What is my problem? (Difficulty solving math and physics questions)

[Adesh]

Hello everyone, Physicsforum has always helped me whenever I get into predicament. So this time again I want the help of experienced people.

I did a great deal of hard work for understanding the true nature of calculus. I read Newton's original works and I found that his sole purpose of inventing calculus was to solve the physics problem. Then I read Gilbert Strang's Calculus and watched his videos on MIT OCW, I built a good foundation for calculus and it was no more a alienated subject to me, after that I went on to strengthen my understanding even more so I read books like Thomas Calculus, Spivak's calculus and little of Apostol's calculus. After this much I watched videos by Professor Herbert Gross which gave a firm knowledge once again. To fulfill my curiosity I read little bit of Riemann Integrals and Cauchy's work on Calculus to get a even better understanding.

It took me two and half years to do above things. But problem comes in Question Solving, even after so much theoretical concepts I'm not able to solve this problem $f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)$, then find f(2).(problem is given in images). I want to know why I'm not able to solve the problem? What I'm missing? What should I do to solve problems because it depresses me a lot when I can't solve a problem?

I enjoyed Feynman Lectures on Physics so much, I read it devotionally and really like physics so much. But again the problem comes in Question solving. Irodov makes use of all the formulas which I studied but then also I can't solve much of it. The important thing is I like theory much more than problem solving but why I'm unable to solve problems which involve the theory I already know?

Please help me! This forum has always helped me and I hope this time also it will do the same.

 

[scottdave]

You probably will get more response, if you type the problem out rather than post an image. I'm working from my phone right now. Remember in Calculus that f(5) for example is the function f evaluated at 5. This will be a number. Similarly f'(1) is the 1st derivative of f, then evaluated at 1. Then f'"(2) will yield a number as well as f'''(3). Think about that for a bit and see if you can figure what the values of these are.

Here I typed part of it for you.
If f(x) = x³ + x² f'(1) + x f''(2) + f'''(3), then find f(2)

 

[Delta2]

All the problems require from us to apply theory and combine different parts of theory to solve them. But for each problem the application of theory and the combination of different parts of theory is sort of unique I would dare to say. The common thing between problems is that we apply and that we combine the theory.

For example there are million problems in thousands of books out there that can be solved by combining conservation of energy and conservation of momentum. With these two parts of theory (the two conservation principles) we can solve million problems, but for each problem the application and combination of these two principles is unique.

Don't get too depressed if you can't solve a particular problem, it might require some application of theory that is not known or forgotten by you. Also there are some problems that require some steps that need application of some theory that we would never expect to apply in this particular case , yet it applies. Generally don't get too depressed with anything in life.

 

[Adesh]

All the problems require from us to apply theory and combine different parts of theory to solve them. But for each problem the application of theory and the combination of different parts of theory is sort of unique I would dare to say. The common thing between problems is that we apply and that we combine the theory.

For example there are million problems in thousands of books out there that can be solved by combining conservation of energy and conservation of momentum. With these two parts of theory (the two conservation principles) we can solve million problems, but for each problem the application and combination of these two principles is unique.

Don't get too depressed if you can't solve a particular problem, it might require some application of theory that is not known or forgotten by you. Also there are some problems that require some steps that need application of some theory that we would never expect to apply in this particular case , yet it applies. Generally don't get too depressed with anything in life.

Your answer is very great. My sincerest gratitude to you, can you suggest me how can I learn to apply theory, is there any book which I can read to learn that?

 

[Adesh]

You probably will get more response, if you type the problem out rather than post an image. I'm working from my phone right now. Remember in Calculus that f(5) for example is the function f evaluated at 5. This will be a number. Similarly f'(1) is the 1st derivative of f, then evaluated at 1. Then f'"(2) will yield a number as well as f'''(3). Think about that for a bit and see if you can figure what the values of these are.

Here I typed part of it for you.
If f(x) = x³ + x² f'(1) + x f''(2) + f'''(3), then find f(2)

Thank you for writing the mathematical expression.

 

[Delta2]

Your answer is very great. My sincerest gratitude to you, can you suggest me how can I learn to apply theory, is there any book which I can read to learn that?

No I don't know of any book that teaches how to apply theory as a meta concept.

The books that you read on calculus like spivak and apostol they have with each piece of theory , examples for how the theory is applied. I mean if a book is well written will always have examples when it presents a new theorem. Study carefully the theorem, its proof, and these examples, and then try to solve some problems that seem similar. I know that for a single theorem there might be 1 million different applications of the theorem, each application carries a unique sub-flavour, though all are applications of the one and the same theorem. After studying the book examples and solving some similar problems or at least try to solve them you ll get more and more familiar with how a theorem is applied.

One thing you might try is for each example problem that is solved by the book, read it carefully, and then after one hour or two or perhaps after 1 day, read the problem again and try to solve it yourself and see if you can find another solution or just reproduce the solution given by the book. This way you test both your memory and the understanding of the problem.

 

[Adesh]

No I don't know of any book that teaches how to apply theory as a meta concept.

The books that you read on calculus like spivak and apostol they have with each piece of theory , examples for how the theory is applied. I mean if a book is well written will always have examples when it presents a new theorem. Study carefully the theorem, its proof, and these examples, and then try to solve some problems that seem similar. I know that for a single theorem there might be 1 million different applications of the theorem, each application carries a unique sub-flavour, though all are applications of the one and the same theorem. After studying the book examples and solving some similar problems or at least try to solve them you ll get more and more familiar with how a theorem is applied.

One thing you might try is for each example problem that is solved by the book, read it carefully, and then after one hour or two or perhaps after 1 day, read the problem again and try to solve it yourself and see if you can find another solution or just reproduce the solution given by the book. This way you test both your memory and the understanding of the problem.

Sir I tried these things but didn’t get any good results.

 

[scottdave]

Some people learn better with different methods. But I think there is only so much you can learn from reading a book. You need to practixe what you've learned, and probably fail several times to help reinforce what doesn't work for you. Try different puzzle sites like ProjectEuler.net to see if you can solve any of them. Get practice solving problems, even if they aren't Calculus. It should help you get better.

 

[Mark44]

But problem comes in Question Solving, even after so much theoretical concepts I'm not able to solve this problem $f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)$, then find f(2).

Since f(x) is given in terms of its first, second, and third derivatives, the most obvious thing to do is to calculate the first, second, and third derivatives, keeping in mind that (1) f'(1), f''(2), and f'''(3) are constants, and (2) the derivative of a constant times a function is the constant times the derivative of the function.

BTW, I fixed your notation in the first post. Here's a link to our LaTeX tutorial -- https://www.physicsforums.com/help/latexhelp/. The same link also appears at the lower left corner in the input pane.

 

[Adesh]

Since f(x) is given in terms of its first, second, and third derivatives, the most obvious thing to do is to calculate the first, second, and third derivatives, keeping in mind that (1) f'(1), f''(2), and f'''(3) are constants, and (2) the derivative of a constant times a function is the constant times the derivative of the function.

BTW, I fixed your notation in the first post. Here's a link to our LaTeX tutorial -- https://www.physicsforums.com/help/latexhelp/. The same link also appears at the lower left corner in the input pane.

Sir I have followed your advice and have got the answer -2. But no one ever taught me (I mean no book or lecture) to solve like this. How can I learn these things (the hint which you have given) ?

 

[Mark44]

Sir I have followed your advice and have got the answer -2. But no one ever taught me (I mean no book or lecture) to solve like this. How can I learn these things (the hint which you have given) ?

They come with practice, and trying lots of approaches, many of which won't bear fruit.

For this particular problem, you are given a formula for f(x) and asked to find f(2). The very first step would be to use the formula for f and replace x in the formula by 2.
From this, we see that $f(2) = 2^3 + 2^2f'(1) + 2f''(2) + f'''(3) = 8 + 4f'(1) + 2f''(2) + f'''(3)$
At this point we don't know f'(1), f''(2), or f'''(3), but we have a formula for f(x), so it would appear that differentiation is in order. What do you get for f'(x), f''(x), and f'''(x)?

 

[StoneTemplePython]

But no one ever taught me (I mean no book or lecture) to solve like this. How can I learn these things (the hint which you have given) ?

My take is that studying some other kind(s) of math will help broaden your perspective and give you more mental models to reach for... and from there it's a mixture of inspired guessing, pattern recognition, and trial & error.

In this case, you first recognize that you're dealing with a polynomial, and you have information about derivatives (at select points). (i) The derivative operator is nilpotent with respect to polynomials (i.e. a degree n polynomial after n+1 derivatives is zero). (ii) With this insight you may be able to see a triangular system of linear equations to solve, which is sort of the goal of gaussian elimination / row echelon form. Both of the above concepts will be at least touched on in a good Linear Algebra course. (I think Strang and Gross have coursework on mit ocw for linear algebra, though you may need to go through a book afterward to get a deep enough understanding. Don't go the historical route and try to read stuff from Cayley or Sylvester though, it will make your head hurt.)

A general problem solving strategy (and frequent proof strategy) is: work backwards. If you asked yourself: how would I work backwards here? You'd might say to yourself -- that means finding $f'''(x)$, then $f''(x)$, then $f'(x)$, and then at the end $f(x)$. So for $f'''(x)$, now how can I find that? (Basically this gets you to be in the neighborhood of what I said in the prior paragraph, without the bigger ideas / fancy terminology involved.)

 

[berkeman]

I enjoyed Feynman Lectures on Physics so much, I read it devotionally and really like physics so much. But again the problem comes in Question solving.

Have a look at this thread from last year here on the PF. Maybe some of the problem solving tips in it can be of help and inspiration to you.

https://www.physicsforums.com/threads/is-problem-solving-an-inborn-skill.958292/

Is problem solving an inborn skill?