Is proving details in calculus important?

[davidbenari]

Right now I'm seeing multivariable calculus and differential equations. I can apply all the rules they teach in class and solve all the problems with the correct "method".

However, I'm starting to worry that I don't know how to prove anything! For example, if I were teaching implicit differentiation in multivariable calculus to someone else, and that person asks me "why is it this way?" I would either tell them "you see, in leibniz notation differentials seem to cancel out" or just simply say "I don't know".

Why is the derivative of sinx=cosx ? I don't know!

Is my limited ability to prove things in calculus indicative of how little I know?

Is it important to be able to prove things in calculus? Or are these things "stupid details" that if I took the time to do the limits I would see where they come from?

Can anyone tell me what's the normal relationship one has with calculus?

I study physics btw.

Thanks.

 

[Blazejr]

I think this is very common route for students to first learn calculations and later learn theory. I guess calculations are more important at first but if you want to go deeper in physics you will need some theoretical background to better understand what is going on. Try Spivak's "Calculus" for first formal treatment of single variable calculus. Of course there are many more great texts out there.

 

[SteamKing]

Why is the derivative of sinx=cosx ? I don't know!

When you learned to take derivatives of functions in calculus class, what did your teacher do? Did he just pass out a table of functions and their derivatives?

IMO, you should have learned the limit definition of the derivative of a function, and thus learned why the derivative of sin x = cos x by being able to show it.

BTW, f'(x) = limit (as delta x (dx) approaches zero) [(f(x+dx) - f(x))/dx]

Is my limited ability to prove things in calculus indicative of how little I know?

IDK, but it shows that there are major gaps in your math education.

Is it important to be able to prove things in calculus? Or are these things "stupid details" that if I took the time to do the limits I would see where they come from?

Should you be able to prove every definition you come across in calculus? IMO, not unless you want to major in math as well as physics. But you should have been exposed to how to go about proving what the derivative of function is, or being able to show what the integral of a function is.

That you are apparently not familiar with the limit definition of the derivative points out some serious oversights in your math education. This becomes more serious the further you go into physics, as the math gets harder and gaps in your math knowledge could hamper your understanding of the physics.

Can anyone tell me what's the normal relationship one has with calculus?

I study physics btw.

Thanks.

IDK what you mean by 'the normal relationship one has with calculus'. You mean like buying calculus flowers on your anniversary? Taking calculus on a date night every once in a while?

Calculus is a tool which is used to study the relationship of various quantities which are changing w.r.t. time, or some other quantity, etc. A lot of scientific and engineering theory is expressed using calculus concepts. If you are not familiar with these calculus concepts, you won't be able to understand why the physics or the technical aspects of a certain subject are expressed that way.

 

[Aichuk]

Oh my god, I'm in the exact same position. My teacher taught me only the formulas but I don't know why they happen. I want to study engineering later on, will I suffer due to this?

 

[davidbenari]

When you learned to take derivatives of functions in calculus class, what did your teacher do? Did he just pass out a table of functions and their derivatives?

IMO, you should have learned the limit definition of the derivative of a function, and thus learned why the derivative of sin x = cos x by being able to show it.

BTW, f'(x) = limit (as delta x (dx) approaches zero) [(f(x+dx) - f(x))/dx]
IDK, but it shows that there are major gaps in your math education.
Should you be able to prove every definition you come across in calculus? IMO, not unless you want to major in math as well as physics. But you should have been exposed to how to go about proving what the derivative of function is, or being able to show what the integral of a function is.

That you are apparently not familiar with the limit definition of the derivative points out some serious oversights in your math education. This becomes more serious the further you go into physics, as the math gets harder and gaps in your math knowledge could hamper your understanding of the physics.
IDK what you mean by 'the normal relationship one has with calculus'. You mean like buying calculus flowers on your anniversary? Taking calculus on a date night every once in a while?

Calculus is a tool which is used to study the relationship of various quantities which are changing w.r.t. time, or some other quantity, etc. A lot of scientific and engineering theory is expressed using calculus concepts. If you are not familiar with these calculus concepts, you won't be able to understand why the physics or the technical aspects of a certain subject are expressed that way.

I do know the limit definition of derivatives/integrals in calculus. I only went through polynomials in class. The rest were just given in some type of table.

 

[SteamKing]

I do know the limit definition of derivatives/integrals in calculus. I only went through polynomials in class. The rest were just given in some type of table.

Well then, you've been given the tools to determine why the derivative of the sine is the cosine. All you have to do now is apply the definition of the derivative to the sine function and see what result you obtain. If it is the cosine, you will have cleared up your doubts about why the derivative of the sine is the cosine.

 

[homeomorphic]

A better reason "why" the derivative of sine is cosine is to do the geometry that gives the why, rather than just the proof, using the limit definition (you should know the definition). Tristan Needham does the derivative of tangent in the intro to his complex analysis book this way, for example. The limit derivation is very forgettable. I suppose you can always redo it if you remember how. But you get more insight from doing it geometrically, rather than as a formal proof. If you are just going to do the formal proof, I don't see that you're losing much by just memorizing the formula, other than just knowing that it can be done and limit practice, because it doesn't give you a feel for why it's true, anyway. Unfortunately, most people these days tend to have deficient visualization/intuitive geometry skills because that point of view is not emphasized enough, so I'm not even sure it will be easy for most people to see my point. The visual picture of what's going is what is valuable, if you are able to grasp it. That's what let's you see the connections with other things, like Euler's formula, rotational motion, what harmonic oscillators have to do with rotational motion, and it helps you remember it.

When I think of why the derivative of sine is cosine, I see a triangle at the origin with its angle at the origin increasing, then another little similar triangle in there, representing the increment of the function as the angle goes up a little bit. This ends up making it obvious to me that the derivative of sine is cosine and helps me remember which one has a minus in front of it. If this seems very non-obvious, it's because it requires quite a bit of practice and possible drawing some pictures to be able to visualize and make sense of it, and possibly someone to lay out the exact argument for you, if you can't come up with it yourself (as well as mastery of all prerequisite concepts). Now, a lot of people might scoff at this because they might think that it's making it complicated, but it actually ends up making it simpler in the end, even though it's more work, and the way I see it, this is how I get a lot of my enjoyment out of learning math, so it's not as if it's some extra chore that I tack on. If you are just going to take the limit formally, it's really not enlightening, except as limit-taking practice, and I suppose it can satisfy the skeptic in you.

If it makes you feel better, when I first learned calculus, I knew nothing about any of that stuff and just memorized it, so that's perfectly normal. I suspect a lot of professional mathematicians, engineers, and physicists don't know the REAL "why" behind it, either, only the formal, un-enlightening symbol manipulation "why", although it's possible some might figure it out, almost by chance, if they get enough practice with things like the physical arguments involving rotational motion or complex numbers and stuff like that. I'm not sure if it's the end of the world for them if they don't, but it sure makes a number of things a lot uglier, less intuitive, less fun, and more mysterious.

 

[FactChecker]

Some proofs give you intuition about why things work. Other proofs are so obscure that they don't help your intuition or understanding. Let's assume that intuition helps everyone -- engineer, applied mathematician, theoretical mathematician. So you should look at proofs that help you understand, no matter what your career goal is. Other proofs are for the theoretical mathematician.

One caveat: If you see the same proof pattern being repeated over and over, there is something fundamental going on there. Make note of those even if they seem obscure. They may eventually become second nature.