# Book recommendations for highschool/undergrad calculus

I am looking for a book recommendations for learning calculus for high school or under graduation level can you suggest me some good books which have the proper theory and can very well be used to self teach yourself.

I am looking for a book in calculus that pose over topics like: [This is the prescribed syllabus for my course]

Differential calculus

Real valued functions of a real variable, Into, onto and one-to-one functions, sum difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions. Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L'Hospital rule of evaluation of limits of functions. Even and odd functions, inverse of a function, continuity of composite functions intermediate value property of continuous Functions. Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions, Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals. Increasing and decreasing functions, maximum and minimum values of a function, Rolle's theorem and I lagrange's mean value theorem.

Integral calculus

Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, fundamental theorem of integral calculus. Integration by parts, integration by the methods of substitution and partial fractions. Application of definite integrals to the determination of areas involving simple Curves Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, lincar first order differential equations.

I am going to study maths in future (I don't want a book that just touches the topics just for the sake of it, but something that goes deep into the concepts to build a very strong foundation which can help me tackle hard problems...

Edit: Actually I am from India , and as far I know in USA , the highschool mathematics is a lot less than what it is in India. I'm still in highschool and just included the Undergrad so that I can get something more . But what I forgot to mention was only the first year maybe... That is why I also included the syllabus . I think something like topological introduction to limits might be too much of an overkill

• Are you going to continue study math in the future or is this just laying foundations for other subjects? – trisct Jul 6 '19 at 6:38
• @trisct Study maths in future. – RandomAspirant Jul 6 '19 at 6:40
• Spivak's Calculus would be my choice for a rigorous introductory calculus course aimed at math majors. However, I would recommend at least some exposure to non-rigorous calculus first, for motivation. – Bungo Jul 6 '19 at 6:45
• Agree with Bungo. My favorite non-rigorous text is Thomas' Calculus and Analytic Geometry. Has to be the 3rd edition though - subsequent editions have extra authors and stray too far from what the text once was. And unfortunately the 3rd edition is out of print but can be found "used" with some digging. – Kaj Hansen Jul 6 '19 at 7:09

So I guess you are looking for a book on mathematical analysis for mathematicians. I would recommend Zorich's Mathematical Analysis. Pretty much everything in your syllabus can be found (and of course there's a lot more). The reason why I like this book is that it introduces limit (which I believe is a notion that distinguishes college math from high-school math) from a topological point of view. This kind of view can be very beneficial to your future study and to understanding what you are learning.

Another book is The Fundamentals of Mathematical Analysis by Gregory Mikhailovich Fichtenholz (Fikhtengol'ts). I usually use this as a reference because it covers a great number of topics.

Of course I've also heard of other great books like Rudin's Principles of Mathematical Analysis, but I haven't read them through so can't really comment on them.

Some non-rigorous calculus or non-mathematician-oriented calculus can also be helpful in intuitive explanations of concepts and motivations, which is rarely seen in the texts for professionals. Examples of such books are Calculus by Larson and Edwards, Calculus: Early Transcendentals by Stewart and Calculus by Thomas. The good thing about them is that they contain a lot of examples, pictures, backgrounds and real-life applications (and they are colorful). The disadvantage is that they are too long (usually one book is more than 1000 pages, but only covers half of the contents of their counterparts for math major students). And they don't teach you to think like a mathematician.

The ones I mentioned may not be suitable for everyone, and I am just pointing out why I like them or not. The bottom line is you should find one you like. And it is never a bad idea to read more than you plan to.

• Actually I am from India , and as far I know in USA , the highschool mathematics is a lot less than what it is in India. I'm still in highschool and just included the Undergrad so that I can get something more . But what I forgot to mention was only the first year maybe... That is why I also included the syllabus . I think something like topological introduction to limits might be too much of an overkill – RandomAspirant Jul 6 '19 at 7:49
• Well, the book doesn't introduce topology the moment it mentions limit. It starts with the usual $\epsilon-\delta$ language and the neighborhoods and stuff. But after it introduces the (very basic) topological language, you will find that different kinds of limits (that of a sequence, that of a function, the one you take in an integral) are unified, and the theorem about when two limits commute becomes very concise and easy to comprehend. – trisct Jul 6 '19 at 7:58
• Zorich would be very unfriendly to newbies. So is the "baby Rudin". – xbh Jul 6 '19 at 9:31
• @xbh Then what would be friendly ?? – RandomAspirant Jul 7 '19 at 16:07

You might consider my book:

Daniel Velleman, Calculus: A Rigorous First Course.

It is intended to be somewhere between Spivak and books by Stewart or Thomas. It is rigorous like Spivak, but focuses on calculus rather than analysis, so the topics covered are more like the topics in nonrigorous books. It covers almost all of the topics on your list (it doesn't cover all the differential equations topics you listed) and it also covers infinite series and power series.

• Would the downvoters please explain their votes? – user1551 Jul 20 '19 at 14:57