Brushing up on Calculus

[JenniferG]

All of AI is based on linear algebra and everybody needs to get a taste of that.  Even the simple Digits Recognition problem is working in 784 dimensional space.  A wee bit hard to visualize...  That problem and its Neural Network solution is the "Hello World of AI".

In grad school there was a required course in Linear Algebra and we spent most of the time talking about solving simultaneous equations.  If we only knew what was coming at us...

Oh, and you absolutely MUST have a command of statistics (which leaves me hurting) to get anywhere with data analysis and machine learning.  I hated statistics!

I recently learned about OpenAI's ChatGTP and Midjourney/Stable Diffusion, and am blown away by what AI can do now.  So I have an interest in it as well as the analog synthesizer electronics.   My major was in Computer Science so I'm good with programming and software engineering (as well as database design and programming).

So it looks like Linear Algebra and Probability & Statistics will perhaps be useful for both interests in AI and electronics.

Here's my plan:
1) Algebra & Trigonometery (6e by Blitzer, cheap used copy)  -- started on this an am 84 pages in so far, doing all the problems
2) Introduction to Mathematical Thinking (Standford Coursera) by Devlin -- free
3) Calculus I (Stewart textbook, used copies) -- will use OpenStax and Larson as well as Khan as backups
4) Calculus II - Stewart "
5) Elementary Linear Algebra -- Howard Anton
6) Discrete Mathematics -- Susanna S. Epp
7) Probability & Statistics (the college level course with precalculus as prerequisite) -- Anthony Hayter
8 ) Differential Equations -- I dunno which one yet.. will worry about this later
9) Physics I & II (mechanical and electricity & magnetism) - Paul A. Tipler
10) The Art of Electronics
11) DSP

I don't know if Calculus III (multivariable calculus) would be that helpful with AI or electronics.

I imagine all this is going to take me a couple years studying a few hours each day.

[westfw]

Introduction to Mathematical Thinking (Standford Coursera) by Devlin -- free

I took that!  A real eye-opener, and rather depressing :-(
("If you're a Math or CS major, you need to be thinking about math OTHER than from the perspective of getting the right answer."  Which is about all the physics/ee curricula math classes I took ever did.)

Discrete Mathematics -- Susanna S. Epp

Let us know how that goes.  I've had a lot of trouble with the more theoretical math classes (and books.)  They seem to start by assuming a more theoretical background than I've got.  "You probably recognize this result as the Golden Ratio, and can see how that makes sense." (NO!)  Sigh.

[jasonRF]

All of AI is based on linear algebra and everybody needs to get a taste of that.  Even the simple Digits Recognition problem is working in 784 dimensional space.  A wee bit hard to visualize...  That problem and its Neural Network solution is the "Hello World of AI".

In grad school there was a required course in Linear Algebra and we spent most of the time talking about solving simultaneous equations.  If we only knew what was coming at us...

Oh, and you absolutely MUST have a command of statistics (which leaves me hurting) to get anywhere with data analysis and machine learning.  I hated statistics!

I recently learned about OpenAI's ChatGTP and Midjourney/Stable Diffusion, and am blown away by what AI can do now.  So I have an interest in it as well as the analog synthesizer electronics.   My major was in Computer Science so I'm good with programming and software engineering (as well as database design and programming).

So it looks like Linear Algebra and Probability & Statistics will perhaps be useful for both interests in AI and electronics.

Here's my plan:
1) Algebra & Trigonometery (6e by Blitzer, cheap used copy)  -- started on this an am 84 pages in so far, doing all the problems
2) Introduction to Mathematical Thinking (Standford Coursera) by Devlin -- free
3) Calculus I (Stewart textbook, used copies) -- will use OpenStax and Larson as well as Khan as backups
4) Calculus II - Stewart "
5) Elementary Linear Algebra -- Howard Anton
6) Discrete Mathematics -- Susanna S. Epp
7) Probability & Statistics (the college level course with precalculus as prerequisite) -- Anthony Hayter
8 ) Differential Equations -- I dunno which one yet.. will worry about this later
9) Physics I & II (mechanical and electricity & magnetism) - Paul A. Tipler
10) The Art of Electronics
11) DSP

I don't know if Calculus III (multivariable calculus) would be that helpful with AI or electronics.

I imagine all this is going to take me a couple years studying a few hours each day.

That is a lot of self study!  But if you have the time I am sure it will be rewarding.  I just have a couple of suggestions.  First, you will certainly need some multivariable calculus for electronics and AI / machine learning, so I would recommend working through those chapters of whichever calculus book to select.  If you want to save some effort, you may not need to go deeply into multiple integration (setting up integrals to find volumes of weird 3-D shapes is not needed in electronics), and if you don't plan on learning engineering electromagnetics then you can skip the vector calculus chapter (with the idea that you can go back and learn it if you need it).  Second, once you know calculus then you should learn calculus-based probability and statistics.  I think it will actually be easier to understand that way, plus it will help reinforce your calculus knowledge. 

jason 

[jasonRF]

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 would highly recommend physicsforums, which also has a lot of math on it.  If you post questions out of textbooks (even if not for a class you are taking) then you need to post them in the 'homework' section and they have a format they want you to use (for example, you need to clearly state the problem and show what work you have done to try and solve it), but they do have a significant number of folks over there who help answer questions from students.  For textbook problems they will not give you the answers as that breaks forum rules, but they will help walk you through how to solve it and will let you know if you got the right answer.  As long as you come with a good attitude and are willing to do the work you will find that most folks over there are pretty nice.

jason

[bostonman]

One thing that baffles me is that we can take functions and their derivative or integrate them, but what if we only had a graph?

Say it's a graph of torque of a motor measured on a dyno. I'm sure the computer can give you an area under the curve, but how is it possible to perform this without the computer if the graph isn't some function?

My friend and I were discussing torque/hp. He said the larger the area under a curve, the more torque or whatever. He showed me a graph of one engine where it was basically a wavy horizontal line (say y=3) and then another that was a curve of basically y=x (where x and y began at zero). He said because it appeared the area under the "wavy line" of y=3 is higher than y=x, then it has more area.

My argument was that the y=x could have more area depending on how far you go on the X axis.

To prove my argument, I took the y=3 to be a perfectly horizontal line and y=x to be a perfect right triangle, so I calculated the area of a rectangle versus the area of a right triangle.

Turned out in his example, the rectangle had more area proving him correct, however, I twisted things around a bit and used the same example with different numbers making the triangle have more area to prove that just because there is a horizontal line doesn't mean it will always have more area under the curve than a triangle.

Anyway, my point is, I took a wavy horiztonal line and made it perfectly flat and a curved line and made it a delta x / delta y.

Mathematically it's wrong to do this because I missed (or added) area. So how can I look at a curve and know the area under it without a computer doing the work for me; and without approximating?

[CatalinaWOW]

One thing that baffles me is that we can take functions and their derivative or integrate them, but what if we only had a graph?

Say it's a graph of torque of a motor measured on a dyno. I'm sure the computer can give you an area under the curve, but how is it possible to perform this without the computer if the graph isn't some function?

My friend and I were discussing torque/hp. He said the larger the area under a curve, the more torque or whatever. He showed me a graph of one engine where it was basically a wavy horizontal line (say y=3) and then another that was a curve of basically y=x (where x and y began at zero). He said because it appeared the area under the "wavy line" of y=3 is higher than y=x, then it has more area.

My argument was that the y=x could have more area depending on how far you go on the X axis.

To prove my argument, I took the y=3 to be a perfectly horizontal line and y=x to be a perfect right triangle, so I calculated the area of a rectangle versus the area of a right triangle.

Turned out in his example, the rectangle had more area proving him correct, however, I twisted things around a bit and used the same example with different numbers making the triangle have more area to prove that just because there is a horizontal line doesn't mean it will always have more area under the curve than a triangle.

Anyway, my point is, I took a wavy horiztonal line and made it perfectly flat and a curved line and made it a delta x / delta y.

Mathematically it's wrong to do this because I missed (or added) area. So how can I look at a curve and know the area under it without a computer doing the work for me; and without approximating?

You can't.  But you can break that curve into small regions and add the areas of those regions.  You can approximate the area of those regions by rectangles, with height either equal to the lowest value of your curve in that region, or the highest value of the curve in that region. (There are other and better estimates of the values which can give better accuracy with fewer divisions, but this choice is the simplest to describe and perform) Obviously the latter choice will give a value greater than the actual area under the curve and the first choice will be smaller.  This is tedious, but totally possible.  And as you make the rectangles narrower the two estimates will get closer and closer to each other.  For any curve drawn by plotting with a real pen or pencil they will eventually converge on the actual value of the integral.  But you don't have to go there for any real problem.  Depending on how "bouncy"  the plot is you only have do a few subdivisions and average the two estimates to get a "good enough" answer.  This is essentially what you were doing with your rectangle and triangle estimate, using just a single division.  It is also what the computer is doing, but the computer is both very fast and infinitely patient so can easily divide a page size graph into several hundred or several thousand divisions and achieve accuracies of tiny fractions of a percent instead of the few percent error bounds on most manual integrations.

As you get into the higher realms of calculus you will find there are functions where these upper and lower estimates do not converge to the same value, but they are esoteric functions that are rarely if ever encountered in "real" life.  But they are useful tools to use to solve real life problems.  The Dirac delta function is one example you may have heard of.  One of the most famous is the function which has value 1 for all irrational numbers and 0 for all rational numbers.  Don't worry, you will be a few years beyond basic calculus before you have to deal with such concepts.

Finally, you can use similar ideas to define and compute the derivative of the graphed function.  And there are real things that you can do with these estimates.  But the real power and the real focus of entry level calculus courses is for standard functions which have functionally defined derivatives and integrals.  Things like the trig functions, and powers of numbers along with their sums (polynomials) and a few other functions.  These functions allow formal and general solutions without the tedium of the manual methods required for arbitrary functions.  And these functions can be used to approximate a huge number of real life situations.

[Nominal Animal]

Mathematically it's wrong to do this because I missed (or added) area. So how can I look at a curve and know the area under it without a computer doing the work for me; and without approximating?

Assuming you mean how to estimate the area, without resorting to approximations:

Overlay a regular rectangular grid (like graphing paper) over the curve, and count the number of square cells strictly included in the area. Also count the number of cells at least partially included in the area.
The correct area is somewhere between the two.

If you look at each partially included square cell, you can estimate how much of it is within the curve.

Instead of estimating each square cell, you can can also subdivide the partial ones into four equal squares, each having 25% of the original square area, and recursively do this until you reach as high a precision as you want.  (Three subdivisions gives you 8×8 subcells, each having 100%/64 = 25%/16 = 1.5625% of the area of the original square.  You don't need to do much recursion, really, in other words.)


Human perception is wonky.  For example, we estimate widths and heights completely differently.  One must consciously adjusts ones own instinctual estimates to arrive at something closer to reality, and this adjustment is something one can work on, by estimating things and then measuring them.
This is what some call "a calibrated eye": it is something you can train yourself in.

One crafty trick is to make one estimate, then rotate the graph 90°, and make a new estimate, completely ignoring the previous one.  The difference tells you how much difference there is between width and height estimation in your instinctual faculties.

[Nominal Animal]

That reminds me of a calculus exercise I liked, and many are familiar with:

What height and radius minimizes the surface area of a cylindrical container with unit volume, \$V = 1\$?

(The volume \$V\$ of a cylinder with radius \$r\$ and height \$h\$ is \$V = \pi r^2 h\$, and the surface area is \$A = \pi r^2 + 2 \pi r h + \pi r^2 = 2 \pi r (r + h)\$.  The answer is approximately \$r \approx 0.54\$, \$h \approx 1.08\$, the cylinder fitting in a cube with each side \$1.08\$ units long.  The exact algebraic values are not difficult to find.)

The method you use to work this out can be used to solve all similar problems, and does extend to a number of "minimize a property while keeping this other property fixed" -type of problems, and I've actually found it useful in real life.

[RJSV]

   Good MATH question!
   I generally respond, when can, to some (first year) Calculus questions.  Partially because that math catagory often (seemingly) lacks common sense or intuitive approach / description.  But, with good teacher it's not so daunting.  I was lucky, first couple of college semesters, to have a good Prof.

   My interpretation has been that there is a definite heirarchy, going up the scale of complexity, and if Calculus is not taught with that orientation, that is taught by just throwing everything into (a textbook), then it's difficult to follow.

   Briefly, the order of teaching that, would start with concept of 'limits', especially the subtle outcomes obtained by repeating a function...even extrapolating to case, when your parameter goes on 'forever'.
Like, for example, take a '1' divide it in half, and keep repeating....that easy example helps understand the non-intuitive concept of limits...

   The other two concepts, up the ladder of expertise, are the derivative, (best taught first), and then the Integral.  Those two are complementary processes, as your derivative, (or rates of change), can be processed, to obtain integral, although first step is incomplete, in that there will be, also, some constants to solve.

   Anyway, the teaching of this math, or mis-teaching, has always been a pet-peeve of mine!
Others, in school, have, actually, hired me, for short stints, helping sort out homework and helping study for tests.
PM me questions OK too, even if I'm busy, (might take day or two for response, lol)

- Rick

[Nominal Animal]

I generally respond, when can, to some (first year) Calculus questions.  Partially because that math catagory often (seemingly) lacks common sense or intuitive approach / description.  But, with good teacher it's not so daunting.  I was lucky, first couple of college semesters, to have a good Prof.

I agree, first year calculus is often approached from odd angles, and can be difficult to grasp because of that.  A good prof can approach it from different directions, and adjusts their approach to suit the students best.  One is very lucky to have that sort of a prof; I know a few (mostly in physics, though).

The reason I push for looking at interesting problems first is closely related.  One discovers the best suited approaches for themselves only through experimentation, and when an effective approach is found, it is much easier to find additional (in depth) material with similar approach.

For example, I learned basic programming before I learned basic calculus, and had written a "Thrust"-like game in Turbo Pascal.  So, when the continuity of a function was discussed, I intuitively grasped how it geometrically meant that if you have two points on different sides of a line, a line drawn between those points must intersect with the original line, something I had battled with to find out when my triangular vessel intersected the cave edges.  From computer graphics I understood how the existence of the differential of some function is related to points where the curve does not have a well-defined tangent, like at the vertices of a polygon.  Hearing about the Bisection Method (for root finding), I immediately understood how it is analogous to binary search so commonly used in programming, and so on.  When things slot into place like that, learning is easy –– and a lot of fun, too.

Indeed, my first tangle with Linear Algebra was much before that, back when I first became interested in descriptive geometry, and sought books on 3D computer graphics; then had to try and find out what the odd notation meant, not even knowing its name!  (That is also at the root of why I keep repeating basics that all participants in a thread already know: it just so frustrated me as a kid understanding the overall concepts, but not being able to unravel them down to the individual arithmetic operations I could write a program to do.  All I would have needed was two pages describing the exact syntax, dammit!  This was in the late 80s, till 1992 or so, so before Internet, and in the era of Libraries with Books made from Trees.)

I first encountered complex numbers in computer graphics and fractal generation –– fractint, anyone? –– so it was very intuitive for me to approach them as something between scalar real numbers and 2D vectors, and how that affected the algebraic rules of the real numbers I was familiar with.  Stepping forwards into calculus on complex numbers, and later wave functions in physics, was no problem; just new useful tools (operations) on complex numbers.

[westfw]

One thing that baffles me is that we can take functions and their derivative or integrate them, but what if we only had a graph?

You probably won't learn that in a calculus class, at least not in any useful way.

It's a field of computer science, though.  Called "Numerical methods", this includes a bunch of matrix-based methods of solving equations, as well as methods for differentiation and integration given a table of (X,Y) values.

You probably learn in calculus class that the derivative is just the slope of the tangent line deltaY/deltaX as deltaX approaches zero, and integration is just a sum of rectangle areas deltaY*deltaX (also as deltaZ approaches zero.)  As far as I remember, a lot of the numerical methods consist of compensating for deltaZ not being very close to zero, usually by taking into account other known points "near" the desired X value.

See for example: https://en.wikipedia.org/wiki/Numerical_differentiation

[westfw]

It's a field of computer science, though

(I guess that technically, this is just math, rather than "computer science."  The algorithms developed are well suited to being implemented on computers, though, and an additional aspect is keeping track of the errors introduced by the limits of your computational methods (like the limits of floating point formats.))

[bostonman]

You probably learn in calculus class that the derivative is just the slope of the tangent line deltaY/deltaX as deltaX approaches zero, and integration is just a sum of rectangle areas deltaY*deltaX (also as deltaZ approaches zero.)

This is basically what I learned. I did all three levels of calculus and moved onto Diff EQ, but I never could grasp the concepts of calculus to solve real world (and theoretical) problems. I did well in Diff EQ too, understood it, but never quite could grasp the differences of calculus versus Diff EQ (I think one solves instantaneous and the other creates a formula over time).

With algebra and trig I understand the goal that is trying to be achieved. Certainly I don't remember all of algebra and willing to bet not all of it is used ever again. Being able to solve for X is the goal, but, if a person doesn't reduce the equation completely because they overlooked a rule, the solution can still be obtained thus making algebra more forgiving. A good example is reducing logs. I don't remember off the top of my head, but something to do with a negative log in the denominator and having to multiply a log over a log. Anyway, regardless, if this isn't done, one can still solve for X. It may be a bit of extra number entry in a calculator, but it can still be solved.

With calculus (and why I stated "theoretical" above), some of it are word problems that wouldn't have the ideal answer, but, calculus is far less forgiving. An example of "theoretical" is my previous question I remember (but for the life of me have no idea how to solve) is a swimmer swimming to shore at one rate while the water current is pushing the swimmer sideways, where on the shore will the swimmer land?

This is theoretical because a human wouldn't swim at a constant rate, the water current isn't constant, etc... but I understand the concept (it's one rate of change with respect to another rate of change). I understand the goal that is trying to be achieved, I can differentiate a function, I can integrate a function, but how to piece it all together is what I never understood.

I remember first starting calculus and thinking why can't the 'd' in dx/dt just cancel leaving you with x/y. Then I saw equations with x d/dt and questioned whether the x is being pulled out of dx and leaving you with d/dt or if d/dt is a calculus "symbol" and the x is just a variable that was never part of d/dt in the first place (actually, now that I think of it, I think it still confuses me). Such as does was 'abc' d/dt really d(abc)/dt (the parenthesis are included only to make it easier to read)?

I had a professor who was very good at teaching calculus, however, his practice tests were the same as his regular tests with the exception of changing numbers. Providing you could do well with the practice test, you were guaranteed to do well on the test. If not for this, I probably would never have passed; or maybe it would have forced me to understand the foundation thus being able to provide answers in this thread and not asking questions.

Sometimes just being able to have someone present to ask "why" helps me understand better rather than just seeing the common introduction lectures on calculus. Those lectures (and usually people I've spoken to who explain it) talk too fast and while I'm processing the initial stuff, they are already on the next step forcing me to get confused.

[TimFox]

Nominal Animal's example "What height and radius minimizes the surface area of a cylindrical container with unit volume, V=1?" of an extremum several posts up is an excellent example of a practical use of calculus.
Finding a minimum or maximum of a function is a common problem in engineering.
A common teen-age gripe is that there is "no algebra in the real world".
My reply to that is "yes, there is algebra in the real world and it's all story problems".

[rstofer]

That reminds me of a calculus exercise I liked, and many are familiar with:

What height and radius minimizes the surface area of a cylindrical container with unit volume, \$V = 1\$?

The one I like is "Gabriels Horn":  imagine a horn constructed by revolving the function y=1/x from x=1 to infinity around the x axis.  Now, find the surface area and volume.

OK, the volume is PI units so that is how much paint it takes to fill the volume.  But the surface area (just the walls) is infinite so there isn't enough paint in the universe to paint the surface.

These values are calculated as the sum of surface rings (for the surface) or disks (for the volume).

https://en.wikipedia.org/wiki/Gabriel%27s_horn

Just fooling around with integration.

With a computer, I kind of like Riemann Sums since I have already coded it in Fortran.  The attached code was written for a demonstration so is hardly minimal.  The cool thing is that it shows the center Riemann Sum is a pretty good approximation.

In any event, it integrates a function:

Code: [Select]

f(x) = (x**2) + sqrt(1.0 + (2.0 * x))
from 3 to 5 with 100,000 slices.  There are 3 Riemann Sums (Left, Center and Right) along with trapezoidal.  It serves no particular purpose other than providing entertainment - I like Fortran (since 1970).

[TimFox]

Of course, the volume of paint required to coat the interior surface is not a physical situation, since paint has a finite thickness and will clog the pipe past a certain small diameter.

[rstofer]

You probably learn in calculus class that the derivative is just the slope of the tangent line deltaY/deltaX as deltaX approaches zero, and integration is just a sum of rectangle areas deltaY*deltaX (also as deltaZ approaches zero.)

I remember first starting calculus and thinking why can't the 'd' in dx/dt just cancel leaving you with x/y.

Of course it can!  It just isn't taught that way; it typically starts out with limits and works down.

Consider y = x² and let's 'nudge' it a little:

y+dy = (x + dx)²
y+dy = x² + 2x*dx+dx²  But dx is tiny and dx² is even smaller - toss it!
Subtract off the original equation y = x² and all we have is
dy = 2x * dx or
dy/dx = 2x

This is known as the infinitesimals approach and was used in the early years of Calculus.  It is discussed in the still popular "Calculus Made Easy" book by Silvanus P Thompson (1910).  There is an updated version with comments by the late Martin Gardner (THE math guy at Scientific American magazine).  Pages 21..24 of the Martin Gardner edition with this discussion on page 52.  I highly recommend the book!

You need to think of 'd' as a 'little nudge'.  dx is a little nudge of the x variable and dy is a little nudge of the y variable.  dy/dx is the little nudge of y as a result of a little nudge of x.

[rstofer]

Of course, the volume of paint required to coat the interior surface is not a physical situation, since paint has a finite thickness and will clog the pipe past a certain small diameter.

Sure, mess up the beautiful math with reality.  Mathematicians live in their own version of reality unencumbered by physical constraints.

[bidrohini]

You can search the specific topics at khan academy.

[TimFox]

Of course, the volume of paint required to coat the interior surface is not a physical situation, since paint has a finite thickness and will clog the pipe past a certain small diameter.

Sure, mess up the beautiful math with reality.  Mathematicians live in their own version of reality unencumbered by physical constraints.

Even mathematicians compute volume with all three dimensions...

[rstofer]

You can search the specific topics at khan academy.

Or you can solve them at symbolab.com or graph them at desmos.com.  Khan Academy is one of my favorite resources.

[westfw]

I had the somewhat interesting experience of having Calculus, Physics, Chemistry, and EE classes all teaching a bunch of the same math at the same time (and throw in that CS class on numerical methods, too.)  The non-Math classes were all sort-of trying to catch students up from whatever they had learned in Math to what was needed to do the EE/Physics/Chemistry work (particularly WRT 3D things like gradients and curls and stuff that I only vaguely remember.)
I mean - Maxwell's equations, right?  Fundamental to physics and theoretical EE:


Sigh.
I guess this left me (temporarily, anyway) with a bunch of problem-solving skills, and less of the math theory (as I've complained about before.)
So you might want to add some physics books to your calculus curricula, to have "practical examples" at hand.

(And then, after years of calc, AC circuit analysis is mostly done using phasors, which more or less combines Fourier theory (all signals are sine waves) and complex number theory (complex numbers and exponentials make sine waves too, and they're really easy to differentiate/integrate) to do away with most of the actual need to do any calculus.  I was SO pissed.  But it was really neat.))

[wizard69]

...Otherwise you might be better served by boning up on linear algebra, boolean logic and other fields.

This is a good suggestion.   Sometimes I have to break out the books for trig or some geometry calculation that I should have remembered.   Boolean logic might not apply if she is interested in instruments but I think your point is the math follows the tech.   

Depending upon what I'm doing at the moment I might grab "Engineering Mathematics Handbook" [Jan Tuma], Machinery's Handbook [Industrial Press], HandBook for Electronics Engineering Technicians [Kaufman and Seidman] and a couple of others.    I find Handbooks are great for refreshing what you already "know", and frankly the Tuma book covers more than I ever knew.   The reality is in the tech world the math does not live alone, if you are doing math it is usually to accomplish something electrically or mechanically.   Given that the mind becomes rusty in both venue, if you have forgotten the math you probably have forgotten the implementation details in the mechanical or electrical world.

As for those "others" some of them are PDF's found on the internet.   One that might be applicable here is "Handbook of Filter Synthesis" [Anatol I. Zverev], which is a challenge to read.   Another is: "Introduction to the Mathematical Theory of Systems and Control" [Jan Willem Polderman & Jan C. Willems].   In both cases found free on the net.   Most of the stuff I've posted is rather old, but the basics don't often change.   In any event this highlights that good info can be found on the net.

[wizard69]

(I plan on doing mostly analog electronics related to music synthesizers.)

This may or  may not require a lot of math on your part, the tendency is to say yes calculus is required.   More importantly if you already know Calculus it might benefit to start searching for books focused on applying math to filter and oscillator design.   The math will come back, especially in the context of an applied circuit.   In a different post I posted the titles of a few books that might be of interest.

It would also be advisable to take advantage of downloads from the lines www.analog.com (LTSpice and other tools).    Www.TI.com has its own resources including a filter design tool plus they run an "Analog Design Journal" (ADJ), that might be right on target (once in awhile anyways) for your interests.   The archive is rather large and I'm not about to search it all.

In any event I'm thinking what you want to do here is to skip the refresh of Calculus and dive into some circuit design.   That would start to refresh memory and highlight what if anything you really need to brush up on.   A lot of the math these days is solved and in the form of software tools.

Now if your goal is to bush up on the math to then create some nice open source tools for electronic music device design I'm sure many people will be happy.   Also I've assumed that the goal here is electronic instruments not software based virtual ones.   AS for the electronic instruments and software you don't even need to build an entire app, just generate designs for spice to digest .   I'm just looking for ways to leverage your programming skills in this endeavor.

[wizard69]

When you say what do your think, I'm not thinking well after being up way to long but I will try.   I really don't see the value in getting a bunch of math text books if you have already taken a full series of classes and did well!   In stead look for hand books that are directed at electronics engineering.   The reasoning is simple, the math will be covered in a way that is compatible with electrical engineering.   Compatible might not be the right word, the idea here is that the math will incorporate electrical engineering concepts, thus you refresh in a way directly compatible with your goals.
 

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'd do neither.   Why?   Because most of what you will need to know will come back real fast and can be refreshed from a handbook or the electronics text you will use.   Beyond all of that a lot of the heavy math is incorporated into various bits of software that you are likely to use.

Besides mistakes in usage often are not pure math.   One of my bigger frustrations in the distance past was evaluating a trigonometric equation wrong and putting a positive 1 where a -1 was needed.   Sure going through a bunch of text might help to avoid such stupidity but it will also put you months away from doing anything related to audio instrumentation.

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.

Yes this is a huge problem.   But again I think you are focusing too much on what you think you have forgotten.   You probably did forget a lot but it is not completely gone, I know that I've forgotten a lot but you can get back into it, by picking up a handbook or a text focused on the electronics.   Walk by any engineers desk and you will find all sorts of reference books.

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.)

Again I wouldn't invest in a teaching text book.   You have already gone through that and will just need refreshing from time to time.    Spending months on getting to 100% mathematically is a big distraction when you can start engineering hardware tomorrow.   I'd spend the money on breadboards, power supplies and instrumentation.