Have you ever used "symbolic logic" to help you learn calculus?

[timmeister37]

TL;DR Summary

The title says it all if you want a mighty brief summary.

I failed to become a mechanical engineer because I could not learn how to do sequences and series in Calculus II. I could get Cs and Bs on all the concepts of Calculus II until I got to Sequences and Series. Then I would get F minuses on any tests involving series problems such as Infinite Series, Power Series, Taylor series, etc. My final grade for Calculus II the first time I took it was an F. My final grade for Calculus II the second time I took it was a D. But in my major, only a C or better in Calculus II was considered passing. So my parents refused to keep paying for my college education, and I quit attending the university.

Years after I took Calculus II at a university, I took an "Introduction to Philosophy" (henceforth, I will refer to this class simply as Philosophy class) class at a Community College. In my philosophy class, I learned about symbolic logic. I don't think I have ever used symbolic logic outside of my philosophy class. But when I was in philosophy class learning about symbolic logic, I remember thinking that perhaps symbolic logic could have helped me to learn sequences and series in Calculus II.

After I stopped taking Calculus II at a university, but before I took philosophy class at a Community College, I learned how to successfully complete sequences and series problems in a Calculus textbook I owned by reverse engineering the problems. My textbook had the answer to all the odd numbered homework problems in the back of the textbook. I learned how to fill out the entire Series problem by starting with the answer and working backward. This felt like a major epiphany at the time. However, I don't know if learning to reverse engineer the series problems would allow me to solve a series problem if I was not given the answer. So I don't know whether or not I could have taken one of the calculus tests I was given on series and pass it once I learned how to reverse engineer the problems.

Have you ever used symbolic logic to help you learn how to solve any calculus problems? If so, what areas of calculus did you learn by using symbolic logic? For instance, was it differentiation, integration, sequences and series, etc?

How did you learn how to solve series problems in Calculus II such as Power Series, Infinite series, Taylor Series, McLauren Series, etc.? Did you learn how to solve these problems by reverse engineering them?

 

[jedishrfu]

Sadly, no. I’ve used Boolean algebra and truth tables to construct a circuit but I’ve never used symbolic logic to do any other math subject.

im assuming by symbolic logic you mean Ps and Qs with syllogisms right?

I remember when I was a teen totally fascinated by Spocks logic on Star Trek that I tried to imitate him. i found a logic book that my dad used in school and studied the syllogisms and thinking but it just wasnt conducive to imitating Spock. However, Boolean algebra seemed to help.

what was it about series and sequences that threw you off. Series are the precursor to doing integrations in Calculus. The hardest part for me was the convergence tests and why they worked or why they were limited.

 

[timmeister37]

im assuming by symbolic logic you mean Ps and Qs with syllogisms right?

yes

 

[jedishrfu]

Perhaps this older thread will help

https://www.physicsforums.com/threads/using-symbolic-logic-to-learn-calculus.863731/

Basically some folks said yes it helped because of the Ps and Qs used in theorems. But I never felt it helped me explicitly as by then my mathematical skills allowed me to read and understand how to use a theorem.

 

[nucl34rgg]

I've never actually needed it. Even in analysis, the most complicated expressions just required a bit of careful thought. I've never dealt with anything complex enough to require that I translate it into symbolic logic. As long as you know contrapositives and how to negative quantifiers, I think you should be ok.

For elementary calculus, you just need a good background in trigonometry and algebra, and a bit of understanding of functions and transformations.

You may find "symbolic logic" helpful in the following way. To express a limit, we say that $$lim_{x \to x_0} f(x)=L$$ if $$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } \forall x \in D,|x-x_0| < \delta \implies |f(x)-L| < \epsilon.$$ (Here $D$ is the domain of $f$. This part of the domain statement is often omitted, but is essential for understanding the logical structure of the statement.)

So, to show the limit doesn't exist is to show $$\exists \epsilon > 0 \text{ such that } \forall \delta > 0, \exists x \in D \text{ such that }|x-x_0| < \delta$$ and $$|f(x)-L| \geq \epsilon.$$

Notice the negation of $p \implies q$ is $p$^~$q$, where $^$ means "and" and ~ means "not".

At first, symbolic logic might help you understand what is being said more carefully.

 

[bhobba]

At first, symbolic logic might help you understand what is being said more carefully.

I loved analysis from the start. The reason was I read a book on symbolic logic out of interest a couple of years before, and what amazed me was when asked to prove something like a limit is unique, it seemed to follow the same pattern. First, you fix epsilon, then try to 'mechanically' formulate the opposite. You often see a contradiction, and the proof sort of pops out. That is at the beginning - after a while, you see how to do it without the detailed symbolic logic. Later still, you use epsilon proofs only if you want to be pedantic.

Thanks
Bill