Any Calculus Starter Textbook suggestions?

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Frank Li
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I would like to start learning my calculus course before in school. Are here any textbooks or science reading books that would help me with the situation?
 

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  • #2
bacte2013
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I strongly recommend George Simmon's "Calculus with Analytic Geometry". He writes very clear, sophisticated exposition for both high schools and beginning undergraduates. He also has excellent problems sets. I also recommend APEX Calculus, which is free to download and also has fascinating exposition.
 
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  • #3
Frank Li
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I strongly recommend George Simmon's "Calculus with Analytic Geometry". He writes very clear, sophisticated exposition for both high schools and beginning undergraduates. He also has excellent problems sets. I also recommend APEX Calculus, which is free to download and also has fascinating exposition.
Thanks I'll try them
 
  • #4
smodak
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Simmons as suggested above is good. I have a lot of other favorites.

There is a small and fun to read book that makes the basics strong. https://www.amazon.com/dp/0883858126/?tag=pfamazon01-20
For Applications, you cannot beat https://www.amazon.com/dp/0992001005/?tag=pfamazon01-20
Great way to quickly learn calculus https://www.amazon.com/gp/product/0471827223/?tag=pfamazon01-20
Great book (Q&A style) https://www.amazon.com/gp/product/0486203700/?tag=pfamazon01-20
A really fun book to read https://www.amazon.com/gp/product/0691161909/?tag=pfamazon01-20
Really good Calculus books: https://www.amazon.com/gp/product/0691130884/?tag=pfamazon01-20 and https://www.amazon.com/gp/product/0312185480/?tag=pfamazon01-20
Oldies but goldies https://www.amazon.com/gp/product/B00GMPZBGA/?tag=pfamazon01-20, https://www.amazon.com/gp/product/0201048108/?tag=pfamazon01-20, https://www.amazon.com/gp/product/0030892686/?tag=pfamazon01-20 - These are much better (along with Simmons) than nay of the current calculus books.
A Tutoring Book https://www.amazon.com/gp/product/0879421835/?tag=pfamazon01-20
An Infinitesimal approach https://www.amazon.com/gp/product/0486484521/?tag=pfamazon01-20
The book that allegedly taught Feynman https://www.amazon.com/dp/1406756725/?tag=pfamazon01-20
For Problems https://www.amazon.com/gp/product/1592575129/?tag=pfamazon01-20
Lastly, there is the best of the best https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20 (although in my opinion it serves the best as the second book rather than the first)

Hope you find something interesting in the above list. If I were you, I would opt for a non traditional textbook and have some fun reading through it. You will have to go through textbooks when they teach you at school anyways. You cannot really go wrong with any of the books I suggested or Simmons.
 
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  • #5
Frank Li
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Simmons as suggested above is good. I have a lot of other favorites.

There is a small and fun to read book that makes the basics strong. https://www.amazon.com/dp/0883858126/?tag=pfamazon01-20
For Applications, you cannot beat https://www.amazon.com/dp/0992001005/?tag=pfamazon01-20
Great way to quickly learn calculus https://www.amazon.com/gp/product/0471827223/?tag=pfamazon01-20
Great book (Q&A style) https://www.amazon.com/gp/product/0486203700/?tag=pfamazon01-20
A really fun book to read https://www.amazon.com/gp/product/0691161909/?tag=pfamazon01-20
Really good Calculus books: https://www.amazon.com/gp/product/0691130884/?tag=pfamazon01-20 and https://www.amazon.com/gp/product/0312185480/?tag=pfamazon01-20
Oldies but goldies https://www.amazon.com/gp/product/B00GMPZBGA/?tag=pfamazon01-20, https://www.amazon.com/gp/product/0201048108/?tag=pfamazon01-20, https://www.amazon.com/gp/product/0030892686/?tag=pfamazon01-20 - These are much better (along with Simmons) than nay of the current calculus books.
A Tutoring Book https://www.amazon.com/gp/product/0879421835/?tag=pfamazon01-20
An Infinitesimal approach https://www.amazon.com/gp/product/0486484521/?tag=pfamazon01-20
The book that allegedly taught Feynman https://www.amazon.com/dp/1406756725/?tag=pfamazon01-20
For Problems https://www.amazon.com/gp/product/1592575129/?tag=pfamazon01-20
Lastly, there is the best of the best https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20 (although in my opinion it serves the best as the second book rather than the first)

Hope you find something interesting in the above list. If I were you, I would opt for a non traditional textbook and have some fun reading through it. You will have to go through textbooks when they teach you at school anyways. You cannot really go wrong with any of the books I suggested or Simmons.

Thanks! Your information is massive!
 
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QuantumQuest
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  • #8
MidgetDwarf
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I can vouch for the Moise Calculus book. Really great book with interesting problems. Shows you the why and how. It is at a bit lower level than Spivak but above Thomas.
 
  • #9
Buffu
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No list is complete without Apostol's calculus.
 
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  • #10
Quantum Aravinth
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M. Spivak
Really an awesome book
 
  • #11
Demystifier
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No list is complete without Apostol's calculus.
But it's really much more than calculus. It's also linear algebra, probability theory, ...
 
  • #12
smodak
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Spivak and Apostol are great books and so are Courant and john's calculus books, I in fact suggested Spivak in my post above with a caveat. However, I do not believe Apostol or Spivak should be used as a first calculus book for High School Students.

Spivak https://www.amazon.com/dp/0883858126/?tag=pfamazon01-20that will be appropriate as a great companion for a real text book - the only reason I did not suggest it above is I do not think this book should be used stand alone and it costs way too much ($48 in Amazon) for what it is.
 
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  • #13
alan2
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You really should stop recommending Spivak or Apostol for high school students who have no clue what calculus is. Those of us who understand it find them to be lovely books but I can't think of a better way to make a new student hate math than to suggest to him that those books are representative of what they will likely do with calculus.
 
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  • #14
smodak
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but I can't think of a better way to make a new student hate math than to suggest to him that those books are representative of what they will likely do with calculus.
:)
 
  • #15
MidgetDwarf
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You really should stop recommending Spivak or Apostol for high school students who have no clue what calculus is. Those of us who understand it find them to be lovely books but I can't think of a better way to make a new student hate math than to suggest to him that those books are representative of what they will likely do with calculus.

He did offer a word of caution about Spivak/Apostol.

My suggestion would be to get Moise: Calculus. It is between Courant and Stewart/Thomas. It goes over topics, like the Induction Principle, and Well Ordering Principle. Has neat problems. The writing in the text is lucid and clear, and Moise makes the ideas connect in elegant and beautiful ways. He also discusses topics that should be familiar from previous math classes, in a very thoughtful and meaningful way. Why do we call a coordinate system right handed? How did people define the trigonometric functions? It talks about the Winding Function, something that was new to me. Limit laws are clear. Proofs are very clear and offer a lot of insight as to why the theorems hold. A very excellent book!

Get Moise and maybe Thomas/Finnly 9th ed. Use Moise as the main text and Thomas as a supplement.
 
  • #16
alan2
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People above are recommending them. They really are inappropriate for a kid asking to learn calculus but every time the question comes up the same answers are given.

No list is complete without Apostol's calculus.

M. Spivak
Really an awesome book
.
 
  • #17
Buffu
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People above are recommending them. They really are inappropriate for a kid asking to learn calculus but every time the question comes up the same answers are given.




.
I am sorry. I recommended that book because other people were recommending all types of books except this one, so I thought I would complete the list.
 
  • #18
vanhees71
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Well, to recommend good university books for freshmen to highschool students is good advice. When I was about in the 8th grade, I had big trouble with mathematics, and I couldn't make sense of my school book nor with the explanations of my teachers. Then I was lucky to find some textbook called "Mathematics for Engineers", which covered the stuff (I think it was elementary Euclidean geometry about triangles, sin, cos, tan, and all that) I had trouble with in a clear and lucid way. It was far from being a strict mathematicians' math textbook, but full with applications and exercises with solutions in the appendix. From that day on math was my favorite subject, and I had good marks too. I even started to self-study math ahead of class getting an idea about calculus (differentiation and integration) quite early. I never could make sense of what was in the school books, but these university books were the door opener for understanding math, and pretty soon I also read introductory textbooks on "true math" with real strict proofs of theorems, a discipline that had not much in common with what's called "math" in school. So good introductory university books can definitely help to motivate high school students to learn a subject, because they teach the subject without hiding it behind well-intended but flawed didactics!
 
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  • #19
Demystifier
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When I was about in the 8th grade, I had big trouble with mathematics, and I couldn't make sense of my school book nor with the explanations of my teachers. ... they teach the subject without hiding it behind well-intended but flawed didactics!
Can you tell now more precisely what exactly was wrong with their way of teaching?
 
  • #20
MidgetDwarf
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Can you tell now more precisely what exactly was wrong with their way of teaching?
I am assuming most of the high school textbook, were full of pictures,diagrams, non related math jargon, and only the calculation aspect. Maybe no explanation from the high school authors as to why we do such and such, and why its important, etc.
 
  • #21
vanhees71
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Can you tell now more precisely what exactly was wrong with their way of teaching?
They taught mathematics as a collection of recipies to solve the standard problems in the books but never explained why these recipies worked, let alone gave proofs (not even heuristic ones). What's even worse is that there was no systematics in teaching the subjects (a contradiction to what imho math is all about). The jumped from one topic (say naive geometry, where we had to construct things with help of a "geo triangle" and a compass; then totally unrelated some algebra like solving for quadratic equations) without any systematics. To me that's a big waste of time: Math should be taught as a way of thinking and a systematic tool to solve problems in everyday life rather than a collection of senseless recipies to solve certain types of textbook problems.
 
  • #22
Buffu
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They taught mathematics as a collection of recipies to solve the standard problems in the books but never explained why these recipies worked, let alone gave proofs (not even heuristic ones). What's even worse is that there was no systematics in teaching the subjects (a contradiction to what imho math is all about). The jumped from one topic (say naive geometry, where we had to construct things with help of a "geo triangle" and a compass; then totally unrelated some algebra like solving for quadratic equations) without any systematics. To me that's a big waste of time: Math should be taught as a way of thinking and a systematic tool to solve problems in everyday life rather than a collection of senseless recipies to solve certain types of textbook problems.

I deeply agree with everything you said except
"What's even worse is that there was no systematics in teaching the subjects (a contradiction to what imho math is all about). The jumped from one topic (say naive geometry, where we had to construct things with help of a "geo triangle" and a compass; then totally unrelated some algebra like solving for quadratic equations) without any systematics."
, what do think ? should someone taught everything from differentiation to vector calculus before starting anything else ?

Though I think physics education is the one that is really messed up. They teach concepts of differential equations and vector calculus with shtty hand waving without any proofs, just gives an impression that physics have no formalism in it and people who study physics just do ad-hoc proofs to get desired result.
 
  • #23
Demystifier
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Though I think physics education is the one that is really messed up. They teach concepts of differential equations and vector calculus with shtty hand waving without any proofs, just gives an impression that physics have no formalism in it and people who study physics just do ad-hoc proofs to get desired result.
You must be a mathematician. :biggrin:
Just recall how calculus has been introduced by its inventors, Newton and Leibnitz. There is a good reason why hand-waving appeared before the proper proofs.
 
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  • #24
smodak
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You must be a mathematician. :biggrin:
Just recall how calculus has been introduced by its inventors, Newton and Leibnitz. There is a good reason why hand-waving appeared before the proper proofs.
+1
I need to teach my 15 year old daughter, who is in 9th grade, AP calculus AB and make sure she does well in the exam. I am surely not going to start with Spivak or proofs. I am going to teach her the basics (mainly the concepts and the applications) of limits, continuity, derivatives and integrals and I will teach her to solve a lot of problems (computation and applications). Heck I am even going to buy her a Barron's guide to do well in the exam - which will actually be more useful for her purpose than Spivak or Apostol or proofs. We cannot have the same solution for different problems.

Now read OP's statement:
would like to start learning my calculus course before in school. Are here any textbooks or science reading books that would help me with the situation?
I do not believe OP in a much different situation than my daughter is.
 
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  • #25
vanhees71
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I deeply agree with everything you said except , what do think ? should someone taught everything from differentiation to vector calculus before starting anything else ?

Though I think physics education is the one that is really messed up. They teach concepts of differential equations and vector calculus with shtty hand waving without any proofs, just gives an impression that physics have no formalism in it and people who study physics just do ad-hoc proofs to get desired result.
I don't think that you should start with calculus in high school. It's not so much the content than the way it's taught I criticize. Math should be taught as a coherent way of thinking about logical "universes", not as a collection of unrelated recipies to solve (often unrealistic and boring) problems in bad textbooks.

Physics is different. I think, my physics education in highschool was way better than the mathematics, but maybe that's an exception, because I had an exceptionally good teacher. She worked as a postdoc on atomic physics before becoming a high-school teacher, and she taught the curriculum in a very coherent way, doing a lot of demonstration experiments and, even more important, letting us do experiments as much as possible. We learnt a lot on classical physics (mechanics and electrodynamics) and even a good overview on "modern physics" (relativity and quantum theory, even up to simple applications of the Schrödinger equation, atomic and nuclear physics and a glimpse on HEP physics).
 
  • #26
smodak
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I don't think that you should start with calculus in high school. It's not so much the content than the way it's taught I criticize. Math should be taught as a coherent way of thinking about logical "universes", not as a collection of unrelated recipies to solve (often unrealistic and boring) problems in bad textbooks.

Physics is different. I think, my physics education in highschool was way better than the mathematics, but maybe that's an exception, because I had an exceptionally good teacher. She worked as a postdoc on atomic physics before becoming a high-school teacher, and she taught the curriculum in a very coherent way, doing a lot of demonstration experiments and, even more important, letting us do experiments as much as possible. We learnt a lot on classical physics (mechanics and electrodynamics) and even a good overview on "modern physics" (relativity and quantum theory, even up to simple applications of the Schrödinger equation, atomic and nuclear physics and a glimpse on HEP physics).
How do you learn all that physics without calculus? Perhaps our definition of 'calculus' is different :)
 
  • #27
Buffu
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She worked as a postdoc on atomic physics before becoming a high-school teacher

Why did she become a high school teacher ?

How do you learn all that physics without calculus? Perhaps our definition of 'calculus' is different :)

Who needs calculus to do physics in high school ?
 
  • #28
smodak
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Why did she become a high school teacher ?
Because she did not know 'real' calculus LOL. Sorry could not resist :biggrin:
 
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  • #29
smodak
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Who needs calculus to do physics in high school ?
In my opinion the best way to learn calculus is by doing mechanics and vice versa. Starting with calculus and mechanics together in high school (as I did some 30 years ago) worked well for me.

OP, I apologize. We took this discussion far away from your original question. I will try not to respond off-topic anymore.
 
  • #30
Buffu
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In my opinion the best way to learn calculus is by doing mechanics and vice versa. Starting with calculus and mechanics together in high school (as I did some 30 years ago) worked well for me.

OP, I apologize. We took this discussion far away from your original question. I will try not to respond off-topic anymore.

What calculus did you do in mechanics ? because most calculus I did was proof of laws of kinematics.
Taking Maths was not compulsory. You can take Biology instead of Maths and still take Physics.
 
  • #31
smodak
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What calculus did you do in mechanics ? because most calculus I did was proof of laws of kinematics.
Taking Maths was not compulsory. You can take Biology instead of Maths and still take Physics.

I was in India and here in the US we do not have the books that I used there. We had different books for calculus and mechanics. We covered most of what's in Savov (perhaps even in a bit more depth) for mechanics. We also did electricity and magnetism and other topics - for example, most of what's in Resnick and Halliday.

As I said, our definition of calculus may be different :oldbiggrin:. When you say calculus, you probably mean what's in Spivak/Apostol or some analysis books which I did not tackle until later. When I say calculus, I kind of mean the calculus necessary to do physics.
 
  • #32
vanhees71
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How do you learn all that physics without calculus? Perhaps our definition of 'calculus' is different :)
Sigh. I didn't say, I don't want any calculus at high school but I don't want to begin with calculus. You need algebra and geometry first. I define highschool as in Germany to start with the 5th grade. Calculus is introduced in 10th (or 11th) grade depending on whether you have 12 (or 13) years until the final exam (Abitur).

Of course, physics without calculus is impossible, although I learnt in the US that they offer courses explicitly marked "calculus free". I had to teach some hours to substitute a colleague, and that was the most challenging lecture I've ever given in my live. I had to introduce velocity and acceleration without derivatives or even to introduce derivatives without being allowed to call them derivatives. It's just rediculous, because it's not simpler but much more complicated in this way, but that's another story.
 
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  • #33
vanhees71
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Why did she become a high school teacher ?



Who needs calculus to do physics in high school ?
I think she wanted to have more time for her family than she could have when pursuing a career as a physics researcher. I and I think anybody needs calculus to do physics in high school (in the higher grades of course). Without calculus most of physics stays qualitative and a lot if then missing from the true "flavor of physics", and in my opinion high-school students should get a realistic flavor of all subjects to be able to choose what to do for the rest of their lives.
 
  • #34
vela
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I used to feel that you had to have calculus to truly learn physics, but I've grown to think the opposite. There's a surprising amount of physics that you can learn without calculus. The level of math required depends on the level of the course. Do high school freshmen really need to know how to calculate the moment of inertia of various solids, or is it more important that they understand what rotational mass means? I'd rather they understand why the solid disk rolls down the slope faster than the hollow ring than being able to calculate the moment of inertia of either shape. On the other hand, for the physics or engineering majors in college, learning how to set up the integral is a technique they definitely need to learn because they'll be doing similar calculations later on. (Plus they can use all the practice they can get in evaluating the integral as well.)

Also, you have to take into account the goal of a course. The life science majors who take physics aren't planning to become physicists. Is it really important for them to get a true feeling for what it really means to do physics, or is the goal to expose them to the ideas of physics so they have a decent understanding of what we currently know and what we don't? I'd say the latter is more important than the former for this group of students.

I do understand and sympathize with those who object to removing math from a course simply because the students are afraid of math–that is, dumbing down the course. If a course has a calculus prerequisite, there's no valid reason to avoid using it in class when it makes the exposition clearer, or expecting students to solve a few problems that require calculus.


When you major in physics in the US, you typically don't learn, say, classical mechanics once. That's not how people learn. Most of us learn through repeated exposure to ideas and concepts. You see classical mechanics in high school physics, in intro physics in college, in an upper division course, and again in grad school. Each time the level of sophistication grows. You wouldn't start with teaching Lagrangians and Hamiltonians to high school students because it would be pointless. They simply wouldn't get it.

So getting back to the original point of this thread, the same considerations apply to learning calculus. Does the student want to learn the basic ideas of calculus, how to apply them, and how to do the calculations? Or should he or she immerse themselves into proving every last detail to get a true feeling for what real mathematicians do? For someone starting out, I'd lean toward the former. Ideally students should understand the reasoning used to reach various results, but I don't think it's particularly useful to spend a lot of time at this level trying to write proofs for everything.
 
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  • #35
smodak
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So getting back to the original point of this thread, the same considerations apply to learning calculus. Does the student want to learn the basic ideas of calculus, how to apply them, and how to do the calculations? Or should he or she immerse themselves into proving every last detail to get a true feeling for what real mathematicians do? For someone starting out, I'd lean toward the former. Ideally students should understand the reasoning used to reach various results, but I don't think it's particularly useful to spend a lot of time at this level trying to write proofs for everything.
Very well said. Completely Agree.
 

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