Choosing the Best Introductory Calculus Textbook: Non-Biased Reviews

[Tri]

Hello, I need help with deciding which textbook to buy. All I want is your non-biased opinions on which is the better introduction Calculus book.

Calculus
by Ron Larson


Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra
by Tom M. Apostol


Calculus: An Intuitive and Physical Approach
by Morris Kline


Calculus
by Michael Spivak

These are my options, what are your thoughts?

 

[A. Bahat]

Some of the choices you listed have very different target audiences, e.g. Spivak is almost a real analysis text, while Kline takes, as the title indicates, "an intuitive and physical approach". Give us some more detail about what kind of book you are looking for if you want a good recommendation.

 

[Tri]

Some of the choices you listed have very different target audiences, e.g. Spivak is almost a real analysis text, while Kline takes, as the title indicates, "an intuitive and physical approach". Give us some more detail about what kind of book you are looking for if you want a good recommendation.

Basically, I would just like a good introduction Calculus that teaches you from the ground up covering all fundamentals.

 

[micromass]

The best calculus book in my opinion is Spivak. It has really good explanations and many challenging (but hard) exercises.
The only problem with Spivak is that you're likely not ready for it. In order to read Spivak, you need to be familiar with calculus and proofs. So it's more like a second text on calculus. The same is true for Apostol.

If you're encountering calculus for the first time, then you should check out "A first course in calculus" by Serge Lang. I liked that book very much and it's not as hard as Spivak (although that means it is less rigorous).
Another book that should be good is "Quick Calculus" by Kleppner and Ramsey.

You should check these two books out!

 

[Tri]

The best calculus book in my opinion is Spivak. It has really good explanations and many challenging (but hard) exercises.
The only problem with Spivak is that you're likely not ready for it. In order to read Spivak, you need to be familiar with calculus and proofs. So it's more like a second text on calculus. The same is true for Apostol.

If you're encountering calculus for the first time, then you should check out "A first course in calculus" by Serge Lang. I liked that book very much and it's not as hard as Spivak (although that means it is less rigorous).
Another book that should be good is "Quick Calculus" by Kleppner and Ramsey.

You should check these two books out!

Already ordered Spivak, I have no exposure to Calculus but I'm pretty good at Algebra,trig,and geometry. Is there anyway I can manage with this colossal book?

 

[micromass]

Already ordered Spivak, I have no exposure to Calculus but I'm pretty good at Algebra,trig,and geometry. Is there anyway I can manage with this colossal book?

I can assure you that you are going to struggle. It's very rigorous book and the exercises are hard. Here are some tips:

• Don't rely on Spivak alone. Use other easier books, use internet as a resource, use physicsforums, use khan academy. If you don't use other sources, then you will likely not get much out of Spivak since it's difficult.
• The book requires you to prove things. You are probably not used to proving things. This is why a proof book might come in handy. Take a look at Velleman's "How to prove it" or at Houston's "How to think like a mathematician".
• Do not expect you to solve every exercise in one minute. Some exercises might take hours or days of thinking. This is normal. Don't expect this to be like a high school textbook.

 

[Tri]

I can assure you that you are going to struggle. It's very rigorous book and the exercises are hard. Here are some tips:

• Don't rely on Spivak alone. Use other easier books, use internet as a resource, use physicsforums, use khan academy. If you don't use other sources, then you will likely not get much out of Spivak since it's difficult.
• The book requires you to prove things. You are probably not used to proving things. This is why a proof book might come in handy. Take a look at Velleman's "How to prove it" or at Houston's "How to think like a mathematician".
• Do not expect you to solve every exercise in one minute. Some exercises might take hours or days of thinking. This is normal. Don't expect this to be like a high school textbook.

Well ok, I mean I'm not completely brain dead. I've gone through numerous University level textbooks and I just started sophmore year. Is the book really that difficult( I'd be surprised considering it's introductory level) or are you assuming I'm at the average high school level and this is too advance?

 

[micromass]

Well ok, I mean I'm not completely brain dead. I've gone through numerous University level textbooks and I just started sophmore year. Is the book really that difficult( I'd be surprised considering it's introductory level) or are you assuming I'm at the average high school level and this is too advance?

OK, so here's an standard Spivak exercise:

If x and y are not both zero, then $x^2+xy+y^2>0$. Show this.

For which numbers $\alpha$ is it true that $x^2+\alpha xy+y^2>0$ whenever x and y are both nonzero?

If you can do things like that, then Spivak might be ok for you.

 

[micromass]

Another one:

Let A be a subset of $\mathbb{R}$. We call x an upper bound of A if $x\geq a$ for all a in A.
The least upper bound of A is called the supremum of A. That is, if x is the supremum of A, then x is an upper bound and if y is another upper bound then $x\leq y$.

Now, assume that x is an upper bound of A. Show x is the supremum of A if and only if for each $\varepsilon >0$, there exists an a in A such that $x-\varepsilon< a$.

This is a pretty standard result that you should be able to prove yourself.

 

[Tri]

Another one:

Let A be a subset of $\mathbb{R}$. We call x an upper bound of A if $x\geq a$ for all a in A.
The least upper bound of A is called the supremum of A. That is, if x is the supremum of A, then x is an upper bound and if y is another upper bound then $x\leq y$.

Now, assume that x is an upper bound of A. Show x is the supremum of A if and only if for each $\varepsilon >0$, there exists an a in A such that $x-\varepsilon< a$.

This is a pretty standard result that you should be able to prove yourself.

This is real analysis, why are you even showing me this? Obviously I have no idea how to do it, that's why I'm getting the book, which teaches you how.

 

[dextercioby]

Not a mathematician, but I'm a big fan of Shilov's book published by Dover under the title <Elementary real and complex analysis>.