Ready to start Calculus (I think), but unsure where to start

[NovaeSci]

Hi all,

Over the last few weeks I've been catching back up with High School Algebra, Trigonometry, along with some Geometry. I'm now looking for the next challenge and unsure where to start.

Due to studying Astrophysics/Physics topics, I'm assuming studying applied mathematics topics is the best way forward? Also, is Calculus the next step after High School maths? I'm in the UK, so for anyone in the US, I've just completed Algebra II.

Should I start at Pre-Calculus? I'm guessing the first steps of Calculus is to start with Differential and Integral Calculus? I have read the self-study mathematics guides, along with a quick look at the textbooks recommended, but I thought best to ask seeing as I'm studying it to apply to Astrophysics, rather than just purely the maths. What would be a good textbook for me to purchase? Looking for an actual textbook, as I prefer to work through it, as to learn from lectures like on Khan Academy. I prefer to use videos as a supplementary tool.

Something which I am really curious to know about, and bare in mind I know nothing about Calculus yet, is if learning it has different approaches? For example is a Pure Mathematics major going to learn Calculus different to, say, an Applied Mathematics Major? If so, is it beneficial to learn both ways? Whilst I'm mainly going to be using Applied Mathematics, I would definitely like to learn Pure Mathematics as well, to have a good overview. Is there a better way to learn topics by way of Applied, then Pure, or vice-versa?

Thanks for you responses in advance )

 

[Frabjous]

Given that your ultimate goal is to study physics, you will need Pre-Calculus and Geometry. Geometry is also important to a pure math path because it is a first exposure to proofs.

You have to learn applied Calculus, because you need the exposure to performing calculations. The pure math version is more abstract, so I would do it second/concurrently.

 

[NovaeSci]

Well Geometry wise, I have done the equivalent of High School Geometry on Khan Academy, is this enough? Also, are the Pre-Calculus topic on Khan sufficient?

When starting Calculus, is it Differential and Integral which should be my first exposure? Any good textbooks that deal with this in an applied way?

 

[Frabjous]

Regretfully, I am too old to make specific recommendations or comment on how in-depth Khan is. I’m sure others will chime in.

 

[fresh_42]

I think that it is sufficient to learn everything about triangles, the trigonometric functions, and some formulas for areas and the volume of regular solids. Most of it can easily be looked up nowadays, but classic theorems Pythagoras, Thales, sine and cosine theorem, etc. should be within your repertoire.

When starting Calculus, is it Differential and Integral which should be my first exposure? Any good textbooks that deal with this in an applied way?

The standard recommendation on PF is Spivak's Calculus.

 

[NovaeSci]

Already gone through Trigonometry. Looking at the Pre-Calculus topics on Khan, it appears I have already gone through the recommended topics. I'll take a look at Spivak's Calculus - is this applied? How does it stand up to the one recommended on the guide? Keisler's Elementary Calculus I believe, followed by Nitecki's Calculus deconstructed.

 

[fresh_42]

I don't know Spivak personally, since I studied such books in my own language. But Spivak has been recommended so often, has so many editions, that it is safe to say that it is a classical standard textbook. It will very likely follow the curriculum at universities, will say: the other way around. If you want to study the high school stuff first, then have a look at the calculus books on OpenStax: https://openstax.org/subjects.

You should learn everything about sequences, continuity, and series prior to differentiation and integration.

 

[PeroK]

Already gone through Trigonometry. Looking at the Pre-Calculus topics on Khan, it appears I have already gone through the recommended topics. I'll take a look at Spivak's Calculus - is this applied? How does it stand up to the one recommended on the guide? Keisler's Elementary Calculus I believe, followed by Nitecki's Calculus deconstructed.

There is an excellent site here for all things calculus:

https://tutorial.math.lamar.edu/

 

[NovaeSci]

Thanks :) I'm just a bit clueless where to start post Algebra/Trigonometry. Will check out the website as well; however, mainly looking for a textbook for the most part.

 

[Vanadium 50]

I'm now looking for the next challenge and unsure where to start.

Can you solve every problem in your trig (and earlier) textbooks?

 

[fresh_42]

A very good collection of exercises in calculus is
B. Demidovich: Problems in Mathematical Analysis
It is old and I don't know whether it is still available for purchase, but it has hundreds of problems, esp. for engineers and phycisists.

 

[NovaeSci]

I'll give it a gander. Also, yes to the solving topics, but at the current time I don't really know if these topics should go deeper.

I own Stroud's Mathematical Engineering. Think it would be useful to work through the rest of this to get a good background on the topics, then I can specialise by picking topics at a time to learn with a deeper meaning and applying them to physical models?

 

[MidgetDwarf]

Moise Calculus. Walks the path between applied and theory. Closer to Courant, but not s difficult.

 

[vela]

When starting Calculus, is it Differential and Integral which should be my first exposure?

Typically, you see differentiation first and then integration, but when I took calculus, the book started with integration. I don't think it really matters that much.

I don't know Spivak personally, since I studied such books in my own language. But Spivak has been recommended so often, has so many editions, that it is safe to say that it is a classical standard textbook.

I don't have Spivak either, but my impression was that Spivak is a bit more advanced and more proof-oriented than the typical intro calculus book. The Amazon reviews seem to reflect this. As the OP is looking for a book that focuses on learning how to solve calculus problems, it may not be the best choice.

 

[NovaeSci]

I noticed this. I'm just wanting to learn the topic with as many problems as possible. Just over the next few years to understand numerous branches of maths. But then can study them in more detail as I need to, or should I need to study it in more depth.

 

[Frabjous]

Back in the day, we used Calculus of a Single Variable by Faires and Faires. I never felt my single variable calculus knowledge was lacking. It is reasonably priced used. I do not believe that there have been any major advances in calculus pedagogy since it was printed in 1989.

edit: I forgot you were in the UK. Find out what good UK universities are using to teach scientists and engineers calculus and choose it. There should be reasonably priced used versions available.

 

[Joshy]

I felt like a lot of people choked in our calculus classes as soon as any trig was involved. If I were to retake the classes again myself, then I'd probably practice a lot of trig and and the identities.

 

[NovaeSci]

Wouldn't the books provided at Universities be on top of lectures and tutorials, though? So may not be the best for someone studying completely independent. I'd imagine to textbooks would have chapters referred to in the tutorials/lectures as well, so different books for different parts. I had a look at a few and there can be up to 5-6 books for each topic. Also, this isn't really helpful when trying to figure out how to order the books as I again imagine parts will be cherry picked and not in linear order.

Regarding Trigonometry, I've had a loo and see what you mean by the topic going a lot further. I've only self-taught myself the areas of Sin, Cos and Tan, along with Pythagoras. What would be ideal topic in Trig that would be good prep for Calculus? As you can tell, I'm a tad clueless and feel like a headless chicken, haha.

 

[Vanadium 50]

I've only self-taught myself the areas of Sin, Cos and Tan, along with Pythagoras.

Oh dear. Why this mad rush to Calculus? It will still be there in a few months or a year when you've better built up your foundations.

If you take a book like Loney's, it sounds like you are at roughly page 40. The book is almost 500 pages long.

 

[Frabjous]

Regarding Trigonometry, I've had a loo and see what you mean by the topic going a lot further. I've only self-taught myself the areas of Sin, Cos and Tan, along with Pythagoras. What would be ideal topic in Trig that would be good prep for Calculus? As you can tell, I'm a tad clueless and feel like a headless chicken, haha.

As a test of your knowledge, take a look at this book (It’s inexpensive or you can probably find a copy online). It’s only 128 pages.
Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons
https://www.amazon.com/dp/1592441300/?tag=pfamazon01-20
If you find all three sections things that you know and have not forgotten, you are ready for calculus. If not you will have identified areas that you need to learn in more depth.

If you can find a copy, his calculus book would also make a good choice.

 

[NovaeSci]

Oh dear. Why this mad rush to Calculus? It will still be there in a few months or a year when you've better built up your foundations.

If you take a book like Loney's, it sounds like you are at roughly page 40. The book is almost 500 pages long.

No rush at all! I just thought Calculus happens after Algebra II. It's usually the case in the UK: after GCSE, you study Calculus in college before you go to University. But you only study the trig basics for GCSE. What is the name of the book by Loney, by the way? Cheers

 

[NovaeSci]

As a test of your knowledge, take a look at this book (It’s inexpensive or you can probably find a copy online). It’s only 128 pages.
Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons
https://www.amazon.com/dp/1592441300/?tag=pfamazon01-20
If you find all three sections things that you know and have not forgotten, you are ready for calculus. If not you will have identified areas that you need to learn in more depth.

Thanks I'll check this out :) Quite a short book as well. Will be useful to find any gaps in my knowledge in order to know what to look deeper in to. Cheers

 

[PeroK]

No rush at all! I just thought Calculus happens after Algebra II. It's usually the case in the UK: after GCSE, you study Calculus in college before you go to University. But you only study the trig basics for GCSE. What is the name of the book by Loney, by the way? Cheers

The UK A-level syllabus is covered here

https://www.examsolutions.net/a-level-maths/

 

[Vanadium 50]

What is the name of the book by Loney

Plane Trigonometry. But there's nothing magic about that book.

 

[Frabjous]

Calculus happens after Algebra II

I do not know the UK system, but in the US it is
Alegebra II
Geometry (includes exposure to proofs)
Precalculus (includes trigonometry and analytic geometry)
Calculus

 

[NovaeSci]

I do not know the UK system, but in the US it is
Alegebra II
Geometry (includes exposure to proofs)
Precalculus (includes trigonometry and analytic geometry)
Calculus

Think that's where I was getting confused. I'll definitely start to take a good look into precalcus and look out for extra trigonometry skill and analytic geometry. Thanks again for the help all

 

[Vanadium 50]

The US schools near here have a different sequence.

1. Algebra I
2. Geometry
3. Algebra II/Trig
4. Pre-calculus

 

[jtbell]

My sequence 50+ years ago was:

8th grade: Algebra I
9th grade: Algebra II
10th grade: Geometry
11th grade: Trig & Analytic Geometry (we didn't use the term "Pre-Calculus" in those days)
12th grade: Calculus (single-variable, basic derivatives and integrals)

This was the "accelerated track" in my steel-mill-and-factory-town's school system. Students in the ordinary college-prep track started with Algebra I in 9th grade, and didn't take calculus until college/university.

In college the math department placed me out of Calculus I, so I continued in my freshman year with

Calculus II: more sophisticated integration techniques, and infinite series
Calculus III: multivariable calculus (which was also presented from a somewhat different perspective in the physics department's Electromagnetism course)

 

[FactChecker]

Algebra, trig, and geometry are what I would call pre-calc, except that pre-calc is a faster-paced review of those topics.

 

[Mondayman]

There are a number of precalculus textbooks by authors such as Larson, Stewart, Axler, Lial, Beecher. They are all big 800-1200 page tomes containing everything you need to know to start learning calculus and usually have tons of exercises. But they are very rote and computational in nature, with very few challenge problems or proofs.

Just a note. When I went back to school, I used Khan Academy, my class textbooks, and Precalculus by Lial. I spent three years mastering algebra and trigonometry before taking calculus. If you really want to be successful, you need to know it cold. I can't tell you how many of my classmates in university did poorly in calculus because they couldn't factor a difference of cubes or remember the unit circle.