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eGuevara

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In summary, the conversation discusses the preferences and experiences of different individuals with mathematical textbooks on calculus, specifically the books Apostol and Spivak. Some find Spivak's books more enjoyable and fun, while others appreciate Apostol's style and precision. It is noted that Apostol's book is more suitable for review and for math students, while Spivak's book is better for learning calculus. The conversation also touches on the idea of not providing solutions to all problems in textbooks and how this can be beneficial for learning. Finally, the conversation ends with a humorous example of a fallacy in induction.

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eGuevara

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Physics news on Phys.org

- #2

verty

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If you own both volumes, read them.

- #3

eGuevara

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- #4

morphism

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I like both so much it's hard to decide which I like more.

- #5

mathwonk

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spivak is a ntural born teacher of tremendous ability and it comes LL THE WAY THROUGH IN HIS BOOK.

apostol i got into later, and began to aPPRECIATE it for the niceties of some arguments. it is very very scholarly and precise, but a bit eccentric in order of rpesentation. it is a superb work and since you already know calculus, should serve you very well.

apostol is relatively dry comopared to the fun of spivak, but ok for serious advanced students. one of my friends took beginning calc from apostol at MIT and loved it.

As one of wine dealers said when asked to choose betwen two fine wines "you are asking me to choose between my father and my mother!".

so there is no way really to prefer one to the other except by personal taste. I enjoy owning both, and also courant, hardy, goursat, and dieudonne!

great books are uniquely different from each other enough not to be interchangeable, but anyone is also sufficient.

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pivoxa15

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mathwonk

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morphism

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- #9

symbolipoint

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What level of Calculus were you teaching when you used this Spivak book? How does it compare the (don't get upset at me...) the Larson & Hostetler undergraduate Calculus book and the Salas & Hill undergraduate Calculus book?i have taught most of the way through spivak and i and the clas really enjoyed it. as i did so i began to realize some of his proofs were roughly the same as the ones in courant, but written up in a much more accessible and fun style

Symbolipoint

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mathwonk

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- #11

mrmusavi

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i think if you studied calculus before, the Apostol is better for review the calculus. buti think spivak's problems is normal and better.

but i think if anyone wants to learn Calculus the stewart's books for some reasons and in my experience is the best. if you do all the problems, you can see that most of the top universities picked up it. but Apostol is good for any who wants to review and be more exact means that Apostol 's books is suitable for math students mostly, and has a unique subject order.

but i think if anyone wants to learn Calculus the stewart's books for some reasons and in my experience is the best. if you do all the problems, you can see that most of the top universities picked up it. but Apostol is good for any who wants to review and be more exact means that Apostol 's books is suitable for math students mostly, and has a unique subject order.

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pivoxa15

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mathwonk said:

Is it better to not know how to do a problem after spending ages on it or read a bit of the answers in the hope of receiving some clue.

Also its good to confirm that you got the right answer - espeically if you are not a confident student.

Also the suggested answers might be different and more likely better than what the student has done so it would be good to see the author's thinking process.

I have the habit of not doing a problem if the answer isn't immediately avaliable in the fear of not being able to do a problem and not getting to see the solution.

- #13

mathwonk

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there is almost no limit to what we can do when we RE NOT AFRAID TO TRY.

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eGuevara

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Describe the fallacy in the following "proof" by induction:

Theorem:Given any collection ofnblonde girls. If at least one of the girls has blue eyes, then allnof them have blue eyes.

Proof:The statement is obviously true forn = 1. The step fromktok+1can be illustrated by going fromn = 3ton = 4. Assume, therefore, that the statement is true forn = 3and letG1,G2,G3,G4be four blonde girls, at least one of which, sayG1, has blue eyes. TakingG1,G2, andG3together and using the fact that the statement is true whenn = 3, we find thatG2andG3also have blue eyes. Repeating the process withG1,G2andG4, we find thatG4has blue eyes. Thus all four have blue eyes. A similar argument allows us to make the step fromktok+1in general.

Corollary:All blonde girls have blue eyes.

Proof:Since there exists at least one blonde girl with blue eyes, we can apply the foregoing result to the collection consisting of all blonde girls.

(Here comes the funny )

Note: This example is from G. Pólya, who suggests that the reader may want to test the validity of the statement by experiment.

LOL, induction is alright, but go get a GIRL!

(I know I know, I laugh at stupid things.)

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CPL.Luke

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In books with large numbers of problems its quite commmon to find mistakes in the solutions, its better to learn how to analyze your solution to determine how correct it is.

So sometimes the author's put in fewer solutions, just so they can check the solutions over a bit more.

So sometimes the author's put in fewer solutions, just so they can check the solutions over a bit more.

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pivoxa15

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eGuevara, is that proof really given by Polya?

I know its meant to be a joke but it is a wrong illustration of proof by induction - there is a fundalmental error.

In proof by induction one must show in general k=>k+1 for any k

In the proof it had "The step from k to k+1 can be illustrated by going from n = 3 to n = 4." This is fundalmentally wrong!? Just because it works going from n=3 to n=4 dosen't mean it will go from n=45 to n=46. So the corollary and theorem is logically wrong!?

Even though its a joke, It might mislead people into thinking this is really how a proof by induction works. I don't find this point funny.

I know its meant to be a joke but it is a wrong illustration of proof by induction - there is a fundalmental error.

In proof by induction one must show in general k=>k+1 for any k

In the proof it had "The step from k to k+1 can be illustrated by going from n = 3 to n = 4." This is fundalmentally wrong!? Just because it works going from n=3 to n=4 dosen't mean it will go from n=45 to n=46. So the corollary and theorem is logically wrong!?

Even though its a joke, It might mislead people into thinking this is really how a proof by induction works. I don't find this point funny.

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- #17

Hurkyl

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pivoxa15: it's simply a more advanced version of "Find the mistake in this attempted proof that 0 = 1" type problems. (incidentally, the method in the inductive step does work from 45 to 46)

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eGuevara

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it IS wrong.

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radou

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- #20

mathwonk

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- #21

pivoxa15

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eGuevara said:

Describe the fallacy in the following "proof" by induction

it IS wrong.

I didn't see that. Now it makes more sense.

This proof is more subtle than I thought. Where was his mistake?

Is it this:

The Theorem is: Given any collection of n blonde girls. If at least one of the girls has blue eyes, then all n of them have blue eyes.

This is true for all collections of set of 1 blonde girls. Let's suppose {G5} where G5 is the 5th blonde girl dosen't have blue eyes. The theorem still stands because its not the case that at least one girl has blue eyes.

But for collections later on say {G1, G5} G1 has blue eyes but G5 doesn't hence a counterexample is found and the theorem disproved.

So his step in showing for any group consisting of k blonde girls if one has blonde eyes all have => for any group consisting k+1 blonde girls if one has blonde eyes all have is incorrect. In the proof he argued "The step from k to k+1 can be illustrated by going from n = 3 to n = 4." That is not general enough and lies his mistake. I found a counter example by going from n=1 to n=2

Is my analysis correct? Any better suggestions?

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loom91

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pivoxa15 said:I didn't see that. Now it makes more sense.

This proof is more subtle than I thought. Where was his mistake?

Is it this:

The Theorem is: Given any collection of n blonde girls. If at least one of the girls has blue eyes, then all n of them have blue eyes.

This is true for all collections of set of 1 blonde girls. Let's suppose {G5} where G5 is the 5th blonde girl dosen't have blue eyes. The theorem still stands because its not the case that at least one girl has blue eyes.

But for collections later on say {G1, G5} G1 has blue eyes but G5 doesn't hence a counterexample is found and the theorem disproved.

So his step in showing for any group consisting of k blonde girls if one has blonde eyes all have => for any group consisting k+1 blonde girls if one has blonde eyes all have is incorrect. In the proof he argued "The step from k to k+1 can be illustrated by going from n = 3 to n = 4." That is not general enough and lies his mistake. I found a counter example by going from n=1 to n=2

Is my analysis correct? Any better suggestions?

The fallacy is more subtle than that. I was speed reading the book to look at some proofs so thoughtfully ommited from our high-school calculus syllabus so I had little chance to do the problems. This was one of the few I did. You found the correct counter-example but your logic is not completely correct. Write all the individual steps in the proof and analyse each step carefully, there's a step that can not be justified on the basis of the previous steps.

Molu

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rocomath

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i completely agree with this. my first week of Calculus, i was so damn scared bc not only was i thinking to myself, ok i didn't take Calculus in HS (so no previous exposure) and now my algebra is kicking my ass. our first test, i studied a lot and it was my highest grade. before Calculus, i never knew about "solution's manuals" and i found out about it, lol. so i bought the manual and many times when i was stuck, i referred to it. it became an addiction ... the first week of Calculus, i would think in my sleep how to solve the problems, Calculus was continually racing through my mind. after that, i felt robbed ... i was no longer thinking as much as i did the first week. but i told myself, this isn't good so for everytime i used the manual, i would spend a little more time analyzing the problem. i notice this problem with my classmates as well, they would even have the manual open while doing their homework ... ultimately, they dropped. anyways, solution manuals are bad, but answers to the exercises aren't so bad.mathwonk said:

there is almost no limit to what we can do when we RE NOT AFRAID TO TRY.

btw, i don't bring my manual with me anymore, i want to relive the first week of Calculus over and over :)

- #24

Shing

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I have a question, would anyone be kind enough to answer me?

Recently, I am self-studying calculus via Apostol's book

but I have totally no idea whether if I have done correctly or not in the excises, paticularly the proofs.

so anyone would give me a few advises?

- #25

renob

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which one of the two books is a better supplementary text for a first year calculus course?

"Apostol or Spivak" refers to two well-known math textbooks, "Calculus" by Tom M. Apostol and "Calculus" by Michael Spivak. Both books are commonly used in university-level mathematics courses.

This is a matter of personal preference. Some people prefer Apostol's textbook for its clear and concise explanations, while others prefer Spivak's for its more rigorous approach and challenging exercises.

Both books are suitable for beginners, but Apostol's textbook may be more approachable for those who are new to calculus.

Both textbooks cover the same topics in calculus, but Spivak's textbook may be more thorough and comprehensive in its coverage.

Yes, both books can be used for self-study. However, it is recommended to have some background knowledge in calculus before attempting to use these textbooks as they can be quite challenging for self-study without prior knowledge.

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