Supplemental topics for precalculus

In summary: Applications of Calculus: Derivatives of Various Types, Integrals of Various Types, Taylor Series, Infinite Series.
  • #1
somefellasomewhere
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TL;DR Summary: I will be reviewing/filling in gaps in my precalc knowledge over the course of 6 months and I think I'll have time to go off the beaten path a bit with the material I cover. What topics could supplement this that maybe I would not be able to learn in a typical precalculus course/book?

New user so apologies if I posted this in the wrong section. I'll start by a long winded and mostly irrelevant explanation of how I got here... So I went to a pretty bad high school at a town that has just barely broken out of its 3 digit population status, and as such I didn't have much opportunities to thrive academically (nor did I want to for the first three years, so this is mostly my fault).

I kinda shuffled my feet through my math classes up until my senior year, where I took my high school's precalculus class and college algebra/trig. While this might be enough preparation for calculus in college, I want to be extra thorough since my hs precal class has been not great (teacher is wonderful, she's the first teacher who hasn't shown complete disinterest in the subject and our education, however she's had to spend so much time teaching stuff we should have already known that it has slowed the class down significantly) and I can already feel my college algebra knowledge fading.

So I've bought a cheap precalculus book off of ebay (the axler book), and I've decided I'll work through the non-trig sections of the book since I feel I have a firm grasp on trig and my main goal is to fill in the gaps in my algebra knowledge. I have about six months to do this, and since a lot of this will be review (and I will have a lot of time over the summer) I think I will have time to supplement this with some additional topics that maybe I would not be able to cover in a typical precalc book/course.

An example of the type of thing I'm thinking of is parametric curves, which has an appendix in the back of my book, and flipping through the pages it seems really interesting but a somewhat obscure topic. Maybe I'm wrong about that though, from what I've gathered it becomes relevant in calc 2. But are there any other topics like this that aren't typically taught in high school but are graspable to high school level students? I thought about learning to code alongside this venture, but I have tried to self-teach myself to code before and I never really got past a very elementary understanding of programing that way.

Sorry for the excessive verbage, any suggestions yall could provide would be greatly appreciated.
 
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  • #2
Oh goodness... I forgot to explain my motivations for doing this. I'll be a college freshman majoring in biology next year and since I've gotten a lot of gen eds out of the way, I'll have the opportunity to add a minor or second major in math (which I've only recently realized the importance of, in biology and in general). I want to build up a very firm foundation since I will be taking a lot of classes in math, and I don't want to be limited by my lack of background.
 
  • #3
I suggest you supplement your reading with online videos like Khan Academy or mathispower4u.com. I like both sites.

However, in your case, the mathispower4u.com videos can be matched to the topics you chose to study in your book, as each video is based on solving a specific problem.

https://mathispower4u.com/calculus.php
 
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  • #4
You say that you have time to investigate new subjects and are considering a Math minor or second major. It sounds like have never had calculus. I would advise you to look at calculus. It never hurts to get a head start and that is probably going to be your next class. Other than that, some introduction to linear algebra and/or probability and statistics would not hurt. All of those subjects can come as a shock if you have never seen them before. Some extra time to get oriented in those would be very helpful.
Do not stop with your programming efforts. It takes time to get skilled at that.
 
  • #5
I used the 5th edition of "Precalculus" by Lial-Hornby-Schneider when I was preparing for university study. The last few chapters are:

8. Applications of Trigonometry: Law of Sines/Cosines, Vectors and Dot Products, Applications of Vectors, Polar Form of Complex Numbers, De Moivre's Theorem, Products/Quotients and Powers/Roots of Complex Numbers, Polar Equations & Graphs, Parametric Equations & Graphs.

9. Systems and Matrices: Systems of Linear Equations, Matrix Solutions & Determinant Solutions of Linear Systems, Partial Fractions, Nonlinear Systems of Equations, Inequalities and Linear Programming, Matrix Properties and Inverses.

10. Analytic Geometry: Parabolas, Ellipses, Hyperbolas, Conics Summary.

11. Further Topics: Sequences and Series, Arithmetic and Geometric Sequences & Series, Binomial Theorem, Mathematical Induction, Counting Theory, Probability.

The topics I underlined were some topics that I ran into in first year math and physics classes. Having had previous exposure really helped. I personally found that knowing trigonometry cold was the most helpful, however. Followed by the systems and matrices.
 
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  • #6
Mondayman said:
I used the 5th edition of "Precalculus" by Lial-Hornby-Schneider when I was preparing for university study. The last few chapters are:

8. Applications of Trigonometry: Law of Sines/Cosines, Vectors and Dot Products, Applications of Vectors, Polar Form of Complex Numbers, De Moivre's Theorem, Products/Quotients and Powers/Roots of Complex Numbers, Polar Equations & Graphs, Parametric Equations & Graphs.

9. Systems and Matrices: Systems of Linear Equations, Matrix Solutions & Determinant Solutions of Linear Systems, Partial Fractions, Nonlinear Systems of Equations, Inequalities and Linear Programming, Matrix Properties and Inverses.

10. Analytic Geometry: Parabolas, Ellipses, Hyperbolas, Conics Summary.

11. Further Topics: Sequences and Series, Arithmetic and Geometric Sequences & Series, Binomial Theorem, Mathematical Induction, Counting Theory, Probability.

The topics I underlined were some topics that I ran into in first year math and physics classes. Having had previous exposure really helped. I personally found that knowing trigonometry cold was the most helpful, however. Followed by the systems and matrices.
This looks like an excellent list and it would give an introduction to a lot of good subjects without getting too deep into any one of them.
But I would not shy away from a brief introduction to differential and integral calculus and would continue developing computer programming skills.
 
  • #7
somefellasomewhere said:
Oh goodness... I forgot to explain my motivations for doing this. I'll be a college freshman majoring in biology next year and since I've gotten a lot of gen eds out of the way, I'll have the opportunity to add a minor or second major in math (which I've only recently realized the importance of, in biology and in general).
In that case, I would insert a short deviation into genetic algebras! They can be taught without calculus and demonstrate the use of mathematics outside of physics.

E.g. problem 4 in here:
https://www.physicsforums.com/threads/math-challenge-march-2019.967174/
Solution available here:
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
 
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  • #8
fresh_42 said:
In that case, I would insert a short deviation into genetic algebras! They can be taught without calculus and demonstrate the use of mathematics outside of physics.

E.g. problem 4 in here:
https://www.physicsforums.com/threads/math-challenge-
Do not stop with your programming efforts. It takes time to get skilled at
march-2019.967174/
Solution available here:
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
I never would have known this had existed if you didn't comment. Thank you! Is there some sort of introductory resource I could use to learn more about this? Due to their conciseness, everything I can find seems to assume previous knowledge of abstract algebras (am I using that term(s) right?). In regards to programming, I remembered a udemy course I bought a long time ago that teaches coding from the perspective of math or the converse (maybe I used that math word right okay I'll stop).
 
  • #9
I have two little paperbacks "Pure and Applied Mathematics" (vol. 1&2) where I found such examples but they are not in English. If I want to find something at a higher level than Wikipedia, I look out for lecture notes, normally with the search key "<subject>+pdf", e.g. "genetic algebras + pdf" or "introduction + genetic algebras + pdf". This usually leads me to university servers where professors (or students) uploaded their scripts.

Another, maybe easier example in named books is error-correcting codes: Caesar, linear, cyclic, etc.
 
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