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eGuevara
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I own both volumes of Apostol, but I must confess that when I bought them I had never heard of Spivak's book. I think it's pretty much a toss up for me now. What about you guys?
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
mathwonk said:my dad used to have a book of riddles and he would not let us see the answers when he read the riddles to us. i think its the same principle. something to hold over your head.
Describe the fallacy in the following "proof" by induction:
Theorem: 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.
Proof: The statement is obviously true for n = 1. The step from k to k+1 can be illustrated by going from n = 3 to n = 4. Assume, therefore, that the statement is true for n = 3 and let G1,G2,G3,G4 be four blonde girls, at least one of which, say G1, has blue eyes. Taking G1,G2, and G3 together and using the fact that the statement is true when n = 3, we find that G2 and G3 also have blue eyes. Repeating the process with G1,G2 and G4, we find that G4 has blue eyes. Thus all four have blue eyes. A similar argument allows us to make the step from k to k+1 in 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 :rofl:)
Note: This example is from G. Pólya, who suggests that the reader may want to test the validity of the statement by experiment.
eGuevara said:pivoxa15: the top of the excercise reads:
Describe the fallacy in the following "proof" by induction
it IS wrong.
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?
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:fear (often of embarrassment or failure) is a big enemy of learning. keep fighting it.
there is almost no limit to what we can do when we RE NOT AFRAID TO TRY.
"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.