I need Calculus book that suits my level. (Or at least primary book which I will follow more closely)
I don't have much formal education but have recently read Lang's Basic Mathematics (cover to cover, doing all almost all of the exercises).

In my search for books I found:
Spivak - Calculus
Apostol - Calculus 1 and 2
Courant - Introduction to Calculus and Analysis vol 1 and 2
Lang - First Course in Calculus

I found info about the other books, and it seems that Apostol's Calculus will be best suited for self learner and covers more than Spivak (also gives some applications), while Courant covers even more than Apostol but has less and harder problems.
I have read a little bit (about derivatives, limits) of Lang's book and it's quite easy to follow.

So, where Lang's book stays? Are the other books too advanced for me?

Which book I should use as primary text if time is a concern and I want to cover more things? (Sorry if that's too many questions)



There is a thing in the college I want to go called "Mathematics and Informatics" (that's in Eastern Europe).

I'm kind of in hurry because I want to get in college and I have the chance to skip the first year if I know enough. (first year is mostly C++, Calculus and little bit of Linear Algebra and I already know enough C++)

So if I'm to skip the first year I should be ready for the Mathematical Optimization, Discrete Math, Differential Equations, Information theory (IDK if I translate correctly). All of these are intro level.

I have around 6-7 months before I try to get the exams. And little bit more before I eventually start college.

BTW, Basic Mathematics was quite challenging and there have been some exercises that got me to look at the back of the book for solutions or search on the internet. Don't wanna make it sound like I've done 100% of them (someone in my situation my get discouraged after reading this), but I tried.

  • $\begingroup$ It's been several years since you asked this question. I wonder how your studies turned out. If you read this and want to let me know you can find me from my stackexchange profile. $\endgroup$ Jan 6, 2023 at 1:06

3 Answers 3


You ask an interesting question.

First: working through Lang's Basic Mathematics on your own and doing all the exercises is an impressive feat for someone with little formal education (in mathematics).

All four of the calculus texts you ask about are more difficult than the average text, but you should be able to manage any of them. Any of them will give you a good foundation for further study.

I know Courant and Spivak reasonably well, Apostol and Lang only by reputation and reviews. Courant is a classic. It will give you the best sense of the depth and usefulness of calculus, and how to think about mathematics. Spivak is probably the most thoughtfully rigorous. I think Apostol would be the most thorough, touching just about anything that might appear in any calculus curriculum. Lang will be straightforward, but not encyclopedic.

You don't say why you are in a hurry, or where you want to go next (more reading? back to school?).

I would suggest that you spend some time working through the first chapter or so of each - I think you can see that material on line. Then decide which suits your learning style best. You might want to study from two of the books, so you can compare the approaches and learn from two views.

Good luck.

Edit in response to the edit.

In six or seven months you should be able to prepare yourself for that exam.

Optimization might require some knowledge of multivariable calculus and linear algebra. That's in the second volume of Apostol, maybe a bit in Courant.

I don't usually recommend studying toward a particular exam, but if you can find old copies of the one you have to take you'll have a little more information on what to be sure to think about.

Is there someone at the school you can talk to now about optimizing your chances for admission?

  • $\begingroup$ Thanks for the answer. I've edited my post. BTW, I will definitely use more than one book, as it proved to be very useful to use different resources when I read Basic Mathematics. $\endgroup$
    – mecalk0z
    Sep 30, 2019 at 13:45
  • $\begingroup$ Indeed, it's intended for second year students to take some multivariable calculus and linear algebra in the first semester (of the second year). The optimization course is designed to take that in account. I talked with a professor there and he assured me that all I need for the exam is Calculus and superficial knowledge of Linear algebra. He assured me that I'll be able to take it with only C++ and Calculus as the second year is more math heavy. Thanks again for the advice! $\endgroup$
    – mecalk0z
    Sep 30, 2019 at 18:16

I have a similar background most of my math has also been self studied. In fact I also started by reading Serge Lang's Basic Mathematics. This gives a great foundation and gives you a lot of nitty gritty stuff that will help you prove things later! If you can handle that book I definitely think you'll be fine reading his calculus book (I also read that book after reading Basic Mathematics).

Being a fellow self studier I highly recommend that you look at MIT open courseware. MIT posts top quality courses with lecture videos, problem sets, exams, and solutions 100% for free online. Sometimes learning straight out of the book can be difficult and this will be very helpful!

Here is the calculus course

Here is the discrete math course

Watching these lectures with the book will definitely help you prepare for the exam. In addition I highly recommend 3B1B's animated short videos that help visualize the concepts from calculus they are amazing!

3B1B videos

Good Luck!

  • 1
    $\begingroup$ I always loved full length lectures (as opposed to what I've seen on Coursera). Seeing someone who took the path I am taking is quite reassuring. Thanks for the advice and pointing that MIT has this kind of courses (they even have Differential Equations and Linear Algebra lectures)! $\endgroup$
    – mecalk0z
    Oct 1, 2019 at 23:10
  • $\begingroup$ Yeah I know they are awesome and free! I've never liked coursera too much either. Also don't forget to check out the 3B1B videos! Out of all the things those have been my favorite and they're some of the best animations I've ever seen for anything really. $\endgroup$
    – PMaynard
    Oct 1, 2019 at 23:45
  • $\begingroup$ MIT also has computer science courses, though I'm not sure if they are in C++ or python. $\endgroup$
    – PMaynard
    Oct 1, 2019 at 23:47

Lewis and Zax Essential Discrete Mathematics for Computer Science for discrete math

https://sites.math.northwestern.edu/~newstead/teaching/290fa18/ for linear algebra

https://mooculus.osu.edu/textbook/mooculus.pdf for calculus

Optimization in single variable is somewhere in the calculus text.

Skip around to the topics you want as you're in a rush.


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