Homework help?

[@rt]

Hello,
Is there an appropriate place for homework help?
Which is early Maths for engineering (where “help” doesn’t mean cheating or asking for answers).
Cheers

[@rt]

I’ll go ahead since I have NEVER done this, and can show my work.
If any problem, just delete.
Basically, I don’t know why my answer is the same as the question.


Write the equation for a straight line with the following gradient and y intercept:
m=2, c=5.

slope intercept form: y = mx + c
y = 2x + 5.

pick some values for x: -1, 0, 1.
2* -1 + 5 = 3
2 * 0 + 5 = 5
2 * 1 + 5 = 7

plot1: x = -1, y = 3.
plot2: x =  0, y = 5.
plot3: x =  1, y = 7.

draw the graph when you get here.



gradient = rise/run = 2/1 = 2.
y intercept = 5.

y = 2x + 5 … (which is also the question)

[Kremmen]

Well. The slope is the coefficient of x and the intercept at x=0 is the constant part so your answer is the only one fitting the premises.

[Neganur]

Well, if the task was to "Write the equation for a straight line with the following gradient and y intercept[...]"

Then you have answered the problem. I don't see what else is required except drawing the graph, which you also did. What level of education is this?


I'd also not call it a gradient when referring to a line, clearly what they mean is the slope. But that's just me getting caught up in semantics.
(gradient is a vector in that it has both magnitude(=slope) and direction, it points towards the steepest increase)

PS: "slope intercept form: y = mx = c" is a typo yes? You meant to write y = mx + c ?

[Ampera]

1, traditionally, or at least in my experience, Y-intercept is given the variable name b (The generic form is y=mx+b).

2, I would have gotten serious points off for a graph that looked like that. If you can't use graphing paper, write the axis, labeling it and using arrows to express an infinite plane, then put perpendicular ticks for your units. Plot points by placing them in open space instead of making this weird graph like thing that just looks confusing.

Besides that, this is all correct. This is a very basic graphical Algebra problem, to answer a previous question.

[IanB]

Basically, I don’t know why my answer is the same as the question.

As others have said, if you have a straight line with the equation y = mx + b then m is the slope of the line and b is the y-intercept.

What the question is asking you to do is to plot a line with the equation y = 2x + 5 and then verify that the slope (rise/run) is indeed equal to 2, and that the y-intercept (where the line crosses when x = 0) is indeed equal to 5.

The reason the answer is the same as the question is they are asking you to verify what you have been told by doing an experiment. This is crucial to critical thinking: do not simply take what you are told on trust, but rather try to verify for yourself to check that what you are told is accurate.

[madires]

Maybe is just about the visualization. BTW, I'm used to call that a function, i.e. f(x). In a more formal way the y intercept would be written as:
   f(0) = 2 * 0 + 5 = 5
And the slope (first order derivative) would be:
   f'(x) = 2

[Ampera]

Maybe is just about the visualization. BTW, I'm used to call that a function, i.e. f(x). In a more formal way the y intercept would be written as:
   f(0) = 2 * 0 + 5 = 5
And the slope (first order derivative) would be:
   f'(x) = 2

That, however, is far beyond this level of basic Algebra. In the US we don't normally teach functions until much later, never mind derivatives.

[@rt]

Hi,
It’s the end of my first week, but this is jumping ahead to the end of my first quarter.
I think it’s about equivalent High school Maths B (or Maths II in older days).

The question is literally this:
"Write the equation for a straight line with the following gradient and y intercept m=2, c=5”.
How is that asking me to plot the graph?

The plot is my own homework scribble.. this doesn’t get submitted or marked.

Yes, I meant y = mx + c.

[Neganur]

How is that asking me to plot the graph?

"draw the graph when you get here."

[@rt]

No, the question is posted in my post directly above.... all of the rest in my first post is my own work, including where I wrote “draw the graph here”.

[Ampera]

Then they just need the equation you already made. I think the problem is that it's a bit too simple of a question as it's just plugging things in.

[IanB]

The question is literally this:
"Write the equation for a straight line with the following gradient and y intercept m=2, c=5”.
How is that asking me to plot the graph?

It's not. But you made it seem like it was. So:

Question: Write the equation for a straight line with the following gradient and y intercept: m=2, c=5.

Answer: The equation is: y = 2x + 5

That is it. Please explain what you are not sure about?

But is that literally the question? I think the question should have been: "Write the equation for a straight line with a slope of 2 and a y intercept of 5."

When posting questions, be very careful not to paraphrase them or reinterpret them in your own words. Write them exactly as given. Precision is important.

I don't know if you changed the wording of the question here, but if you didn't then the question given to you was a poorly written question.

[@rt]

Ok, back in 2 mins.

[@rt]

I’d post the image except Google have already canned my hosting again.
I did change it a bit to eliminate part b, since I’d only do the same thing again.
The quotes enclose the exact question, and I’m dealing only with part a.

“
5. Write equations for straight lines with the following gradients and y intercepts.

a. m = 2, c = 5
b. m = -1, c = 0.5
“
If it’s as easy as you say it is, isn’t that stupid?

[IanB]

If it’s as easy as you say it is, isn’t that stupid?

Sometimes questions like this are just for revision, to make sure people are following along, and sometimes to set things up for questions that follow.

Also, "easy" is in the eye of the beholder. Some students in 8th or 9th grade may struggle with such concepts.

[@rt]

This is a tertiary preperation course conducted at a University.
I’ve got a year of this for not taking Maths B in high school.
That’s dissapointing, but thanks.

[IanB]

This is a tertiary preperation course conducted at a University.
I’ve got a year of this for not taking Maths B in high school.
That’s dissapointing, but thanks.

You don't have a location shown. May I ask which country you are in?

My sense is that straight lines, slopes and intercepts are usually covered around age 14 or 15 in high school. It seems you might have missed out on a lot of high school math if you are catching up on it now.

[Ampera]

This stuff was technically covered in my College Algebra and Trig course, but only in a cursory form, and we moved onto more advanced stuff, before I swiftly moved onto precalc which was an online course and the greatest waste of my academic time I have ever experienced. Calc 1&2 were really good though, doing stats now as a easy one-off, before moving onto full time education including physics, general sciences, and likely Calc 3.

If you don't know how to do this, and you've just hit uni, I'd say you're in for a bit of a crash course here, as I learned this stuff when I was ~11, maybe 12. (I started my first college course, which was College Algebra and Trig when I just about turned 14, and it was largely review).

[mtdoc]

My oldest son is 11 and in the 6th grade and they have covered this.  Granted, he is in an advanced program, but I think in the US this is something usually covered in 7th or 8th grade (12-14 year olds).

[@rt]

I’m in Australia. The degree I want requires English and Maths B high school pass. I did Physics and some easier Maths subject.

Some of this stuff now is as simple as applying the order of operations and stuff like that. I don’t have any choice but to do it because this will give me the equivalent high school pass for Maths B.

[rstofer]

I’d post the image except Google have already canned my hosting again.
I did change it a bit to eliminate part b, since I’d only do the same thing again.
The quotes enclose the exact question, and I’m dealing only with part a.

“
5. Write equations for straight lines with the following gradients and y intercepts.

a. m = 2, c = 5
b. m = -1, c = 0.5
“
If it’s as easy as you say it is, isn’t that stupid?

Well, the negative slope throws a slight wrinkle in the equation and in the plot.

At least the equations are linear.  That will change and when it does, sketching a graph can become something of a PITA.

Graphing:  www.desmos.com
Calculating: www.symbolab.com

I'm using these every day with my grandson while he works through Calculus I.  It is ALWAYS helpful to have a picture.  And symbolab gives the complete solution, not just the answer.  There are times when a complete solution is educational.

As you get farther along, you will want to use some kind of math program.  Right now, I am fond of MATLAB and wxMaxima.  There are free alternatives to MATLAB but given the reduced price for students, it's a handy thing to have.  Besides, in the world outside academia, math problems aren't solved by hand.

[IanB]

At least the equations are linear.  That will change and when it does, sketching a graph can become something of a PITA.

Graphing:  www.desmos.com
Calculating: www.symbolab.com

I'm using these every day with my grandson while he works through Calculus I.  It is ALWAYS helpful to have a picture.  And symbolab gives the complete solution, not just the answer.  There are times when a complete solution is educational.

As you get farther along, you will want to use some kind of math program.  Right now, I am fond of MATLAB and wxMaxima.  There are free alternatives to MATLAB but given the reduced price for students, it's a handy thing to have.  Besides, in the world outside academia, math problems aren't solved by hand.

Although I recognize the appeal of computer algebra and graphing tools, I am not sure they are good for students.

Students went through their studies for centuries without the aid of such tools. The disadvantage: they had to think a bit harder. The advantage: they became mental athletes with minds that were fitter and better trained than people that did the mental equivalent of riding around in cars and taking the elevator instead of the stairs.

I never had the advantage of graphical calculators or tools like MATLAB when I went through my studies, and I never missed them because they didn't exist yet. Use MATLAB in your professional life by all means, but don't use it while you are learning. Learn how to sketch a graph by hand so you can intuit the shape it is supposed to have. Intuition is a massively valuable tool to have for the practicing engineer.

When you are studying you want to be doing the mental equivalent of going to the gym. Always remember, no pain, no gain.

[@rt]

Nothing I’ve done so far is any more than a line or two of C.
Unfortunately, although I could show understanding of the problems, it needs to be done on paper with non-programmable sci calculator.

Besides, in the world outside academia, math problems aren't solved by hand.

[Ampera]

Personally, I think computerized mathematic tools aren't a bad thing so long as your class doesn't let you get lazy with it.

In my Algebra and Trig class, (which required a TI-83/84 series, but I used an HP-48G moving onto an HP-50G because screw TI and go HP) calculators of course were useful for number tables and basic plots, but as long as it wasn't a CASulator, all I could do was bang out of a few numbers really fast, which definitely helped speed up the learning process. Once Calc 1/2 hit, I upgraded to an HP Prime, and yes, while the 50G is a CASulator, it wasn't something I used in any extent until pre-calc which was just a really shitty online class.

That HP Prime, let me tell you, because of the rules of the class allowing you to use cheat sheets on the test full of whatever examples or generics you want, I could fill that baby up with as many notes as I needed. The CAS functionality REALLY helped me get through the often very thick Algebra that I knew how to do anyways, to quickly check the answers of my problems. Of course there was the possibility of human error, but the HP Prime didn't give you any steps. You either had to give it some, or have everything go in one fell swoop, which only goes so far.

While the other students had TI calculators which cost about as much as my fancy HP Prime (was bought for around 110 USD), I could bang out the answers to entire maths problems in moments. Now, you have to understand in a modern maths class, the answer is 120% worthless. If you only give the answer to a problem, you will be lucky to get 50% on the test. As stated, the HP Prime just lets you check yourself, make sure that you didn't make a stupid mistake on paper. I still had to compute, evaluate, and even graph by hand, using the same tricks and techniques I would use otherwise. It was a GREAT learning tool and assistant, something that I wouldn't have done as well in the class without, not because it helped me cheat, but because it helped me learn.

Today, the HP Prime is the best calculating tool. It even gives PC CAS engines a run for it's money with Xcas being a PC implementation of the same CAS library. However, without RPN, which is the second best feature by a close margin of the HP Prime, seriously RPN is awesome, I always keep an HP Prime emulator on my desktop to use for quick maths.

[@rt]

If you can store any notes in those, they probably aren’t allowed in any exam in my country.
I had planned to buy a TI graphing calculator just for this course, but it has to be even simpler than that.
You can’t take a smartphone or anything like that either (just because it has a calculator App).

For my own stuff I’d just take a 2D graphics library and do small graphic programs for each of the problems.

[ferdieCX]

I teach an introductory electronics course at the local technical university. It is for students that have only made the traditional high school, called "liceo", instead of doing the technical one. All of them are at least 18.
They have learned this stuff when they where 14, but they seem to have just learned how to automatically solve the exercises, without really thinking what was going on. They seem not to understand, that they also can graphically solve a system with two linear equations. 
I have to re-explain it, so they can understand the meaning of the characteristics curves of a FET or BJT, and that graphically finding the working point is just the solution of an equation system, although one of them is not linear.
If it where up to me, no-one under 14 should be allowed to touch a calculator and no-one under 18 a computer.
P.D. I have worked 19 years in computer field-service, 12 of them repairing boards (DEC and Olivetti).

[Neganur]

I would not necessarily blame that on the students alone

[ferdieCX]

I would not necessarily blame that on the students alone

I don't blame the students at all, I blame only our educational system.
The "liceo" was supposedly conceived as a preparation for the university, but it is really only a conglomerate of almost unrelated subjects. The students don't learn, what are these subjects useful for.
Even worse is, that in the last 23 years, our educational authorities have been trying to lower the level of our secondary technical education, to make it more like the "liceo".

You are from Finland, so you have one of the best educational systems in the world. Congratulations

[hans]

Graphing:  www.desmos.com
Calculating: www.symbolab.com

And everything else: wolframalpha.com
My university also hands out student licenses for Mathematica as well.

I think it requires some discipline not to abuse the tools given. During Calculus classes I used WolframAlpha a lot to check my answers. I'm not going to earn any points or 'mechanical skills' with WolframAlpha. The answer is not the solution, the intermediate steps are. In later cases numbers will be stripped away anyway, and just the relationships between parameters is more important than getting a numerical result.

Mathematica is a step up in programmability, which is useful for solving e.g. a set of equations in mesh current analysis (linear algebra and matrices). In that course the professor really didn't care which tools I used. As long as the answers are sane and accurately represent the phenomena in the circuit (compared with simulator).

In other courses these tools can also assist if a problem is sanely solvable. Sometimes the intention of a question is not completely clear, e.g. do I need to rewrite this formula or not? Wait, is that even sanely possible? Putting it into Wolfram to quickly check is a quick answer to last question. If it is not, I just saved 2 hours massaging an equation, just in order to come to the same conclusion. If it is, then I need to spend 2 hours massaging that formula.

Personally I don't see this as "lazy" but rather trying to be pragmatic and efficient with time. If you look at many engineering classes, in so many cases we will derive solutions by bounds or removing irrelevant terms to make equations solvable by hand.

But this is just my experience at university on a graduate level. I think undergrad and secondary school math courses will be way more focused in creating good foundation of hand solving problems. E.g. my Calculus course was full of integration by parts, triple integrals and Stokes' theorem, which I haven't done by hand ever since, but if it comes across it should be no problem to solve it.

---

Also regarding coming across this level of math at a later age: don't worry. I haven't solved my first equation till I was like 17, because I was misplaced in my first years of secondary education. I've had class mates in my university premaster that didn't have any maths since secondary school (their bachelor computer science programme was advertised as being "math free"). We both passed the premaster with good grades and no resits, given hard disciplined work of daily training.

[ferdieCX]

 If you look at many engineering classes, in so many cases we will derive solutions by bounds or removing irrelevant terms to make equations solvable by hand.

Reducing the solutions by removing irrelevant terms, helps also to understand what is the principal element that makes something work or not.  With today computing resources, we can calculate a theoretically exact solution, but doing just that obscures the fundamental inner works of a system.
IMHO, Octave and similar programs are very useful, but they are just a calculation help for complex problems.

[rstofer]

Although I recognize the appeal of computer algebra and graphing tools, I am not sure they are good for students.

Students went through their studies for centuries without the aid of such tools. The disadvantage: they had to think a bit harder. The advantage: they became mental athletes with minds that were fitter and better trained than people that did the mental equivalent of riding around in cars and taking the elevator instead of the stairs.

I never had the advantage of graphical calculators or tools like MATLAB when I went through my studies, and I never missed them because they didn't exist yet. Use MATLAB in your professional life by all means, but don't use it while you are learning. Learn how to sketch a graph by hand so you can intuit the shape it is supposed to have. Intuition is a massively valuable tool to have for the practicing engineer.

When you are studying you want to be doing the mental equivalent of going to the gym. Always remember, no pain, no gain.

I did the program, with a slide rule.  The HP 35 was invented when I was in my last year of undergrad.  I couldn't afford it...

I disagree completely!  There is no place in the real world where anybody is going to solve systems of equations (for example) by hand.  And nobody would trust the solution if they did.  Today's approach to calculus is centuries old and was obsolete even then.  Just because it was always done that way is no reason to continue it.

What I would rather see students learn is how to USE the concepts and the tools.  As far as I am concerned, all the time spent studying limits is a complete waste.  Who in the real world ever needs them?  As EEs, most, if not all, of our signals are continuous by definition.  Oh, sure, we toss in a Heaviside Step Function now and then so we don't have to deal with negative time but mostly we assume continuity.  I can't recall ever having to prove it.

Name any possible use for the epsilon-delta proof of a limit.  It needs to be a use that might actually occur in the real world of engineering.

I would rather see the students spend more time on applications and less time on theory.  Two reasons:  Calculus is taught for math majors and that filters out a lot of engineering types who will NEVER need the underlying theory.  Who has ever used the limit definition of a derivative?  Better to spend more time on Related Rates problems and less on pedantic nonsense.  Second, Calculus is just a tool.  It many ways it is like a sophisticated slide rule.  Pump in functions and grind out derivatives.  Machines do that very well.  What machines can't do is think!  How do I solve this problem?  How do I describe the problem for the machine?  Calculus may, of may not, be involved.

I have a Great Courses calculus video series presented by a professor at the University of Florida.  Apparently, they have two math tracks:  One for math majors and another for engineers.  Kudos!

Consider the problem of AC circuit analysis.  Just the usual Rs Ls and Cs where the impedance is given.  Now write a system of equations to solve either by mesh or nodal analysis.  Let's say it is, oh, 6x6.  We had one of those the other day!  OK, let's back off to a more reasonable 4x4...

Solving this by hand with complex numbers will inevitably wind up with simple algebra errors.  What is learned?  Well, you learn to avoid this analysis at ALL COSTS.  You learn to HATE complex numbers!  You begin to question why you picked this major!  Instead of dumping the equations into a number cruncher, you have to spend an inordinate amount of time to get the wrong answer!  But you could have learned to more effectively use a number cruncher - something that might actually be useful out in the world.  Wrong answers in the real world have consequences.

[rstofer]

 If you look at many engineering classes, in so many cases we will derive solutions by bounds or removing irrelevant terms to make equations solvable by hand.

Reducing the solutions by removing irrelevant terms, helps also to understand what is the principal element that makes something work or not.  With today computing resources, we can calculate a theoretically exact solution, but doing just that obscures the fundamental inner works of a system.
IMHO, Octave and similar programs are very useful, but they are just a calculation help for complex problems.

Again, I disagree!  I remember the mathematical solution to a second order ordinary differential equation and I know that I will probably come up with an exponentially decaying envelope and some kind of sin() or cos().  But I can't see it.  How fast does it damp out?  What is the resonant frequency?  It's only when I have MATLAB (or equal) plot the solution that I can make any sense of what is going on.  Specifically, I'm talking about the step response of an RLC circuit or Damped Harmonic Motion.  And, yes, I can probably sketch it out.  But MATLAB is faster and more accurate.

I'm not talking about digits, we can't measure to 12 decimal places anyway.  What I am talking about is shape.  Engineers ALWAYS draw pictures.  We can't even talk without a pencil and paper.  We think in terms of sketches, not digits.

We invented the hammer when we got tired of driving nails with a rock.  Same thing with math.  There's a reason people invented things like MATLAB. They got tired of the nonsense of solving problems by hand.

[rstofer]

And if you get stuck on a homework problem, it is better to have Symbolab get you unstuck before you just chuck the whole program.  There is something to learn from their detailed explanations.  So, learn it!  Sometimes it's just a little brain jog.  Other times they come up with a solution you didn't even consider.  And you learn even more!

What doesn't work is to get stuck and give up!

I am not suggesting a student use Symbolab to DO the homework but I am suggesting they check their work if there is any doubt.  You really only learn from right answers.  Wrong answers are counterproductive.

[@rt]

As far as I’m concerned, anyone should be able to lug in whatever tool helps do something, or demonstrates the understanding of something, so long as it can’t do the thing, rendering the student a bystander.

Pencil and paper are a technology. A shovel is a product of a technology. If I’m at home I don’t dig a hole in the yard to take a dump, but I don’t necessarily take issue with anyone who wants to.

[ferdieCX]

As a practitioner engineer, I should to be crazy to not use MatLab, octave and the likes. They can draw a graphic faster and better than anyone. I am talking about learning, you have to draw for example a Bode plot at least once in your life by hand, to really understand it.
As for complex numbers, I would never pretend that my students solve anything more complex than a two mesh circuit by hand, but they have to be capable of that. Why ? Because they have to understand that impedance is a vector entity and have also to learn the concept of phasor. In real life, I use octave to solve mesh circuit equations.
I am not a math fan, it is for me only a tool, and I don't like to learn any math that has no application in electronics
Some time ago, I wondered that in a company in Germany, the mechanics apprentices were learning to draw by hand. I was told, that they use CAD programs in the second year. They had the experience, that the apprentices that started doing it directly with the computer, had later problems interpreting mechanical diagrams.

[rstofer]

One of the other things machines do well is to iterate over changing parameters.  Once I have hand drawn a plot of, say, that RLC step response, great, I understand what it looks like for a given set of parameters.  Now, how does a change in the resistor value influence the response?  A change in capacitance?  How about adding a little real world resistance to the inductor.

Iteration is where computers really excel.  Way back then I knew that I didn't want to solve these circuits by hand.

[ferdieCX]

At this point, it is now the moment of using your preferred tool, be it Spice, MatLab or anything else, and adjust the values or just play with them.
You have already drawn it once in your life, and you know that the computer is not playing magic, it is just doing the same calculations that you have once done, but faster.

[hans]

I would rather see the students spend more time on applications and less time on theory.  Two reasons:  Calculus is taught for math majors and that filters out a lot of engineering types who will NEVER need the underlying theory.  Who has ever used the limit definition of a derivative?  Better to spend more time on Related Rates problems and less on pedantic nonsense.  Second, Calculus is just a tool.  It many ways it is like a sophisticated slide rule.  Pump in functions and grind out derivatives.  Machines do that very well.  What machines can't do is think!  How do I solve this problem?  How do I describe the problem for the machine?  Calculus may, of may not, be involved.

I have a Great Courses calculus video series presented by a professor at the University of Florida.  Apparently, they have two math tracks:  One for math majors and another for engineers.  Kudos!

Consider the problem of AC circuit analysis.  Just the usual Rs Ls and Cs where the impedance is given.  Now write a system of equations to solve either by mesh or nodal analysis.  Let's say it is, oh, 6x6.  We had one of those the other day!  OK, let's back off to a more reasonable 4x4...

Solving this by hand with complex numbers will inevitably wind up with simple algebra errors.  What is learned?  Well, you learn to avoid this analysis at ALL COSTS.  You learn to HATE complex numbers!  You begin to question why you picked this major!  Instead of dumping the equations into a number cruncher, you have to spend an inordinate amount of time to get the wrong answer!  But you could have learned to more effectively use a number cruncher - something that might actually be useful out in the world.  Wrong answers in the real world have consequences.

I think this a pretty common joke already:

en-gi-neer
[en-juh-neer]
1. A person who solves problems that you did not know you had, using methods you do not understand.
2. A person that has forgotten more math than you ever knew.

 

But to be honest, I somewhat disagree with your post. Mathematics and engineering are married, that's just the way it is.

How much you're exposed to mathematics is a matter of choice, and also the level you're solving problems at. In my opinion and probably the majority of the folks on this forum, building electronics on PCBs and writing firmware typically does not require lots of math. But there are also plenty of engineers hard at work creating new components, quantitatively testing and verifying systems, and writing "middleware" libraries that one often takes for granted.

How are you going to be sure a cryptographic algorithm is implemented correctly?
How is an analog IC designer going to determine the bias for his circuit, know for certain his circuit is not going to oscillate given a set of load impedance, and what stages of his amplifier should be modify in order to do so?
Who actually invented some of these fancy analog circuits in the first place? I mean some of the ones that are counter intuitive; like an amplifier circuit that cancels out it's own noise.
How is a digital designer going to decide on the best binary adder design in an ASIC?
Or perhaps the reverse question: why do we care about lookahead adders when it's slower on most FPGAs anyway?

I think any college program is trying to accomplish 2 goals. One path is onto the job market (that wants highly practically trained students so they are valuable straight away) and the other is onto further education programmes (up to bachelor, master and PhD). The latter requires students with a more formally trained background in maths, continuity between programmes is preferred.

Additionally I think one should learn skills where they are taught best. In academia the courses are taught by professors of which you expect to have a very thorough understanding of the material, and can teach it most rigorously and efficiently. Practical skills can be obtained with hobby projects (show you're autodidact) and internships, and aboveall; every day on the job itself.

But I think rarely it will happen that an employer is going to pay you a degree because you need to know more about theory of microwave engineering, but have never seen or worked with a Maxwell equation before.

[rstofer]

Additionally I think one should learn skills where they are taught best. In academia the courses are taught by professors of which you expect to have a very thorough understanding of the material, and can teach it most rigorously and efficiently. Practical skills can be obtained with hobby projects (show you're autodidact) and internships, and aboveall; every day on the job itself.

But I think rarely it will happen that an employer is going to pay you a degree because you need to know more about theory of microwave engineering, but have never seen or worked with a Maxwell equation before.

Fully one third of Calc I is focused on stuff that will NEVER be used but provides an elegant introduction to the derivative.  Why not use all those hours to extend the study of the applications of Calculus and handwave past all the derivations.  Yes, we need to spend a few minutes on the secant line and tangent line and it might be useful to find the tangent line parallel (or perpendicular) to a function at a point.  Beyond that, it's time to move on!

Doing the epsilon delta proof of a limit is just about useless.  Related Rates, OTOH, is what Calculus was meant to be.  How changes in one variable effect another variable like the famous ladder sliding down a wall.  How fast is the top of the ladder fallng if the bottom is moving at 'v' away from the wall.  That will be important when you hit the ground.

I wouldn't argue about Maxwell's Equations.  I will admit to struggling with the Calculus and that's a shame.  I decided then and there that if I passed the course I would never get involved with RF or any other application that required those equations.  Yes, I know they refer to the current flow in a wire but I can work with wire not knowing anything about Maxwell - at frequencies that interest me.  Motors, Rotors and Dynamos was another struggle.  But I actually care about motors so I have had to go back and look at it.

Cointrol Systems with a Spirule!  Are you kidding me?  MATLAB can plot the poles and zeros a lot faster than I can using the Spirule.  And it's likely to be the right answer for the functions given.  And I can play 'what if' with the transfer functions; I can use Simulink and add knobs and dials to control the transfer functions.

Instead of wasting time on side issues, we could be turning out engineers that LIKE the subjects rather than dreading them because of the math required.

Yes, engineering is math.  Essentially all math.  But that doesn't mean we need to do it with a pencil and paper.

[IanB]

Fully one third of Calc I is focused on stuff that will NEVER be used but provides an elegant introduction to the derivative.  Why not use all those hours to extend the study of the applications of Calculus and handwave past all the derivations.  Yes, we need to spend a few minutes on the secant line and tangent line and it might be useful to find the tangent line parallel (or perpendicular) to a function at a point.  Beyond that, it's time to move on!

Perhaps it's lucky then, that I never had the misfortune to take Calc I 

When I was introduced to calculus in the British education system, the high school curriculum briefly went over the concept of limits and how you end up with a derivative, but it handwaved past all the rigorous proofs. Our teacher said that if we chose to study mathematics at university this would all be made very rigorous, but for now we don't need to care about that and let's just accept it by intuition and focus on the application.

It turns out that I did not choose to study mathematics at university, I studied engineering instead. In engineering school nobody had time for rigorous proofs, we just had to learn how to use the tools to get answers. So I never did see the epsilon delta proof. There was no time for that stuff. We had more important things to learn.

[ferdieCX]

Excuse me, but I have to disagree, if you can't rigorously demonstrate something, you are not really sure about it. To have seen that something is absolutely right, gives you a high level of confidence. It doesn't matter if five years later you don't remember the demonstration.
In the 70s, our high school math exams had two separated parts: i) A practical one, where you had for example to study and draw a function ii) A theoretical one, where you could be asked to demonstrate on the blackboard any theorem of the course. I had the luck not to have a simple secondary school math teacher, but a real graduated mathematician, who was very strict about that.
Although I demonstrate most things, I have not the necessary time to demonstrate everything to my students on the blackboard. So, together with two colleagues, we wrote a PDF where everything is rigorously written down. This way, the students know that nothing comes from inside a "hat", and are also trained in methods to tackle new and unknown problems. Sadly, I known about some colleagues that oft pull things out from their "hats".
I am absolutely against to teach unnecessary subjects, like " projective geometry " , but the really necessary math should be rigorously taught. The " Calculus " from Tom Apostol is one of my best friends