Thanks for the suggestions. Sorry for the emotional insight in my previous posts. This is not the place for that. I will certainly look at your suggestions. I am always on the lookout to expand my knowledge.
Problem solving and so frustrating
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Steven G
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You, not me, suggested that I think you want me to know it conceptually too when you responded to my wanting you to be able to prove the product rule. All I was saying is that no one being asked to compute the derivative of the product for the 1st time can say-oh it is obvious to me that the product rule is .... . Maybe some brilliant person can see the proof in their head and then know the product rule. Some things one can see conceptually but the product rule is not one of them. Again, I did mean that I want you to know the proof of the product rule. I would never think of not showing the derivation of the product rule in a calculus 1 course. In my opinion it is unaceptable.
You put me on the spot here by saying that you want to know calculus like the back of your hand. If your goal is to learn how to think, then Linear Algebra is the course. The prerequisite for Linear Algebra is calculus 3, not because you use calculus in linear algebra but you do need the mathematical maturity of calculus. Some community colleges have a pre-calculus prerequisite for linear algebra which shows that one does not need calculus for linear algebra. The students in this course get destroyed in this course (if their highest math was really precalculus).
Yes, that Amazon link is to the correct book. They really have some challenging questions and topics not in other calculus books. I do not know about now, but MIT used this book for years. Here is a link to download the pdf of the book-https://docs.google.com/file/d/0B2RI691ABQasY0hLQ29EWFh3cGM/view
There is no reason to start with Zill until you are finished with calculus. I can't speak about after the 1990's but the bible for a Physics book was one by Halliday and Resnick. I can't believe that anyone topped their book.
Respond back.
@Jomo
If that was your central point, I entirely missed it before. Obviously I agree with that though I might have said “intuitively” rather than “conceptually.”
Steven G
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I used conceptually because the OP used it. Sure intuitively was the better word.
First thanks for the investment of your time and I hope that I am not irritating you (I try to be as honest as possible), also thanks for the link to the book.
If you ask me now to derive the product rule then I must say I cannot do it. And I am sure that I have seen it on video and in the book.
Now, where is the problem here? Is it because I cannot memorize it anymore or that I don't see it conceptually or I am not intuitive to find out? And what route should I take. Do I have to memorize these things more, or be more intuitive or know more about the concept?
It is like you go for example to this thread here https://www.freemathhelp.com/forum/threads/every-x-is-making-f-x- positive.129084/ . I was first like I don't know this too. Then there came the suggestion to expand e^x^2 and go from there. I can't follow that (and I surely wouldn't come up with that). Then Dr Peterson suggested to find the minimum and maximum. And that's when it clicked for me and I could solve it (I had put the graph in the computer before to check on my thoughts).
So what is the problem there? I don't know. Why can I not come up with this? Is that intuitive or conceptually?
And I can point out many many cases like this.
It is how to tackle problems. Having the insight or handle to put the methods I have learned to work and in the right order.
I don't know how to do that.
It seems like many here can see what is going on in the problem, know the direction to solve it, and see to for example expand e^x^2 and what to look for.
And this happens much more with problem-solving because you are not confined to a method to solve it (which is helped by being at the end of a chapter so you know what they want).
And if you look at the physics page you see she takes a big road to know physics (a part I am sure). Where she recommends a Calculus book for the math part and when you are at a chapter in the physics book to hopefully finished the Calculus book and switch over to the Zill book. But the road is very long. But I like it that the road is specified. And yeah I know the Halliday and Resnick book is a good one. I now use the University Physics with Modern physics.
Again I hope I am clear and I am not frustrating you.
One way to think about calculus is to start from difference equations.
[MATH]r = p * q \implies \Delta r = (r + \Delta r) - r = (p + \Delta p)(q + \Delta q) =[/MATH]
[MATH]\cancel {pq} + q \Delta p + p \Delta q + \Delta p \Delta q - \cancel {pq} \implies[/MATH]
[MATH]\Delta r \approx q * \Delta p + p * \Delta q[/MATH]
[MATH]r = p * q \implies \Delta r = (r + \Delta r) - r = (p + \Delta p)(q + \Delta q) =[/MATH]
[MATH]\cancel {pq} + q \Delta p + p \Delta q + \Delta p \Delta q - \cancel {pq} \implies[/MATH]
[MATH]\Delta r \approx q * \Delta p + p * \Delta q[/MATH]
Steven G
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You need to make things happen (sometimes) in math. Anyone who can derive the product rule without know the results (f'g+fg') is clever. Now knowing the results makes things much easier. You want f'g but you are missing some pieces, so you add and subtract those pieces and everything comes together. Same for the quotient rule--just make it work. If you can do that, then you added something new to your arsenal to prove new theorems that you could not prove before. This is how you learn to think mathematically.
Dr Peterson probably doesn't realize this but he taught me a few new tools in solving problems. This upped my thinking level. The more you add to your tools, the more problems you can solve and the better you think.
Now go and try to use the definition of the derivative to prove the product rule. Make it work piece by piece. The nice thing about math is that all the pieces must fall into place (unless you are doing research).
Dr Peterson probably doesn't realize this but he taught me a few new tools in solving problems. This upped my thinking level. The more you add to your tools, the more problems you can solve and the better you think.
Now go and try to use the definition of the derivative to prove the product rule. Make it work piece by piece. The nice thing about math is that all the pieces must fall into place (unless you are doing research).
I just can't follow here. And that is the frustrating part. But one thing first. How did you come up with this? Was it something you have memorized or something that you could pull from other information?
Second. I just can't follow it. I see r=p∗q and to get Δr I see you doing (r+Δr)−r (I would never come up with that) but I see of course why it works. Second then I also cannot follow with (p+Δp)(q+Δq). I just don't know where that comes from. Maybe it is easier to see visually. But that's where I already cannot follow anymore. Then you get
pq+qΔp+pΔq+ΔpΔq−pq
Which I also don't get when you distribute things. I know it exposes me as pretty stupid but if I want to move forward then I have to admit this.
Then in your final line, I see the similarity with the product rule.
Now the question is first how did you come up with it? And second, how do I come up with it and also get it. And this is only a little piece. I want this to happen over the whole spectrum of Calculus and beyond.
I hope I am clear.
Steven G
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Here is the first part of a proof of the product rule.
Suppose h(x) = f(x)*g(x).
Then [math] h'(x) = \lim_{Δx->0}\dfrac {h(x+Δx)-h(x)}{Δx} = \dfrac{f(x+Δx)g(x+Δx)-f(x)g(x)}{Δx}[/math]
Now we know the answer! We want to make it so we get f'(x)g(x+Δx) (note that we have [math]\lim_{Δx->0}g(x+Δx)=g(x))[/math]. So what do we need to add/subtract (and then subtract/add) to make this happen?
Suppose h(x) = f(x)*g(x).
Then [math] h'(x) = \lim_{Δx->0}\dfrac {h(x+Δx)-h(x)}{Δx} = \dfrac{f(x+Δx)g(x+Δx)-f(x)g(x)}{Δx}[/math]
Now we know the answer! We want to make it so we get f'(x)g(x+Δx) (note that we have [math]\lim_{Δx->0}g(x+Δx)=g(x))[/math]. So what do we need to add/subtract (and then subtract/add) to make this happen?
Steven G
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I have a brother who is a super genius. He earned his PhD in math (actually operations research--a branch of math) at the age of 23 from Stanford University.
Whenever he helped me with calculus I thought that I knew nothing. He did everything differently then the conventional way. I was extremely depressed and thought that I would never understand math. I will never ever be as good as my brother in math but I realize that I can hold my earn. No one would ever say that I am not good at math. The point I am making is that if you do not understand JeffM's post now, but could still proof the product rule another way that is fine for now. To be honest, if I read JeffM's proof while I was in Calculus 1, I too would not have understood it. JeffM, myself and others on the forum are well beyond calculus 1 and see things differently now. We always write posts about how things could have been explained better than the previous poster/helper. I am not necessarily saying that is true with JeffM's case, in a way you are saying that. I tend to like proofs that follow the definition as one can understand it more easily. JeffM is just using a different definition--he uses differentials which is perfectly fine.
Whenever he helped me with calculus I thought that I knew nothing. He did everything differently then the conventional way. I was extremely depressed and thought that I would never understand math. I will never ever be as good as my brother in math but I realize that I can hold my earn. No one would ever say that I am not good at math. The point I am making is that if you do not understand JeffM's post now, but could still proof the product rule another way that is fine for now. To be honest, if I read JeffM's proof while I was in Calculus 1, I too would not have understood it. JeffM, myself and others on the forum are well beyond calculus 1 and see things differently now. We always write posts about how things could have been explained better than the previous poster/helper. I am not necessarily saying that is true with JeffM's case, in a way you are saying that. I tend to like proofs that follow the definition as one can understand it more easily. JeffM is just using a different definition--he uses differentials which is perfectly fine.
I had tried it yesterday and today. I am embarrassed and that is also one of the reasons why I don't post here very often.
I couldn't solve it. It is so frustrating. I don't know what steps I should take to get better at it this.
I like the way this is presented here https://www.mathsisfun.com/calculus/derivatives-dy-dx.html . It is so clear and shows every step very clearly in a way that I can also get it deeper. Does anyone know if there is a book like this that shows every step in a clear way?
When I read this I almost cried. Like your brother is like this super genius and he probably does hard work to get there but he gets results.
To give you an example. I always go very deep when I try to learn something. Once I wanted to do a triathlon. I trained myself greatly in swimming. I looked at every resource out there. I followed everything I could find with plans, etc..
Now my ankles are very inflexible. That means when I only peddled with my feet I would go backward. I kept training with fins (not all the times).
Then came a guy, we chatted a bit and he never swam before. He jumped in and he was immediately quicker than me (and he wasn't some sort of hidden athlete or something, just a regular guy).
That is very hard to swallow. I don't mind putting in the hours but if you don't get much better after so much dedication it is very frustrating. Especially when then someone jumps in and has no problem.
The same thing happened to me in the gym.
To me, intellect is one of the most important things I cling to and I see the same thing repeating again.
It would be nice for once to be good or great at something you have a passion for.
Steven G
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My brother never studied. He read the textbook the night before his test and then aced the test the next day. I never met anyone before who earned a PhD and was never challenged, which was the case with my brother. He just had the gift of learning and it just came naturally to him.
You want me to fall into a deeper depression do you
. I so wish I was like him. If you are that intellectual then there is nothing that can stop you imo. Yes everyone can have health issues or bad luck. But he doesn't need to have any doubt or fear. He can do as he pleases. Sigh.
Steven G
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You want f'(x)g(x) OR f'(x)g(x+Δx)
We already have f(x+Δx)g(x+Δx) so subtract f(x)g(x+Δx) which gives us [f(x+Δx) - f(x)]g(x+Δx). The bold part along with the denominator, Δx, gives us f'(x). So we have the 1st piece we need which is f'(x)g(x+Δx)-->f'(x)g(x). I did mention that g(x+Δx) goes to g(x) as Δx goes to 0.
Here is a proof that I made for the product rule.
We already have f(x+Δx)g(x+Δx) so subtract f(x)g(x+Δx) which gives us [f(x+Δx) - f(x)]g(x+Δx). The bold part along with the denominator, Δx, gives us f'(x). So we have the 1st piece we need which is f'(x)g(x+Δx)-->f'(x)g(x). I did mention that g(x+Δx) goes to g(x) as Δx goes to 0.
Here is a proof that I made for the product rule.
I have seen the proof. So thanks for the link. I was also distributing and adding and subtracting. But I was way off the mark and wouldn't have come up with the solution.
And now we have a fine example. And that brings us back to the beginning.
When I said I don't believe you that everyone can do it, and it needs intellect and that you are too modest about yourself.
I can of course follow you in the video but tell me how I can learn to do that too and just as important what they mean more as a concept and intuitively? Must I look at more proofs, must I repeat Calculus again and would I learn to get more insight that way? It all looks so natural when it is laid out in front of you. But I can't do it. I just don't know how. Most of the time I can do exercises ok if they are separated by chapter.
And again doing these proofs is not enough you have to know what they do conceptually and visually too.
And I don't know any resources or books that do that. Unless I need to keep repeating Calculus but rotate many books.
But it is nice to have this example to show what I mean.