. It is good to have a basis in it. It gives you more flexibility in future opportunities.

Your proposed basic program is very good. I personally would have taken lots more math, but that is just me.

I don't solve differential equations on a regular basis, but that's not necessarily the point. Hopefully, after several years of struggling through calculus and differential equations, a student will come to have an intuitive understanding and appreciation of the mathematics which governs the world around us. It's that understanding, much more than the ability to solve the equations themselves, which is necessary in an engineering career. Studying math also helps develop the discipline and reasoning skills which are needed to solve engineering problems.

When I was in community college, I took three semesters of calculus, one semester of differential equations, and one semester of matrix algebra. That was the extent of the math courses which were offered, but my calculus professor, Dr. John Wegner, gave an informal class to a small group of students in which he covered some more advanced topics. As far as I know, he taught this class for free. I am grateful to him for his time and dedication.

As soon as I transferred to a four-year university, I was eager to take a more advanced math class. I chose a course in complex variables. This turned out to be a mistake; as an engineering major straight out of community college, I was not quite ready for a course usually taken by fourth-year math majors. I managed to struggle through the class with a C, but really I understood very little. It was, however, a good lesson in my own limitations. (I still have the book, and am hoping to give it another look some day).

Later, for some advanced physics classes which I took as electives, I had to learn a few things about tensor analysis and calculus of variations. I had much more luck with these subjects than I had with complex variables. Calculus of variations, in particular, is a fascinating topic, which allows you to do all kinds of things you can't do with ordinary calculus.

Without a doubt, the most important math subject which I didn't learn in college is statistics. I took a "probability and statistics for scientists and engineers" course, but the professor resented having to teach non-math majors, and as a consequence, did a horrible job. Of course, I can't put all of the blame on the professor, since it's a student's responsibilty to learn, regardless of the quality of the teacher. But I essentially had to teach myself statistics several years later, when I was unexpectedly thrust into a job as a quality manager.

Since graduating, I've also studied a number of mathematical topics such as logic, number theory, and graph theory -- not necessarily because they're particularly useful to me as an engineer, but because they're interesting, and keep my brain cells from atrophying.

How many math courses should be required in an engineering curriculum? Given all of the other material which needs to be packed into four years, it's hard to say. Certainly one outcome of an engineering education should be a solid understanding of the mathematical principles which underlie the world we live in, and calculus and differential equations are essential to that.

Might one consider all the calculus courses as indirect training for problem solving? Granted you might not use calculus directly in most work, but the learning process, and all the homework did train you to solve problems, right?

I went through the same sets of classes, but I'm not ready yet to dump them.

You're right, of course, TJ. Calculus did teach us to solve problems. Whether we use it directly as engineer is probably not the issue. I do wonder about my vector calculus course, though...

"I do wonder about my vector calculus course, though..."

LOL, I'm an electronics engineer (an analog dinosaur) and I WISH I had a better vector calc background. Not for analog design, rather for orbital dynamics.

What do analog design, orbital dynamics and vector calc have to do with each other? Normally, there is not much at the crank-the-handle analog design level.But when called on to support an attitude control system on a spacecraft, there are elements of both worlds that co-mingle. The understanding of both makes anomaly resolution 22,300 miles from the anomaly that much easier. I speak from experience.

As for usefulness of foreign languages (or grammar skills or anything else not "engineering"), aside from the snootiness factor (IMHO, Americans are language deprived compared to their European kin), look at it like this: breadth of knowledge excites different areas of the brain. Much like practical music knowledge music enhances math and science skills, learning multiple languages expands the brain, the database engineers draw on to bring practicality to science. Problem solving uses different parts of the brain -- and the parts of the brain used appear to be dependent on what the problem is.My experience with truly brilliant engineers is that the smarter they are, the more their breadth of knowledge in areas other than engineering.

Thinking out of the box requires that there is some knowledge out of the box. If not, box barriers will rarely be breached.

You've got me there, Sparks. You make a strong case for greater breadth of knowledge. I have to admit, though, I never thought I'd bump into anyone who wished they had had more vector calculus.

Whether or not one actually uses calculus heavily throughout one's career is I think irrevelant because there are two factors I'd give greater importance to. One, mathematics is one of the foundations of engineering, so it's incumbent upon people who want to be engineers to learn math. (Eye surgeons still go thru anatomy classes and learn about the heart, lungs, leg bones, etc.)

Secondly -- and this is where I have a real bug in my bonnet (to mix metaphors so I don't write anything too sour) -- if you want to call yourself an engineer, you should have to make it through engineering school, and that means taking classes which are hard to pass. The PC and the Internet has killed the little respect the profession had by enabling every idiot who's written an Excel macro (or even a line of HTML) to think that they're technical people who are "engineers."

I know we can't put the genie back in the bottle, but I think we should try to keep up some barriers to entry, at least in the "real" engineering world.

I know engineering and science majors must take difficult courses, and we all have. But I draw the line at courses that have no practical value in terms of knowledge or techniques. Years ago science and some engineering students had to take a foreign language for 2 or 3 semesters. Supposedly those courses would help the students translate scientific papers from German into English. I took two semesters of German as a sophomore and one more semester as a senior. I used my "knowledge" of German exactly once. At the time a MS in chemistry required passing a translation test, but examinees could use a technical German-English dictionary! I never used German again and can't say I feel constrained without fluency in a spoken language other than English. As far as I know, most colleges and universities have given up foreign language requirements for science and engineering students, although a few might hold on. Students would do better to learn C or LabVIEW.

I agree that, with the sheer volume of material which needs to be covered in four years, it's unrealistic to require science and engineering students to learn a foreign language. But it ought to be strongly encouraged. Being bilingual in English and Spanish has greatly enriched my life, both professionally and personally. (For one thing, I doubt that I would have married my wife, who is from El Salvador, if I didn't speak Spanish). I wish I knew French and German, since I regularly work with colleagues in Quebec and Austria. Of course, most educated people in those places speak English, but if they went to the trouble to learn my language, why shouldn't I do the same? Mandarin Chinese would be a good one to know, too. I've met a number of second-generation Americans who seem to be proud of not knowing their parents' language. This seems to me to be a foolish attitude. The more languages a person can speak, the better.

While I don't completely disagree w/ your position, I for one am very grateful at having taken all those Mathematics sequences in college. As a "pre-engineering" major at a 4-year accredited college in the early 1960s, I was enrolled as a Mathematics major, w/ a concentrated minor in Physics. Not only did I take the entire Calculus sequence, but as other respondents have mentioned, I also took the advanced Algebra sequence, including ABSTRACT & LINEAR Algebra, and NUMERICAL ANALYSIS. In addition, I also have LINEAR & PARTIAL DIFFERENTIAL EQUATIONS & COMPLEX VARIABLES to my credit, and finally, a course of PROBABILITY & STATISTICS. All these courses, taken as a whole seemlessly meshed w/ the PHYSICS curriculum. There's no way that I could have been successful in all the PHYSICS courses, had it not been for the well-grounded basics of the Mathematics program.

As an accomplished electrical/electronic engineer today, I think I'd be less than successful had I not had a good foundation in these curricula. Furthermore, I'd probably NOT have been as effective as a "radio" engineer, had it not been for the course in COMPLEX VARIABLES, given the basic need to deal w/ capacitive & inductive reactance, admittance. It was always about "a+jb"!

About the only courses that have served no useful purpose to me was AN INTRODUCTION TO ALGEBRAIC TOPOLOGY. Ironically, I still have that textbook to this day.

First, I have been a part of new product development for over 15 years, I have never had to use calculus. Force caluclations? Yes. Complex geometric interactions? Yes. Mixed materials? Yes. Etc... No calc though. I also have no issue with the classes because they do teach you stuff the layman never would understand and that expansion of thinking makes you more an engineer than the CAD jockey who gets promoted.

The issue is does Calculus classes take away from hands on product design engineering learning we all do day in and day out? Industrial designers have to make what they design in classes for 2-3 years, and is about half their classes. What amount of time does the average engineering student spend in his/her undergraduate actually designing, making, and testing their ideas? There is the issue that Calc and other foundation classes have, they take away from what the corporations would like to see from graduates (undergrad degrees), some practical experience...

I can only talk from a UK perspective, where the standard of mathematics of graduates is frequently dangerously low; I'm a Mechanical Engineer.

Much first degree work is aimed at proving a level of intellectual capability- sometimes the actual subject matter is less important. My head of school used to say that the most important thing was that he taught us how to learn.

Many good points have been made about analysis software and so on that may negate the need for the user to understand maths, but unless you know what is behind a piece of software, then you and it in combination may merely be a dangerous weapon, like the FE packages that are attached to CAD systems, in the hands of CAD jockeys and not engineers. After all, we did not stop learning arithmetic on the basis of electronic calculators being introduced.

Not everyone in Engineering will use calculus regularly- or at all- but given that we cannot specialise from day one, then everyone will have to learn a bit of everything.

At my last employer, while recruiting, I expected any Engineering candidate not to be fazed by a simple ordinary diferential equation. I'd allow some book work to brush up, but a designer, stressman, analyst of any kind, reliability engineer and so on ought to be able to appreciate calculus and in most cases to be able to do some calculus, otherwise we're just regurgitating formulae with no idea of their background and limits of application, and are then genuinely dangerous.

Even in Project Management and Quality Engineering, procedures are often drawn from theories with limits, for which calculus is critical, and it can be useful, believe me.

The question is- how far do you go- but no one should be entitled to call themselves a professional engineer without having achieved a reasonable grasp of calculus, algebra, statistics, geometry and no doubt others I've forgotten to list.

What we must do is to engage early with real applications, and show the sheer beauty of mathematics as well as its usefulness.

I took the required Calculus courses (and the required foreign language courses!) when I got an EE degree. But I went into software engineering after that and didn't take more than the required Calculus courses. I know there are a few engineers who really think in poles and zeroes and LaPlace transforms, they have a different vocabulary to think with. A bought a book on LaPlace transforms, I know we never covered what's in there. I doubt I'll ever be really comfortable in that domain. Do I NEED that? Apparently not. But I recognize that it's something I could have studied and probably should have if I wanted to pursue a career in electrical engineering, it would have been useful.

I agree with the idea that an engineer's education should contain a richer mix of mathematics. But calculus does train the mind to pursue other math disciplines, (on our own, if necessary) and I think that's one of the primary purposes it (should) serve in the undergraduate curriculum.

And don't forget that somebody had to master the numerical solution of systems of linear differential equations in order to make our CAD programs work.

One of the joys of graduate school is that you suddenly find yourself with the bakground and freedom to explore those other corners of math that you couldn't when you were deep into trying to understand thermodynamics and Maxwell's equations.

A story: I did my most interesting work with calculus after I quit doing engineering and started working on long-range planning and forecasting. You just never know when that stuff is going to be useful!

I struggled with Spanish. Looking back it may have been my poor hearing, or then again perhaps the way my brain is wired. Language, in my not so humble opinion should not be a requirement. My favorite book over the years is "Calculus Made Easy", Silvanus Thompson, released in Dec 1911. Yes, over 100 years ago! A new and updated release is now in print, and in my old book I found several errors that I have corrected in the margin. It is very easy reading. I keep my old "Calculus Problem Solver" and use it mostly as a weight when I'm gluing parts together!

Still 90% of my math needs are met with algebra and trig. Stats I have used mainly in self promotion exercises, accounting for another 7% of my needs.The remaining 3% involve infrequent use of calculus at work or for blue sky thinking.What I really would like to do is understand Maxwell's work (not Heaviside's rework) and vision. (I'm not as smart as I like to think I am.)

I also believe in no wasted knowledge.While learning about logarithms I recall thinking, that this is a nonsense effort without any practical use.Today, in the RF field, I use it daily.Of course my TI-36X beats the hell out of the old paper log tables or my slide rule.

At my school, Lwrence Institute of Technology, we had four levels of calculus plus one about differentials and matrices and a lot of that stuff, all taught by a brilliant professor who had it all in memory. The calculus was a great foundation for understanding physical reality, and it was also quite a hard class, as I recall, but not nearly as hard as my dynamics class was.

I once did watch a contract engineer calculate the exact heat sink required for a project using calculus on the waveforms of the pass transistor. I was even more impressed when the thing ran a constant ten degrees below the maximum allowable temperature limit. If we had been selling thousands of that instrument the 75 cents that he saved on the heatsink would have meant something, but for a one-off product it did not make sense.

In other areas the calculus was the foundation for setting up the equations, but the math was always just simple arithmatic, multiply and divide type of stuff.

A course in the rules associated with valid statistics would have been very handy, along with a handbook of methods. The class would need to be about the methods to assure that the results were correct. The incorrect application of statistical deviation data has caused a whole lot of problems, in my experience. The problem is that the results can not be more accurate than the noise level will allow the data to be. And data accuracy can't be better than the noise level. In other words, when you have a scope trace that is a quarter inch wide with high frequency noise, you can't expect to accurately read the center of the trace with a measurement curser.

This relates to a subject that has been bugging me: I think you need calculus just to read Wikipedia. It used to be a pretty good place to get a handle on new technology but now every page is filled with derivations and complex notations and it has become a place where you need a graduate degree in mathematics to read.

Taking four semesters of calculus was a lot of work for a skill I don't often need. Mostly, I just translate: integrate means I need an opamp and a capacitor, summation = an opapamp and a couple of resistors, differentiate = an opamp and....

I took a thirty-one week statistics class at work and the understanding and tools I got out of that are far more useful day-to-day in test & measurement.

I graduated with a BSME in 1988. I had to take calculus I-IV and a separate differential equations course. Those were offered through the Dept. of Mathematics. Then I had to take a further 8 hours of "Advanced Engineering Mathematics" which was through the engineering department. It was all calculus. That's 23 hours worth. I've used it some in my career, but not nearly as much as they crammed into my brain.

Any subject, to be absorbed by a student, requires a good text and a good teacher. I have had my share of both in every possible combination of good and bad. My basic calculus text was obviously written to impress the author's peers, not to edify the students. My professor was good, but my progress wasn't great until the public library had a 5¢ a pound book sale and I came across Elements of the Differentialand Integral Calculus, by W. A. Granville (1904)—an ancient text, but presented so clearly, logically, and without assumptions, that you HAD to absorb and understand the subject. (This book has been of material assistance to two children, two grandchildren, and a high-school-teacher friend.)

I had a course in advanced engineenring math in which the instructor was so bad that we students ignored him and just read the fine text.

How much calculus? I believe that this should depend on you course of study. The minimum should be basic differential and integral calculus and differential equations. After that, it would depend on whether you're studying Chemical, Civil, Mechanical or other Engineering, sciences, etc.

Languages? The more you can. I think that these days, Mandarin Chinese is an essential. Have you noticed that everything seems to be made in China? Learning a language has the side benefit of exposing one to that language's culture and mental attitudes—a necessity if you're going to do business with those people.

All of this is aside from expanding your knowledge and mental flexibility. Isaac Asimov once said that the breadth of your knowledge is directly related to your creativity.

You certainly convinced me – I just ordered Elements of the Differentialand Integral Calculus, by W. A. Granville on amazon.com. My teenage son is entering the math, engineering, technology science academy for high school next year so hopefully this will help him out...Thanks for the tip!

This is slightly off-topic, but I was recently reminded of something which happened when I was in college, in the first week of my differential equations class. One of my classmates had apparently managed to get through two semesters of calculus without learning any actual calculus. I'm not quite sure how he was able to do this, but he had a TI-89, and used it extremely proficiently. However, he had absolutely no grasp of any of the underlying math concepts. He literally only knew how to punch things into his calculator.

In the first week of class, the professor announced that there would be no calculators allowed on any of the tests. My classmate panicked. The look of horror on his face was visceral. At the end of the class, after the professor walked out, my classmate sat slumped at his desk in a state of shock. He never showed up to class again.

I never saw him in any of my other classes, so I strongly doubt that he stayed in engineering. Reflecting on the fact that he basically wasted a year of his life and presumably had to completely change his career goals made me feel better about having nothing but a cheap solar calculator. I also wondered whether things would have turned out differently for him if he would have put the same effort into learning calculus which he obviously put into learning how to use the TI-89.

I hope I don't sound like I'm ridiculing this guy, because I actually feel sorry for him. I strongly suspect that his overreliance on his calculator started in high school. One of his high school teachers should have identified this problem and helped him with it. By not doing so, they set him up for failure.

(Of course, I'm sure that some people reading this will argue that knowing how to solve calculus problems on a TI-89 is all the calculus you really need to know anyway).

1) Jon- I, too, took two semesters of SCIENTIFIC GERMAN in undergraduate school. It was a course designed to aid students in reading technical literature written in German. Since I was the product of German immigrant parents, those courses were an easy A+. My first vehicle was a 1960 (GERMAN) FORD (TAUNUS) which my aunt & uncle brought back from an extensive trip to Europe in 1959. They unknowingly left the new car brochure under the driver's seat. When I got the vehicle and was cleaning it one day, I saw this pamphlet, and kept it for nostalgia purposes. When my German class professor gave us a course project, I showed him the brochure, and told him I would translate it into English. On the day the projets were due, I stood in the class, and read the entire brochure without missing a beat. After class, he said to me, "don't bother coming back here. You don't need my help." Not only did I get an A for that semester, but for the sequel course as well. Too bad that didn't also happen for E&M & a few other "toughies"!!!!

2) In the 1960s and beyond, just about every desk in an I-B-M office wherever you went there was a simple fixture on it. It was a shaped like a name plate, but all it said was "THINK". Surely a timeless admonition!!

3) Dave Palmer - I had the SAME problem in many of my Mathematics courses. It never seemed to fail, no matter how cautious I was that the batteries in my K&E slide rule ALWAYS failed the night before a final exam, and there were no stores open @ 2 in the morning to get replacement batteries. What a drag!!!!!

Ah, the sliderule, my constant companion until about 1973 when I bought a Commodore scientific calculator. I love sliderules and have several boxes of them awaiting sorting, as well as three in my desk. My senior year in high school, Mom and Dad gave me a Post chemical-engineering slide rule (Model 1491) with temperature-conversion scales, atomic weights of elements, gas-pressure scales, and other scales I never learned to use. What a wonderful tool.

I only have two..... my trusty K&E full-featured rule, and a no-name cheap one that only has the A,B,C,D scales printed on it. It was accurate enough for tests, and since it was so inexpensive, I didn't worry if it was stolen at school. I haven't used it for years, but I don't think I can get batteries for it any more..... used two PX-625 MERCURY batteries. (Ha! Ha!)

My first "electronic" slide rule was the TI SR-50 calculator. Was betwixt & between whether to buy brand new H-P 35 "scientific" calculator w/ its RPN entry method, OR stick w/ the more common TI model. Still have the TI, but ir doesn't work because the non-replaceable battery pack is kaput.

When I die, the world is gonna have tons of fun transporting all these relics into the Museum of Ancient History.......

I had 3 semesters of calc and one of differential equations. The biggest hole was the fact that I didn't have any statistics (at least until I went for my MBA). I think the class list in this article does seem to be more practical.

Because the program that I was enrolled in at the time was a cooperative program (early 1960) w/ the University of Dearborn (a Jesuit school), my formal major was listed as Mathematics w/ a Physics concentration. The parenthetical major was "Pre-Engineering". So, due to the Mathematics concentration, the curriculum included a sequence of PROBABILITY & STATISTICS. Two courses of each, but they were presented from a pure Mathematics point of view, so they left little desire in most to pursue a more practical understanding in daily design & analysis in the "real world" atmosphere.

@OLD_CURMUDGEON: My probability and statistics course had the same issue, forty years later. The course I took was called "probability and statistics for scientists and engineers," so it was ostensibly focused on practical applications. In reality, it was just a condensed version of a two-semester sequence in the math department. It was taught by a math professor who was probably an excellent statistician, but who resented having to teach what he regarded as a dumbed-down course for non-math majors. We talked a lot about the central limit theorem (which admittedly is important), but very little about how to actually apply statistics to solve real-world problems. I think an applied statistics course taught by someone from the industrial engineering department would have been much more valuable.

We had the same professor for the entire sequence of P&S. He seemed to rush through the text & core of the class, assigning a lot of "self-study-solve" problems. In the 2nd half of the course, at one class meeting he mentioned a book titles, INFERENCE & DISPUTED AUTHORSHIP, THE FEDERALIST PAPERS. From that moment on, we knew we were doomed!!!! The rest of the semester was devoted to the mathematical analysis of the text of Alexander Hamilton's treatise, THE FEDERALIST PAPERS. Some of us in the class pondered whether the professor was somehow misplaced..... that he should have been assigned to the HISTORY DEPT. instead of the Mathematics Dept. Although I did well in the class (as a grade point statistic), I learned little of practical value. The only saving grace is that in my career, P&S has never been an issue of concern. And, unfortunately for me, I was to meet up with him again in senior year in a course titled, MODERN ANALYSIS, with an accompanying text about 600 pages, which seemed to be a compendium of mathematical oddities & questions that no one really wanted to investigate. In fact the ramblings in class were so far off topic that most of the 20 students in the class didn't attend on a regular basis, a heretical concept in the early 1960s, when attendance in class was part of the "college experience." The final exam was 10 questions taken from the end of various chapters from that book. We were given the question numbers about 2 weeks before the official end of the semester.

However, except for these examples & one or two others, I'd go back to the same colleges, and the same curriculum again, given that I had a "time machine" to do so. I thoroughly enjoyed the learning experience from the first day I bunked in the dorm until the graduation ceremonies.

I once asked an engineering professor if any school had ever tried teaching on an "as-needed" basis, where you'd start with the things you wanted to learn and then move backwards into pure science and calculus classes as needed. For example, if I want to learn how to design a bridge, I would start with basic bridge design, then learn calculus and structural mechanics when the complex analysis finally called for it. To my surprise, the answer was , "Yes, we tried it." The school was Illinois Institute of Technology. They reportedly had several students go on for Ph.D.s after their undergrad coursework. But the program didn't work for many reasons, mainly because it was too hard to manage and too many students didn't really know what they wanted to major in, so they didn't know where to begin. My point is, that's the way we all learn in life. I think if I could have learned that way, the application of calculus would have made more sense to me.

My first ever lab session in University was about fourier transform on image smoothing duirng 2nd year first semester. But I was supposed to take signal processing course on next semester. It was the same for my classmate and our instructor had one of the busiest days. I had no idea what is going on with this lab. But when I reach final year, doing some mini project about computer vision applications, then only my first lab session is crystal clear.

I like the idea of students taking their major first and then learn backward for necessary subjects. It is more precise and efficient. I think it is impractical in this day due to insufficient data or information about current technology, job perspective and economic.

Suppose you wanted to create a FIR filter with your own requirements. How would you find the necessary coefficients, and how many of them would you need?

Switched-capacitor filters have a few disadvantages. They exhibit greater sensitivity to noise than their op-amp-based filter siblings, and they have low-amplitude clock-signal artifacts -- clock feedthrough -- on their outputs.

The Machinist Calc Pro computes speeds and feed rates for milling, turning, and drilling: cutting speed, spindle speed, feed rate (inches/minute), cutting feed, etc.

Focus on Fundamentals consists of 45-minute on-line classes that cover a host of technologies.
You learn without leaving the comfort of your desk. All classes are taught by subject-matter experts and all are archived.
So if you can't attend live, attend at your convenience.

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