Skipping the proof of the power rule how inadvisable?

[Femme_physics]

It just seems too complicate to bother to comprehend. I know you guys are math/physics fundies, but I'm really just trying to get the hang of the material and be able to solve problems, I'm not looking for some sort of cosmic math super-understanding. Should I really bother with the proof of the power rule?

 

[mathwonk]

I'm a little worried that you think it is complicated. If that is so, probably you should make especially sure you understand it, since something is missing in your grasp of calculus or algebra, one or the other. I assume you mean the rule for derivative of x^n.

Try proving the derivative of x^2 at x=a is 2a. Nothing could be simpler:

you have the difference quotient (x^2 - a^2)/(x-a) and you want to take the limit as x-->a. First of all you can divide, making it easier to take the limit.

I hope everyone knows this gives (x+a). Then as x-->a we get the limit a+a = 2a.


now for x^3, we have (x^3-a^3)/(x-a) which gives us x^2 + ax +a^2, and as x-->a,

we get a^2 + a^2 + a^2 = 3a^2.


for x^4 e have (x^4-a^4)/(x-a) = x^3 + x^2a + xa^2 + a^3, and as x-->a, you get

of course 4a^3.

...etc... (x^n - a^n)/(x-a) = x^n-1 + ax^n-2 + a^2x^n-3 +...+a^n-1,

and we get the limit n.a^(n-1).

...everyone needs to know this much basic algebra.

 

[Femme_physics]

Hmm, well I got an A on the last algebra test, so don't you shake my confidence! :P

You know, just because you know a certain subject doesn't mean that proofs using that subject are going to be easy for you to grasp...but I'll take a long breath and dive into that proof pool again... I'll see what I can fish using difference resources... this proof seems annoying and useless though... but I'll see...thanks for replying...

 

[Pengwuino]

Not knowing how to do easy proofs is usually an indication of a lack of knowledge or understanding of a subject. Being able to prove things like that should be a check on your knowledge. The last thing you want to have happen is for this gap in knowledge to manifest itself as another problem on a test in some class who knows when and you'll kick yourself thinking "why did I skip over that!". Happened to me a lot, it's sadface town when it happens.

 

[micromass]

Instead of just reading the proof. Try to find the proof by yourself. You said you want to solve problems, well, take it as a problem

What other answer did you expect from math fundies

 

[Femme_physics]

Good point micromass ;)

Thanks guys... I'll see how to work it out

 

[mathwonk]

Let me get this straight, you have changed from saying it is complicated to saying annoying and useless. My proof above was designed to address your complaint about complicated, so i tried to make it seems simple.

I don't know how not to be annoying, but I think i can also address "useless".

My proof was meant to remind the student of the useful fact that (x^2-a^2) =
(x-a)(x+a), for example, and similar useful facts about the simplest polynomials.

If you believe these algebra facts are useless, I don't know quite what to say, except to quote bob dylan, and just say good luck.

 

[pergradus]

Sometimes you don't really need to bother with a proof. Often proofs are intricate and may require much deeper mathematical knowledge than the actual result does. In physics you really can't be a mathematician AND a physicist (unless you're one of the rare brilliant ones, but most of us are not) - so you just apply stuff other people have proven and trust the mathematician know what they are doing.

This is not such a case. The proof can be shown trivially using only the definition of the derivative and the binomial theorem. You will use the power rule thousands of times, its good to know why it works. You should study the definition of a derivative - and if that means re-examining limits then DEFINITELY study it. Then take a look at the binomial theorem, then take the general case of x^n.

 

[Femme_physics]

Let me get this straight, you have changed from saying it is complicated to saying annoying and useless. My proof above was designed to address your complaint about complicated, so i tried to make it seems simple.

I don't know how not to be annoying, but I think i can also address "useless".

My proof was meant to remind the student of the useful fact that (x^2-a^2) =
(x-a)(x+a), for example, and similar useful facts about the simplest polynomials.

If you believe these algebra facts are useless, I don't know quite what to say, except to quote bob dylan, and just say good luck.

I don't understand the hostility, and my opinion is subject to change. If you treat your students with that attitude you're not going to make a really inspiring teacher (whether you are one or not). I didn't say that algebra is useless. Where are you getting on with this? I find it actually quite offending and the first time I'm being treated in this forum like that... I'm not sure what I said to deserve it. Forget the fact that I'm the top student in my class, and that we're only going to start calculus next semester, and forget the fact that I worked some time to actually understand it, not thanks to you.

Oh, and everybody claims they have the "simplest explanation" -- thanks for trying though...

 

[chiro]

It just seems too complicate to bother to comprehend. I know you guys are math/physics fundies, but I'm really just trying to get the hang of the material and be able to solve problems, I'm not looking for some sort of cosmic math super-understanding. Should I really bother with the proof of the power rule?

I think you should think of calculus in two ways:

The first way is to understand what calculus really is. Its just a way to do maths with things other than linear definitions. Its like when you figured out things in high school like length, area, volume and so on with objects that had straight lines, you're now figuring out things that don't have constant change (ie non-linear).

So you take that apply it to many different fields. Many physical sciences are concerned with things that are not simply made up one one quantity but often quantities that are related. Best example is velocity (relating distance to time) and acceleration (relating velocity to time).

The second way is to think of the results as part of a massive library.

Applied maths is like writing code. When people write complex code, they don't always write it all: they use libraries written by other people to make their job easier.

Applied maths is the same. You understand what calculus is and what its used for (In the first part that I explained), and then after that, you use the results that you need. If you are pure mathematician then your job is to figure out these rules and extend the rules with a lot of rigour. If you're just using the results (like an engineer or a scientist would), then leave it to the pure mathematicians to verify that its correct and you just use them to solve whatever problem you have.

Having said that proving the power rule shouldn't be too difficult even if you are not pursuing a pure math degree. If the value of n is a positive integer, all you need to prove it as basically the binomial theorem and the definition of a limit. For generalized n (ie not just an integer but real numbers), you used a generalization of this.

 

[Fizex]

I don't understand the hostility, and my opinion is subject to change. If you treat your students with that attitude you're not going to make a really inspiring teacher (whether you are one or not). I didn't say that algebra is useless. Where are you getting on with this? I find it actually quite offending and the first time I'm being treated in this forum like that... I'm not sure what I said to deserve it. Forget the fact that I'm the top student in my class, and that we're only going to start calculus next semester, and forget the fact that I worked some time to actually understand it, not thanks to you.

Oh, and everybody claims they have the "simplest explanation" -- thanks for trying though...

You implied algebra is useless when you said the proof was useless since it uses an algebraic theorem. You're also being rather cocky by saying you don't deserve this. Mathwonk gave you a proof and you're too lazy to take the time to read it. He was only trying to help you until you called it annoying and useless. Math is math, if it is annoying you just have to deal with it. As for useless, how would you have ever known about the power rule without the proof existing? The power rule proof by mathwonk's algebra that (x^n-a^n)/(x-a) always reduces the power by one is simple. If you don't understand this then being at the top of your class is rather meaningless.

 

[micromass]

I don't really see why you think that mathwonk was being hostile. He simply adressed your comment about the proof being annoying and useless.
Maybe he wasn't that friendly, but he just took the time to write out the proof of the theorem and explain it to you why it wasn't complicated. And then you said it's annoying and useless. I don't think I would be "mister friendly" after such a thing too... But hostile? I didn't see any hostile comments from him...

 

[wisvuze]

I thought mathwonk was being helpful and even friendly; that's a lot of tedius math-typing he did up there, for you

 

[pergradus]

I don't understand the hostility, and my opinion is subject to change. If you treat your students with that attitude you're not going to make a really inspiring teacher (whether you are one or not). I didn't say that algebra is useless. Where are you getting on with this? I find it actually quite offending and the first time I'm being treated in this forum like that... I'm not sure what I said to deserve it. Forget the fact that I'm the top student in my class, and that we're only going to start calculus next semester, and forget the fact that I worked some time to actually understand it, not thanks to you.

Oh, and everybody claims they have the "simplest explanation" -- thanks for trying though...

The "hostility" probably stems from the fact that you asked a question, then rejected the answers people gave you. You obviously already made up your mind that you weren't going to bother to learn the proof, and then when people told you you should and even took the time to explain it, you simply dismissed them.

Being "top of your class" is one thing, learning how to respect people and deal with them is another.

 

[thegreenlaser]

Never, ever skip proofs/derivations if you can help it.

Always take the time to derive formulas and prove to yourself why something works the way it does. There's a couple advantages to this. 1) When you understand where a formula comes from, you understand exactly why it works the way it does, so you know when you can and can't apply it. You'll be able to 'wield' it much more effectively. 2) When you start having huge numbers of formulas that you need to know (which you will), you'll often find that all you really need to know are a few basic principles and relationships, and everything else can be derived from those. If you practice the ability to derive things, you'll do way better when given a long list of formulas that you need to know. 3) When you understand where a formula comes from, it sticks with you much longer. You may not remember the specific formula after a year of not using it, but if you took the time to really understand it at the time, chances are you'll be able to derive it fairly quickly from the basic principles you remember.

There are some times when the derivation is above your math ability, and you'll have to just straight up memorize, but in almost all cases, knowing the derivation is much better than just memorizing a formula. If it is indeed a simple formula or one you use a lot (like the power rule) you'll probably find that you'll memorize fairly quickly regardless, but it's still good practice to understand where it comes from.

 

[Femme_physics]

You implied algebra is useless when you said the proof was useless since it uses an algebraic theorem. You're also being rather cocky by saying you don't deserve this. Mathwonk gave you a proof and you're too lazy to take the time to read it. He was only trying to help you until you called it annoying and useless. Math is math, if it is annoying you just have to deal with it. As for useless, how would you have ever known about the power rule without the proof existing? The power rule proof by mathwonk's algebra that (x^n-a^n)/(x-a) always reduces the power by one is simple. If you don't understand this then being at the top of your class is rather meaningless.

Maybe my attitude wasn't fair towards math, I'm willing to accept it. It's always important to understand things thoroughly one can argue. Nevertheless, I wasn't "too lazy" to take the time to read his explanation, it wasn't any better than the book's. I just think there's a way to deal with people without insulting them! Math is often frustrating for students, giving them attitude when they're unsuccessful is not a quality of a good teacher, if you ask me. Rather, encouraging them to try. If I was being arrogant-- I apologize, but does it really need to be reciprocated with a hurtful attitude? The fact you don't think it is doesn't matter-- it hurt my feelings.
Being the top of my class without even having started Calculus is rather meaningful, I think, and the fact that I understand the proof now because I've actually taken the time. I only asked "how inadvisable".. I didn't say that I could never understand it. The replies made me ponder it's very inadvisable. I never asked anyone to actually try to write the explanation here.

The "hostility" probably stems from the fact that you asked a question, then rejected the answers people gave you. You obviously already made up your mind that you weren't going to bother to learn the proof, and then when people told you you should and even took the time to explain it, you simply dismissed them.

Seeing how this thread changed my mind, I don't see how you can say that!

Being "top of your class" is one thing, learning how to respect people and deal with them is another.

Oh believe me, I respect so many people in this forum that help me. This place is a damn life-saver and a blessing on the internet. I just don't like when I express my frustration over an issue and then being attacked! :( I'm sorry if it came off as though I was shooting this frustration over certain someone, that wasn't at all my intent! It's one thing that if it would happen IRL with a fellow student as a part of a joke (he telling me what mathwork did), that's cool, but since I actually REALLY respect mathwork for his knowledge and see all of you as mentors and teachers, this kind of dismissal really hurts :( ... The last thing I would want to happen is you guys quit helping me with future threads (and believe me, I'll have future threads) because you think I'm a disrespectful whiny *****. I just had a moment.Can I say I apologize if I upset anyone with my dismissive attitude and let's move on though? I rather get on with more math/physics, especially proofs of math/physics, since they're really really important-- and less of this "feelings" talk. :)

 

[Fizex]

What is so inadequate with the proof? It is the simplest I can think of and also very short compared to the math proofs you could be doing later. If you're going to overreact like this to a tiny bit of math you need to find yourself a new hobby. I bet no one here can tell you how 'inadvisable' it is to learn the power rule because we have never given a second though about learning it or not.

Besides me, no one was 'attacking' you and it makes you seem paranoid. If you respected mathwonk, then you shouldn't have shot him down by calling his help annoying and useless because that is really offensive.

I don't really think there is anything more to say about the proof. If you don't understand something about the proof say so, but don't call it useless. If the topic was about whether you should learn any proof in general then you should've made a better topic. Also, if your intent wasn't to shoot frustration then you really do have to learn to deal with and respect people.

 

[Femme_physics]

See, Fiz, this is exactly what I'm talking about. Every little struggle a student present does not = "You suck at math. Go home." It reminds me of the "famous failures" clip on youtube. You only inspire me to prove you wrong.

I didn't call his help annoying and useless! I was saying the power rule proof seems... wasn't referring to his post at all... was that the big misunderstanding here? And, I have a lot of passion towards math and physics, if it appears when I post-- great! I think it's a positive thing.

Now I've already changed my mind about it, and I can follow the logic of the power rule... I just really wanted to know the importance of it... nothing was said personally!... I'm not that kin of person... so I really don't see the use in keeping this debate alive... seriously, does everyone have to beat the dead horse, or can we move on?

 

[JohnT89]

I've been studying Calculus I in the last few months (I still struggle because I didn't give it the needed effort) and I think I can understand what Femme Physics is trying to say.

Personally, when I encounter a proof I am sometimes tempted to just skip it for two reasons:

- Most of the times I find proofs like 'needed formalism' more than 'a reason to believe the theorem', altough I've read some with genuine interest.
- Sometimes I can't just follow it easily and I refuse to mechanically memorize something not clear to me. (Yes, I know I am still into the easy stuff)

I am not proud of my attitude and I actually discourage it to anyone serious about doing maths.
I understand that trying to learn mathematics without proofs is most likely doing it wrong, but students are not always interested in what lies 'under the hood' unless they have a genuine attraction to maths in general.

I don't know the actual situation of Femme Physics, but I guess she have the same problem that we have here in Italy:

We 'have to' take Calculus I, often along with other heavy classes, and we have to study it in a 'hurry'. Where I live, Calculus I is like an official 'cram time', while it should be taught slower and learned in a more relaxed and spontaneous manner. Spontaneous learning leads to spontaneous questions, which leads the student to genuinely 'search for' proofs.

 

[Femme_physics]

Hi John, and thanks for the lovely reply. :) I can completely relate to everything you wrote. We actually haven't started on Calculus yet, I'm just floating around it-- so maybe it's even more advisable for me to go over the basics of the basics (which is why I made sure I understand the proof of limits!). Perhaps my knees a little buckled when it came to the derivative and I didn't give up the proper attitude, and I also discourage it to anyone serious doing math to do what I did and even CONSIDER skipping proofs. I agree with the rest, anyway.

Sometimes it's the schools fault as your case JohnT, in which case you ought to take less of the blame...good post anyway.

 

[holomorphic]

My 2p to the OP...

You probably wouldn't go to a professor, ask him to explain how to prove a theorem, and then call his explanation useless and annoying. You owe Mathwonk an apology.

 

[Femme_physics]

My 2p to the OP...

You probably wouldn't go to a professor, ask him to explain how to prove a theorem, and then call his explanation useless and annoying. You owe Mathwonk an apology.

This is titled "how inadvisable", not "explain me the power proof." This is why it's posted at the Academic Guidance and not Calculus section. This section I use for how I should study, not trying to understand my studies-- which the other sections are for. And while you're at it, why don't you show me EXACTLY where I called Mathwork's Explanation, "useless and annoying"-- and please prove to me (while we're on the issue of proofs) that I was referring to his explanation of the proof.

Besides, I already sent Mathwonk a PM to clear the air. I hope he understand that comment had nothing to do with his post...I think he's taken offense because he thought it did. Anyway, please stop beating a dead horse here...jeez...

 

[Fizex]

See, Fiz, this is exactly what I'm talking about. Every little struggle a student present does not = "You suck at math. Go home." It reminds me of the "famous failures" clip on youtube. You only inspire me to prove you wrong.

I didn't call his help annoying and useless! I was saying the power rule proof seems... wasn't referring to his post at all... was that the big misunderstanding here? And, I have a lot of passion towards math and physics, if it appears when I post-- great! I think it's a positive thing.

Now I've already changed my mind about it, and I can follow the logic of the power rule... I just really wanted to know the importance of it... nothing was said personally!... I'm not that kin of person... so I really don't see the use in keeping this debate alive... seriously, does everyone have to beat the dead horse, or can we move on?

Well then you need to tell us what you're not understanding instead of dismissing it. You did call his help annoying and useless, here's how: his post was mainly about a proof, and you said, "this proof seems annoying and useless though". His proof = his post = his help. I'm glad you apologized though, so no harm done.

I don't think we need to move on because they've already said:

Sometimes you don't really need to bother with a proof. Often proofs are intricate and may require much deeper mathematical knowledge than the actual result does. In physics you really can't be a mathematician AND a physicist (unless you're one of the rare brilliant ones, but most of us are not) - so you just apply stuff other people have proven and trust the mathematician know what they are doing.

This is not such a case. The proof can be shown trivially using only the definition of the derivative and the binomial theorem. You will use the power rule thousands of times, its good to know why it works. You should study the definition of a derivative - and if that means re-examining limits then DEFINITELY study it. Then take a look at the binomial theorem, then take the general case of x^n.

And now since you say you already understand the proof, there's really no point in discussing whether you should learn it or not.

 

[Femme_physics]

There you go-- I never specifically referred to Mathwonk's phrasing of the proof...rather, that's what some have inferred from that. I admit, there was some room for misunderstanding which I take the blame for...but that's my indiscretion...no harm done... anyway, it's been tagged as misunderstanding already... seriously...let's move on...

 

[mathwonk]

pardon me femme for acting ticked off and leaving you to twist in the wind. and thanks for the kind PM. we are cool. my best to everyone.