What would it take to learn Calculus at a practical level?

S Haque

Joined Jan 2, 2017
12
The highest math classes that I took WAY back were technical math (for electronics technology) and pre-Calculus. I recall not enjoying pre-Calc so much, but I think I did like things like trigonometry. Playing with micro-controllers has rekindled my interest in electronics. How would I go about gaining functional ability in Calculus so I could work with and understand circuits better? Or would I be better off just cobbling things together?
If I may - about two years ago, I was at a point where I needed to increase greatly my calculus skillset, which I had hated since my undergraduate days. Then I barely scraped through (yes, I admit, I got C's in each class), and stayed away as much as possible - and avoided it where possible.

Well, now I love it, and enjoy taking the time to do the calculus steps by hand in many problems all the way to vector integral and volumetric calculations. I was looking for a book to use (to add to my other books on calculus/pre-calculus) and came across a recommendation on amazon.com, which has been one of the most amazing finds for me: Calculus: An intuitive and Physical Approach by Morris. If you are kindle user (recommended) also try out the kindle edition so your big fat book (heavy) is absolutely portable wherever you need it. And do the step wise chapter exercises please ! It is fun to do the math for complex surfaces/geometric objects and processes. Wow!
 

S Haque

Joined Jan 2, 2017
12
Calculus was an aha moment for me, like pulling the curtain away from the wizard of Oz. So for me personally, I'd recommend learning it even if you barely use it. It's a great tool for viewing the world, like learning some history or philosophy.
Yes, now I am able to address the problems I needed to understand to describe the mode of electromagnetic emissions from a plasma jet. Why? Well, I helped develop a miniature electric propulsion system from scratch for a cubesat --- did all the work, etc. from concept to flight hardware, and AFTER it was done realized I didn't have a clue of how to describe the RF/EM being emitted from the system's plasma jet. And that was after 2 years of hw/fw/sw development -- and the RF/EM being emitted was a primary cause of communications blackout whenever the system was energized which cost me an arm and a leg to learn how to mitigate :)

I don't have Thompson's book. Does it explain what a derivative or integral are, or does it just show how to evaluate expressions that contain them?
Please see the book reference I posted just now in my earlier message. Morris took the time to write out in long hand all of the reasoning of early calculus and his book was written in the sixty's for students at college level.
 
This has been an interesting thread, that I have been wanting to reply too. It turns out that Calculaus has a lot of memorization. It takes on an entirely different meaning when taught by an engineer. So, taking it continuing education rather than mainstay was more helpful. I still have a hard time using it to solve problems from scratch, but at least I understand it.

So, yea, you can use it to find the formula for the volume of a cone, but you can also look up the formula because someone did the work for you.

Simulation, might avoid the use of Laplace transforms.

Complex numbers makes AC circuit analysis a lot easier.

Calculus could come in handy for understanding spectrum analysis.

There may be a lot of stuff missing like the Nyquest theory and say FFT theory.

You can play with calculus concepts at www.wolframalpha.com

Numerical integration and differentiation will allow you to write/understand a PID controller. I wrote a couple of unusual PID controllers. In one case I used the setpoint of one controller (or the proportional band) to actually control another temperature. e.g. The temperature controlled was the ambient in an environmental chamber, but I wanted the surface of my sample controlled. So, I found the proportional band of the OEM controller which turned out to be +-10 degrees and used that to be my -100 to 100% output of my PID. Reset windup happens in software, but not an analog controller. So, it's numerical integration and differentiation.

Now, I needed a formula that essentially needed to relate the rotation of a shaft with a take up spool,on it to amount of material exposed. It did not have to be 100% accurate. The idea was to move a piece of clear teflon about 6" every so often. The 6" was the opening. UV exposure caused deposition on this curtain and it had to be moved out of the way during the process. So, you given initial diameter and thickness. Your starting equation is based on s=rθ and t = the thickness. s is the exposed material, r is the radius and theta is the degrees to move . You need the initial diameter. It's kinda like the VCR problem. I did not have access to anything measuring the deployed material. The packing would not be perfect and the deposition and wrinkles would change the diameter slightly, but that wans't important. A synchronous motor was used too. Motor torque was small. Initially we contemplated a tension controller which was not needed that I made from scratch. There was a slight glitch that I did not accommodate, You have to have a slight negative current when operating at low tensions. We didn't need the controller.

I actually used a turns counter and I had to move a synchronous motor a variable amount of time like every 3-4 minutes. So, yep, it's a simple programming problem given the formula and some knowledge of trig, but it's a much harder "engineering problem" just given the "task". Mechanics had to be solved too. This used feedthroughs in a vacuum system. The PTFE was about 0.001 thick and about 5" wide.

The next task was timing. It turns out that a IBM PC could not do it because of the timing resolutions. It needed extra hardware, I used a PDP-11 for one system. All because we had it.
 

WBahn

Joined Mar 31, 2012
26,398
This has been an interesting thread, that I have been wanting to reply too. It turns out that Calculaus has a lot of memorization.
I found it to be very much the opposite. But I was very fortunate to have teachers, both in high school and in college, that emphasized understanding the fundamentals and learning how to apply them to problems in general. Take your cone example. You say that, yes, you can use calculus to find the volume but you could also just look it up. Well, if you need to know the volume of a cone, look it up. Learning to do so using calculus isn't with the aim of repeating common results that are tabulated in tables, but to solve similar problems that AREN'T in the tables.
 

Thread Starter

bullzai

Joined Jan 19, 2015
42
If you are kindle user (recommended) also try out the kindle edition so your big fat book (heavy) is absolutely portable wherever you need it.
Sometimes Kindle books don't reproduce equations and images well. I like paper sometimes, but, yeah, if the book is huge, the Kindle version is probably more practical!
 

Thread Starter

bullzai

Joined Jan 19, 2015
42
I've heard that the proper reference is "the Calculus." If that's true, why do people rebel and just say "Calculus?" hehe
 

Thread Starter

bullzai

Joined Jan 19, 2015
42
Numerical integration and differentiation will allow you to write/understand a PID controller. I wrote a couple of unusual PID controllers.
I did come across the concept of PID for motor control and a heater control and noticed the the "C" connection. heh As I play with an Arduino more, I run into more math. So, if it isn't at the circuit level, it's in the code. Can't escape. heh
 

WBahn

Joined Mar 31, 2012
26,398
I did come across the concept of PID for motor control and a heater control and noticed the the "C" connection. heh As I play with an Arduino more, I run into more math. So, if it isn't at the circuit level, it's in the code. Can't escape. heh
Nope. I'm writing neural network code and the core of the learning algorithm is error back-propagation which requires using what is known as the chain rule and the partial derivatives of the outputs with respect to not only each of the inputs but also each of the weights. Considering that each layer typically has well over a thousand of these, it is important not only to be able to take the partials, but also to be able to generalize them so that they can be put into a loop structure.
 
Wbahn:

The memorization just killed me. Stuff like the basic integrals. Or even stuff like the sin of 45 degrees. I had to spend most of my time memorizing to be able to do the problems without the book in front of me. It took me 2-3 times longer to learn. There wasn't enough time on an exam to derive them. So, if you asked me what's the indefinite integral of sin(x); I'd know it contains cos(x) and it might have minus sign or what is the formula for integrate by parts.

Algebra has rules and since I learn kinesthetically or by repetition, every subsequent math class required Algebra so repetition was satisfied.
 

WBahn

Joined Mar 31, 2012
26,398
Wbahn:

The memorization just killed me. Stuff like the basic integrals. Or even stuff like the sin of 45 degrees. I had to spend most of my time memorizing to be able to do the problems without the book in front of me. It took me 2-3 times longer to learn. There wasn't enough time on an exam to derive them. So, if you asked me what's the indefinite integral of sin(x); I'd know it contains cos(x) and it might have minus sign or what is the formula for integrate by parts.

Algebra has rules and since I learn kinesthetically or by repetition, every subsequent math class required Algebra so repetition was satisfied.
I don't know about your specific case, but things like the sin of 45° shouldn't be too dependent on memorization. If you sketch (even just in your mind) a 45°-45°-90° triangle with sides of length 1 then the hypotenuse is, by the Pythagorean Theorem, equal to √2. Now you use the definition of the sine of an angle and you get 1/√2 (or √2/2). I almost always have to do an exercise like this if asked for the values of the basic trig functions for 30° of 60°. About all I remember reliably is that either the sine or the cosine is 1/2, but a quick mental sketch immediately lets me figure out all of them with confidence.

Now, it can certainly be argued that to do this you had to memorize a list of nontrivial concepts, such as the sum of angles in a triangle adding up to 180°, what the Pythagorean Theorem is, and that the sine of an angle is the ratio of the lengths of the side opposite the angle to the hypotenuse in a right triangle. Others (myself included) would counter that concepts at this level are foundational and just like we "memorize" that 2 + 2 is 4, we should be able to internalize these concepts so that they become core knowledge that we can recall decades after last using them as things we simply "know" as opposed to things we have "memorized" and that thus have a shelf life before we forget them.
 
I know that trick now, but I wasn't taught it. I was given a table to memorize. If someone would have taught me in 3rd or 4th grade that the digits in the 9x table add up to 9 in 4th grade. then I would not have had trouble with 54, 56, 63 and 64 for the 8x and 9x table. e.g 6*9 or even the product of 2 even number has to be even. Those sorts of rules would have helped me a lot. No, we had to memorize the times tables.
 

WBahn

Joined Mar 31, 2012
26,398
I know that trick now, but I wasn't taught it. I was given a table to memorize. If someone would have taught me in 3rd or 4th grade that the digits in the 9x table add up to 9 in 4th grade. then I would not have had trouble with 54, 56, 63 and 64 for the 8x and 9x table. e.g 6*9 or even the product of 2 even number has to be even. Those sorts of rules would have helped me a lot. No, we had to memorize the times tables.
I really feed sorry for you -- you were done a serious disservice by the educational system you were in.

I've looked at recent (well, actually about 15 years old at this point) trig texts and was very disheartened to find that how they taught trig functions was via a picture of a TI-83 calculator showing which buttons to press. Nowhere that I could find did they even mention the notion that the sine of an angle was defined as the ratio of two particular sides of a right triangle. I didn't have time to look to see what kinds of problems they expected students to be able to solve with that level of understanding.
 

Thread Starter

bullzai

Joined Jan 19, 2015
42
I've looked at recent (well, actually about 15 years old at this point) trig texts and was very disheartened to find that how they taught trig functions was via a picture of a TI-83 calculator showing which buttons to press. Nowhere that I could find did they even mention the notion that the sine of an angle was defined as the ratio of two particular sides of a right triangle. I didn't have time to look to see what kinds of problems they expected students to be able to solve with that level of understanding.
Sounds like learning how to take a test using a TI-83 calculator!
 

Papabravo

Joined Feb 24, 2006
16,183
I don't have Thompson's book. Does it explain what a derivative or integral are, or does it just show how to evaluate expressions that contain them?
It explains them in clear, concise, and accessible language. It was my intoduction and helped immensely when I encountered the more theoretical approach replete with theorems and delta-neighborhoods (shudder!).
 
For my memorization issue, and non-course related, this http://www.wolframalpha.com/input/?i=integral(sinx) works.

BTW: I got kicked out of EE after two years because of the memorization issue, then I went to a 2-year school and graduated with a 4.0 GPA using some tricks like being a "ghost" in classes and being thrown out of class with "If you have better things to do, don't bother coming to class" I did get a 4-year generic engineering degree as well. I happened to graduate from that school while on a leave of absence.

Basically because I had to be able to do the course work without the book to pass the exam. Doing the course work with the book in front of me was much easier except for Dynamics. The Thermodynamics professor had 1/2 the exams closed book and the second half open book.
 

S Haque

Joined Jan 2, 2017
12
I don't know about your specific case, but things like the sin of 45° shouldn't be too dependent on memorization. ... Others (myself included) would counter that concepts at this level are foundational and just like we "memorize" that 2 + 2 is 4, we should be able to internalize these concepts so that they become core knowledge that we can recall decades after last using them as things we simply "know" as opposed to things we have "memorized" and that thus have a shelf life before we forget them.
After being re-introduced to Calculus properly by Morris (see book reference above), I learnt a great way to learn the subject matter: Practice by hand, literally on pen and paper. And do this from time to time to focus your brain on the steps -- not at once, only in certain chunks at certain times. He literally gives narration about topics from various subject areas, and then gives the ANSWERS to the questions, leaving readers to take out their NOTEBOOK and pen (suggest, pencil and rubber :) to work out he steps -- if you succeed, you get the answer, if not, you work through the preceding examples contained in the associated chapters !
 

wayneh

Joined Sep 9, 2010
17,152
OMG, where I went to school there was nothing so dreaded as an open book exam. It meant you had to know things at a much higher level.
 

Papabravo

Joined Feb 24, 2006
16,183
OMG, where I went to school there was nothing so dreaded as an open book exam. It meant you had to know things at a much higher level.
No kidding. An open book exam meant you had to know things much better and deeper. Anybody searching the textbook for a piece of information was already lost, and you knew who these people were immediately.
 
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