Critique My Proof-Writing: Tips from Analysis 4th Ed. by Steven Lay

[opticaltempest]

I tried to prove this more mathematically to the best of my ability. Would anyone familiar with writing proofs like to critique this in order for me to improve it and my proof writing? Is this even considered a proof since I guess it could be considered somewhat trivial?

There are so many problems in calculus and physics that ask for proofs. I have not taken any proof writing classes yet. However, I am reading Analysis 4th ed. by Steven Lay in order to introduce me to proofs. Are there any better resources for learning how to write better proofs?

Thanks


http://img167.imageshack.us/img167/4931/questionxe0.jpg


http://img99.imageshack.us/img99/8108/proofmd5.jpg

 

[Pseudo Statistic]

Looks logical to me, especially when you brought Rolle's Theorem in.
Can't comment much on the format though... but looks good enough for me to understand. :) (Proof newbie here :P)

 

[quasar987]

Mathematically, it is impecable. But I find there are many misuses of words and symbols, and also a "chronological inconsistency". I don't know if any of those will be significant enough in the eyes of the corrector to cost you points, but here goes...

1° x(t) does not represent the DISTANCE traveled as a function of time. It is the position along an axis with respect to some origin (coordinate system). The distance traveled as a function of time is the function

D(t)=\int_{t_0}^t|v_x(t')|dt'

2° In the end of your first paragraph, you write "Let x(t0), [...] and x(t1)=x(t0)" .. ?!? Why do you write that so soon? This equality is a consequence of the average velocity being zero, as you will find later. But you don't know that yet. It makes no sense to write that. This was the "chronological inconsistency".

3° Second paragraph, last word: the correct word is "if", not "since". By definition, the set S is said non-empty IF S contains 1 or more elements. By definition, f is differentiable at x=a IF the limit (...) exists. etc.

4° Why do you sudenly start talking about [a,b] and (a,b)? I would suggest you substitude all those a and b by t0 and t1 respectively.

5° The correct application of Rolle's thm (aka intermediate value thm) is that "\exists c \in (t_0,t_1) such that x'(c)=0".You write well.

 

[opticaltempest]

In reply to #2 - Why did I define "Let x(t0), [...] and x(t1)=x(t0)" so soon in the proof.
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I thought I must to do this in order to establish what kind of function I will be dealing with. I need a very specific function in order to apply Rolle's Theorem / IVT. You said that the equality is a consequence of the average velocity being zero. But isn't it also correct to say that due to the defined equality of x(t0)=x(t1) the consequence is that the average velocity from t0 to t1 will be zero?

How can I later prove to the reader that the average velocity is zero from time t0 to t1 if the reader doesn't know that x(t0)=x(t1)?

It seems like it would be awkward to say:

The average velocity of this function is zero. (Then do calculations here) with the last step showing that x(t0)=x(t1). When done alternatively, I could let the reader know before hand that x(t0)=x(t1) and then show why the average velocity of this function is zero. How and where would you let the reader know that x(t0)=x(t1)? I guess I need to start analyzing more proofs.
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I will change all occurrences of "a" and "b" to "t0" "t1". Also, am I correctly using "if and only if" ?

The feedback helped immensely,
Thanks

 

[quasar987]

In reply to #2 - Why did I define "Let x(t0), [...] and x(t1)=x(t0)" so soon in the proof.
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I thought I must to do this in order to establish what kind of function I will be dealing with. I need a very specific function in order to apply Rolle's Theorem / IVT. You said that the equality is a consequence of the average velocity being zero. But isn't it also correct to say that due to the defined equality of x(t0)=x(t1) the consequence is that the average velocity from t0 to t1 will be zero?

It is certainly correct! But the stament of the theorem you set out to prove is: Given an object whose average velocity is 0 over some perdiod of time, show that its instantaneous velocity is zero at some time during this interval. The hypotesis is that the average velocity is 0, NOT that there are two times t0, t1, such that x(t0)=x(t1).

You then have to show that this hypothesis implies that...

1° ...if [t0,t1] is the interval over which the average velocity is zero, then x(t0)=x(t1). Then, you are equiped to use Rolle's them to...

2° ...show that this implies the existence of a vanishing instantaneous velocity for some time t0<c<t1.

How can I later prove to the reader that the average velocity is zero from time t0 to t1 if the reader doesn't know that x(t0)=x(t1)?

It seems like it would be awkward to say:

The average velocity of this function is zero. (Then do calculations here) with the last step showing that x(t0)=x(t1). When done alternatively, I could let the reader know before hand that x(t0)=x(t1) and then show why the average velocity of this function is zero. How and where would you let the reader know that x(t0)=x(t1)? I guess I need to start analyzing more proofs.

I'm repeating myself but that can never hurt: You don't have to prove that the average velocity is 0; it is your hypothesis. They want you to start with the assumption that the average velocity is 0, and show it implies the existence of a vanishing instantaneous velocity.

Also, am I correctly using "if and only if" ?

Yes.