Brushing up on Calculus

[rstofer]

Now as to checking your work, you will find that a lot of books have practice questions but not all of them have the answers in the book itself, sometimes you have to get a second answer book to go with it.  What else helps though is automatic software.  The stuff of today can beat down a lot of calculus even some differential equations.  You can use that software to check your answers to the practice questions.

Oh and you can also ask questions on forums like this one but im not sure if this one has a section for math.

I linked the sites above but Desmos.com for graphing and Symbolab.com for answers are highly recommended.  At Symbolab, they will give the solution without subscription but to get all the intermediate steps seems to require one.  I have had mine for a long time so I don't recall the details.

We seem to get away with posting math questions in the Beginners forum and since that is at the top of the forum list, it's as good a place as any until the mods complain.

I don't think there is enough interest in math to have a separate forum.  Personally, I like the math questions, particularly if they lend themselves to machine solutions.  Mesh and nodal equations are especially easy with Octave or MATLAB.

There are an enormous number of good sites on the Internet that provide tutoring.  I haven't found a courteous place out in the wild to ask questions.  EEVblog would rank very high in courteous responses.

I don't tend to use them for calculus but many scientific calculators can do calc problems.  They might be useful for checking work.

[westfw]

It takes Octave less than a second

Maxima is based on 40y-old code and has no problem, either:

Code: [Select]

(%i9) partfrac(x^2/(x^3 - 3*x + 2), x);
                          4           5           1
(%o9)                 --------- + --------- + ----------
                      9 (x + 2)   9 (x - 1)            2
                                              3 (x - 1)

[DonKu]

Quick Calculus - A Self-Teaching Guide by Kleppner and Ramsey.

https://www.goodreads.com/book/show/231705.Quick_Calculus

[JenniferG]

Doing differential single variable calculus right now on Khan Academy..  Sal Khan seems to be doing a good job so far with it.. super easy.. on limits right now    If I run into any problems I'll check out Moocows, or other youtube channels on a particular topic.   I also ordered the Calculus 3rd Edition by James Stewart for $11.84 from abe books.. allegedly in very good condition; I'll have that to reference as well .. perhaps do some practice problems from it.

[rstofer]

For wxMaxima (Maxima with a GUI), I have a book "Introduction to wxMaxima for Scientific Computations..." by Dr. M Kanagasabapathy.  It is a terrific compendium of the internal functions of Maxima.

From the Wiki

Maxima is based on a 1982 version of Macsyma, which was developed at MIT with funding from the United States Department of Energy and other government agencies. A version of Macsyma was maintained by Bill Schelter from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release his version under the GPL. That version, now called Maxima, is maintained by an independent group of users and developers. Maxima does not include any of the many modifications and enhancements made to the commercial version of Macsyma during 1982–1999. Though the core functionality remains similar, code depending on these enhancements may not work on Maxima, and bugs which were fixed in Macsyma may still be present in Maxima, and vice versa. Maxima participated in Google Summer of Code in 2019 under International Neuroinformatics Coordinating Facility.

Without the original schematic (courtesy of our moderator, Simon), this code may not be interesting:

Code: [Select]

--> ratprint    : false$
fpprintprec : 4$

eq1  :   0 = -V1 + Z1*I1 + Z4*(I1-I2) ;
eq2  :   0 = Z4*(I2-I1) + Z2*(I2-I4) + Z5*(I2-I3) ;
eq3  :   0 = Z5*(I3-I2) + Z3*I3 + V2 ;
eq4  :   0 = Z2*(I4-I2) + V3 ;
eq5  :   VA = V1 - I1*Z1 ;
eq6  :   VB = VA - V3 ;
eq7  :   Z1 = 2 ;
eq8  :   Z2 = -5*%i ;
eq9  :   Z3 = 4 ;
eq10 :   Z4 = -5*%i ;
eq11 :   Z5 = 4*%i ;
eq12 :   V1 = 120 ;
eq13 :   V2 = 120*%i ;
eq14 :   V3 = 14.14*%i + 14.14 ;

res  : solve([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12,eq13,eq14])$

results : expand(float(res))$
lngth   : length(results[1])$
sorted  : sort(results[1])$

print("")$
for i:1 thru lngth do
    print(sorted[i])$

(eq1) 0=(I1-I2)*Z4+I1*Z1-V1
(eq2) 0=(I2-I3)*Z5+(I2-I1)*Z4+(I2-I4)*Z2
(eq3) 0=(I3-I2)*Z5+I3*Z3+V2
(eq4) 0=(I4-I2)*Z2+V3
(eq5) VA=V1-I1*Z1
(eq6) VB=VA-V3
(eq7) Z1=2
(eq8) Z2=-5*%i
(eq9) Z3=4
(eq10) Z4=-5*%i
(eq11) Z5=4*%i
(eq12) V1=120
(eq13) V2=120*%i
(eq14) V3=14.14*%i+14.14

I1=16.81-22.88*%i
I2=25.96-40.15*%i
I3=18.06-22.1*%i
I4=28.79-42.98*%i
V1=120.0
V2=120.0*%i
V3=14.14*%i+14.14
VA=45.76*%i+86.38
VB=31.62*%i+72.24

Z1=2.0" "
Z2=-5.0*%i" "
Z3=4.0" "
Z4=-5.0*%i" "
Z5=4.0*%i" "

So, here's the original thread.  We get a little clever as things move along.  Several programs are discussed:

https://www.eevblog.com/forum/beginners/mesh-analysis/msg1100020/#msg1100020

wxMaxima is a terrific tool.

[JenniferG]

Have you played with SageMath?  It's in Python. Playign around with it.. pretty cool -- Opensource.

[rstofer]

Have you played with SageMath?  It's in Python. Playign around with it.. pretty cool -- Opensource.

I'll have to install it on another machine.  It pretty much filled the disk on my main machine.  I'm going to uninstall WSL anyway, I need the space.  Never buy a computer with less than 2 TB of disk space...

WSL := Windows Subsystem for Linux for those who haven't used it.  It's based on Ubuntu.

I did a few command line examples and the first problem is that the developers got cute with coloring without spending a moment considering people with less than stellar color vision.  Red on black is invisible, blue on black is only slightly better.  Green on black is probably ok but, really, why not keep everything cyan.  It's been working for 50 years, why mess with it?

I haven't tried SageMath with a GUI or command file and both are necessary.  I think it's doable with Jupyter but I haven't gotten that far.  There are limitations to WSL in terms of graphics (and there's no desktop) so I'll reinstall SageMath on one of my Linux machines.  It'll be worth spending some time with the package.

Bottom line:  I'm going to stick with MATLAB as my go-to math tool and try to keep my code compatible with Octave.  Both systems have fully fleshed out GUIs.

And then there is MATLAB's SimuLink toolbox (and a half dozen others) that I don't want to lose.

[mathsquid]

I like Stewart's Calculus book. I've taught out of it for 25 years, and have no complaints. You might also look at the calculus books on openstax.org if there's something you want more exercises on. Another great resource online is http://www.ilectureonline.com.  It has a lot of videos of problems being worked out.

[Sherlock Holmes]

"Calculus Made Easy".  Originally written by Silvanus Thompson, but modernized by the inimitable Martin Gardner.

That is a very good book indeed, I have it and used it decades ago when I was learning this. Another excellent old book is "Teach Yourself Calculus" these two book together provide excellent coverage of the subject for a learner.

[JenniferG]

I'll have to install it on another machine.  It pretty much filled the disk on my main machine.  I'm going to uninstall WSL anyway, I need the space.  Never buy a computer with less than 2 TB of disk space...

I haven't tried SageMath with a GUI or command file and both are necessary.  I think it's doable with Jupyter but I haven't gotten that far.  There are limitations to WSL in terms of graphics (and there's no desktop) so I'll reinstall SageMath on one of my Linux machines.  It'll be worth spending some time with the package.

I just found out about this resource.  A bunch of reviewed and approved Opensource Math books listed on the following site:
https://aimath.org/textbooks/approved-textbooks/

There are several books on SageMath there as well.. for example under the Numerical Analysis section or Linear Algebra. 
One of them is "Sage for Undergraduates:
https://aimath.org/textbooks/approved-textbooks/bard/

[JenniferG]

I've decided I need to take a step or two backwards before tackling the James Stewart Calculus book I ordered the other day.

I can't decided if I should just buy a Precalculus book or buy the combination of a College Algebra book and Trigonometry book.

I am thinking the College Algebra and Trig books because perhaps they'd each be thinner over all (easier to handle) and perhaps more comprehensive.

The way I progressed in college back around the early 90's was, I took Intermediate Algebra followed by Trig straight into Calculus.  Which was a mistake really. I should of take Precalc or college algebra as I really struggled with Calculus I despite getting an A in it.  (I got an A in both Intermediate Algebra and Trig as well.)

What do you think?  I also see another flavor of a combined book called "Algebra and Trigonometry" which I thought was Precalculus. LOL.. too many different varieties of Algebra and Trig classes if you ask me.

I was considering the College Algebra book by Kaufman, an older cheaper edition.   I don't know which Trig book to get.   But I figure I better go through both of them again before tackling Calculus I & II followed by Differential Equations and Linear Algebra.  (Guessing I can skip multivariable calculus perhaps.. I only ever recall find volumes of 3D objects as well as line integrals.. don't remember the differentiation part of calc iii.)

[BillyO]

Since we're entering into show and tell, these are some of the best texts for applied calculus in my collection.  Last is most basic and a good restart, then 1st, 2nd, 3rd .

The Elements of Calculus is a really good starting point.  Just above high school level stuff.

(There is something seriously wrong with the image attachment mechanism!!!)

[JenniferG]

Nice book collection there

I found what looks like a decent College Algebra textbook with table of contents in the PDF.

https://openstax.org/details/books/college-algebra-2e

What do you think?  I can just read and do problems from my iPad.  Free.

[BillyO]

Nice book collection there

It's only the applied stuff from my past.  I also have a couple of volumes of Schaum's notes on applied calculus, but you can look that stuff up yourself.  I didn't bother with the more abstract real and complex analysis texts.  Not really pertinent to EE.

I found what looks like a decent College Algebra textbook with table of contents in the PDF.

https://openstax.org/details/books/college-algebra-2e

What do you think?  I can just read and do problems from my iPad.  Free.

Yes.  That looks like a good compilation of applied algebra concepts.  It should be fine to do on-line.

[HighVoltageLP]

I can absolutely recommend David Easdown's courses on Coursera starting from this one https://www.coursera.org/learn/introduction-to-calculus

I have a Master's in Systems Engineering and had my uni calculus courses exactly 10 years ago and I found his course EXTREMELY enjoyable for brushing up my calculus. Even went through them again with my GF 

Also, khanacademy might be great, used it to brush up on Probabilities and Statistics, didn't touch the Calculus on there though, as I got a chance to do the Coursera course for free.

[rstofer]

It's interesting that you should bring up Pre-Calc.  At the local junior college, they test incoming students before they let them enroll in Calculus and if they don't pass, they get 2 semesters of Pre-Calc.

Calculus is easy!  Pre-Calc is the problem.

It only takes a couple of weeks of Calculus before you get into trig forms and some of the identities take some thought.  The most elegant of which (IMO) is:
sin² + cos² = 1  Think about the unit circle and the Pythagorean Theorem.  They all tie together with this identity.

Locally, I believe the JC uses "California Algebra 2" by Bellman et al.  There are many versions by state.

I would start at Khan Academy:  https://www.khanacademy.org/math/precalculus

Or, I would spend the money for CalcWorkshop - I tend to like the author's approach and presentation.

Khan Academy has little quizzes in most of their programs.

I can't lay my hands on it at the moment but there is a separate volume in the Prentice Hall mathematics section for Geometry.  Google for 'prentice hall mathematics' to get a more complete view of the offerings.  Some are reasonably priced.

Here's a brief 1413 page volume on Pre-Calculus:

https://www.ilearnacademy.net/uploads/3/9/2/2/3922443/precalculus__edition_5f.pdf

It's a place to start!

[JenniferG]

Thanks for the replies. I started on the following Openstax 1500 page Algebra and Trig PDF.  (Would be equivalent to a Pre-calculus course).  It looks really good, peer reviewed and free    Many colleges use this textbook to keep costs down for the student, since it's very good.

https://openstax.org/details/books/algebra-and-trigonometry-2e

[rstofer]

I spent just a few minutes looking at Trig Identities and it looks like a very good book.  It seems that the answers are given for every homework problem.  That's helpful!

The book does the same as every other book I have seen, it gives the answers to the odd numbered problems.  Still, there are a LOT of problems so there will be plenty of practice.

[mathsquid]

What do you think?  I also see another flavor of a combined book called "Algebra and Trigonometry" which I thought was Precalculus. LOL.. too many different varieties of Algebra and Trig classes if you ask me.

Yeah, "Precalculus" and "Algebra and Trigonometry" are pretty much interchangeable terms when you're looking at titles of textbooks.

[bostonman]

I took calculus and did well, but never could grasp the concepts which has annoyed me for years.

Personally, when I don't grasp the concept, I can't retain what I'm learning. The most basic is why taking the first derivative gives you position, the second gives you velocity, etc... besides reading this in a textbook or online, I don't understand why. In fact, I had to look online just to type that sentence because it never clicked with me.

Word problems was where I couldn't do any calculus. One word problem I remember was a swimmer swimming to shore at a certain speed while the water current was pushing him sideways at another speed. Where on shore would the swimmer end up after X time (I believe this is the correct question that was asked). While I understand the problem in hand, I don't understand how to apply calculus to solve it.

In my case, with electronics, such as inductor (v = L (di/dt)) I get the linear aspect of this. If the current went from 0 to 1A in 1s, and the inductor is 10mH (milli Henries), then the voltage is 10mV [(1A/1s) * 10mH]. But what does this really tell me if it's a 60Hz sine wave (AC voltage source)?

Anyway, sometime ago I came across a video from this guy, but didn't get too far into as I found following him was easy until I closed the video and tried to think about it further:

[JenniferG]

I took calculus and did well, but never could grasp the concepts which has annoyed me for years.

Personally, when I don't grasp the concept, I can't retain what I'm learning. The most basic is why taking the first derivative gives you position, the second gives you velocity, etc... besides reading this in a textbook or online, I don't understand why. In fact, I had to look online just to type that sentence because it never clicked with me.

Word problems was where I couldn't do any calculus. One word problem I remember was a swimmer swimming to shore at a certain speed while the water current was pushing him sideways at another speed. Where on shore would the swimmer end up after X time (I believe this is the correct question that was asked). While I understand the problem in hand, I don't understand how to apply calculus to solve it.

In my case, with electronics, such as inductor (v = L (di/dt)) I get the linear aspect of this. If the current went from 0 to 1A in 1s, and the inductor is 10mH (milli Henries), then the voltage is 10mV [(1A/1s) * 10mH]. But what does this really tell me if it's a 60Hz sine wave (AC voltage source)?

Anyway, sometime ago I came across a video from this guy, but didn't get too far into as I found following him was easy until I closed the video and tried to think about it further:

I watched the video above I liked how he came up with the integral using the area of a triangle formula

I don't know if you've seen the following video playlist or not but I just watched the first two videos in it yesterday.  The two videos are great.. really make it easy to understand:

[rstofer]

In my case, with electronics, such as inductor (v = L (di/dt)) I get the linear aspect of this. If the current went from 0 to 1A in 1s, and the inductor is 10mH (milli Henries), then the voltage is 10mV [(1A/1s) * 10mH]. But what does this really tell me if it's a 60Hz sine wave (AC voltage source)?

If you want to think about the AC response (as opposed to the transient response) then you would just work out the AC impedance and work with that in the frequency domain.  Maybe you vary the frequency and look at the change in output as the impedance changes with frequency.  Think about a low pass filter and what happens at the break frequency - you're down -3dB and falling at -6dB per octave (-20dB per decade)

https://electronics.stackexchange.com/questions/95956/rc-filter-falloff

If you want to see the transient response then you play in the time domain.  Charging/discharging a capacitor is viewed in the time domain.

[TimFox]

Yes, normally one uses the AC calculation for such a sinusoidal problem, which gives the steady-state response (after a transient from the switch-on of the sine wave decays).
For calculus, you still can use V = -L dI/dt, remembering that the derivative of sin(x) is cos(x), where x = 2(pi)f t for a sine wave.

[Nominal Animal]

Finding problems that interest you to solve using calculus etc. is the key.

That way, instead of memorizing details, you'll focus on how the tools can be used in anger.  Granted, this is the applied math approach that probably infuriates proper mathematicians, but since we're here, I'm assuming the topics interest you because you want to use math as a tool, not as a research topic.

In electronics, RLC circuits are an obvious choice for calculus exercises, but there are things like heat equation, and telegrapher's equations used for transmission lines.

I do not bother to memorize details, but prefer to develop an intuitive understanding instead.  Instead of recalling exactly how to do e.g. partial fraction decomposition, I try to understand when it is an useful tool.

Of the various computer algebra systems (CAS) I've used, I've liked Maple the most, but it is commercial.  Of the free/open source ones, I like Maxima slightly better than Octave or Sagemath, but that's purely a personal opinion with no technical basis; on the same order of importance as my preference in pens or pencils.  I very often find myself optimizing expressions by hand (Maple had pretty darn good tools for that), so it is not like the CASes will give you the answers with zero effort: you do need to know how to ask, and what assumptions have been made.

[rstofer]

I watched the Strang video above and it is the only time I have seen an introduction to Calculus that didn't start with a discussion of the secant line connecting two points on a function to derive the average slope followed by moving the points arbitrarily close together to get the tangent line which is the slope at that point.  This brings up the discussion about limits and while I never thought of the concept as useful, I was wrong.  It is a vital concept to understand and perhaps maybe more important than the derivative itself when you're trying to compute the gradient descent of a curve in N dimensional space (machine learning...).  The reason it might be more important is that there is a step size to what would be dx and it is called the learning rate.  If that rate approaches zero, the compute time approaches infinity.  We will always calculate average slopes in terms of a step size.  That learning rate (step size) might still be a number on the order of, say, 0.01.  But it is nowhere close to zero!

I can only pass along what I learned along the way - draw the picture.  If you understand the question, drawing a picture is easy.  If you can't draw the picture, you don't understand the question and you darned sure won't get the correct answer.

Here's the beginning of Khan Academy's discussion of the derivative and how it is derived from limits.

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/v/derivative-as-a-concept

There are a bunch of videos in the series.

Differential Calculus is all about change. Rate of change of distance is speed, rate of change of speed is acceleration. Rate of change of acceleration is called jerk and the rate of change of jerk is called snap.

Integral Calculus is all about area and volume - we're always adding up slices to find an area or volume.

A good introduction to integral calculus always begins with something like Riemann Sums.  Adding up the area of rectangles under a function.

https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/v/simple-riemann-approximation-using-rectangles

We have the function so we can get the value at the corners of the Riemann rectangles and all we need to do is add up the area of a bunch of rectangles.  Maybe you want to use really narrow rectangles and you need to calculate the area of 100,000 slices but that's easy with a computer.

A lot of time has been spent on different ways of adding up slices.  Soon after the slice discussion we get into analytic integration.  There are a bunch of identities to memorize but other than that, it's pretty easy.

Draw the picture!