What Does the Derivative Notation in Calculus Represent?

[Line]

In Calculus what's that big symbol that's before a function stand for. I kinda looks like a big long f.

And then there's that qoutation mark looking thing. It appears beside the F in a function. It kinda looks like this...F'...what's it mean?

 

[d_leet]

The f looking symbol you refer to that appears before a function is called an integral.

and about the quotation mark thing '.

if f(x) is a function
then f'(x) read f prime of x is the derivative of f(x)

 

[Line]

And just what is an integral?

 

[Integral]

I find it interesting that you know enough to post this question in the calculus section?? Hmmm??

 

[Tide]

Incidentally, the $\int$ symbol originated as an elongated letter "S" and stood for "sum."

 

[El Hombre Invisible]

And just what is an integral?

A derivative of a function is the rate of change of that function as some other variable changes. For instance, the velocity of an object is the rate of change of position with time. You can determine this to arbitrary accuracy by taking intervals of distance traveled ($$\Delta x$$) and divide by the time taken ($$\Delta t$$):

$$v = \frac{\Delta x}{\Delta t}$$

The larger the interval $$\Delta x$$ taken, the larger the interval $$\Delta t$$ is. Also, the less accurate the value of the velocity at time t will be, since any changes in velocity during $$\Delta t$$ will effect the value you get. But as the intervals get smaller and smaller, the value for the velocity you get for that interval of time becomes more accurate. As $$\Delta x$$ tends towards zero, then, the value for the velocity at time t becomes correct. A derivative of a function does just this, so if the function of position at some time t is f(t) then the derivate will give you velocity as a function of time:

v(t) = f'(t).

An integral is an anti-derivate - it will undo the derivative to give you the original function, so:

$$\int v(t) dt = \int f'(t) dt = f(t)$$

 

[Line]

You must be an american. Americans give long complicated answers to things.

 

[HallsofIvy]

You must be an american. Americans give long complicated answers to things.

As opposed to short, simple, wrong answers?

 

[HallsofIvy]

My father used to call the $$\int$$ symbol a "seahorse"!

 

[Jameson]

Agreed. The response wasn't even that long. If you'd like to read a real solid definition of the integral and understand all parts of it, get ready to read over 100 pages of material.

 

[JasonRox]

You must be an american. Americans give long complicated answers to things.

With that attitude, you'll make very far in life.

Personally, I always thought Americans gave short lazy answers.

 

[Line]

The Japanese know how to keep it real short. Did you know that they found a reason that american kids aren't doing better. The books are so darned thick. Compared to countries like Germany and Japan which score higher our books are extremely thick. Seems the answer is to keep it short and to the point. Through having less words you cover more material and not just lalalala. They also found that the other countries have more diagrams in their textbooks and we don't.

 

[Math Is Hard]

Calculus will teach you patience. Either you will learn patience or you will not survive into further mathematics courses. You can't expect a 12 week (or longer) course to be summarized into a couple of short sentences. El Hombre did a great job distilling a large amount of information into a condensed form for you.

 

[cronxeh]

Calculus will teach you patience. Either you will learn patience or your will not survive into further mathematics courses. You can't expect a 12 week (or longer) course to be summarized into a couple of short sentences. El Hombre did a great job distilling a large amount of information into a condensed form for you.

Actually that's exactly what they do in Real Analysis - at one point they just summarize your entire 2 classes of Calculus into a few equations and you end up with a nice expression on your face

 

[-Job-]

Most of the time you can find a lot of nice webpages online from universities, covering all kinds of topics, that are very to the point, so you have some options, that's what i do when a book is very bad.
I'm pretty sure though it's not really because of how thick the books are that people don't do very well, that's just a lame excuse in my opinion.

 

[incognitO]

Also, basic courses are pretty much the same in every university in the world, and i bet you every good mathematitian, no matter where in the world was born or formed ,has read the classical texts.

 

[Math Is Hard]

Actually that's exactly what they do in Real Analysis - at one point they just summarize your entire 2 classes of Calculus into a few equations and you end up with a nice expression on your face

I can certainly believe that. But I hope that Line won't misconstrue that as a trivialization of the fundamental knowledge needed before entering a Real Analysis class!

 

[JasonRox]

Also, basic courses are pretty much the same in every university in the world, and i bet you every good mathematitian, no matter where in the world was born or formed ,has read the classical texts.

I wouldn't be so sure about that.

 

[Treadstone 71]

The Japanese know how to keep it real short. Did you know that they found a reason that american kids aren't doing better. The books are so darned thick.

I refer you to "Surely You're Joking, Mr. Feynman!" chapter "O Americano, Outra Vez!" near page 211. High test scores does not mean actual learning.

 

[Line]

I refer you to "Surely You're Joking, Mr. Feynman!" chapter "O Americano, Outra Vez!" near page 211. High test scores does not mean actual learning.

But it's more likely.

Look everyone around the world kows that the US educational system is in a serious crisis. We need to know just exactly what the other countries are doing.


Even more so it's like anything, you learn the basics first before you go into high detail.
It's the same was a street goes through all the stop signs and little intersections a highway only stops at major intersections. By making fewer stops you get further faster... effeciency. It's been said that a person with a shallow broad understanding will fair better than a person with a deep narrow one.

 

[arildno]

Look everyone around the world kows that the US educational system is in a serious crisis. We need to know just exactly what the other countries are doing.

Look to Norway!
That's the way the US education system is heading..

 

[JasonRox]

But it's more likely.
Look everyone around the world kows that the US educational system is in a serious crisis. We need to know just exactly what the other countries are doing.
Even more so it's like anything, you learn the basics first before you go into high detail.
It's the same was a street goes through all the stop signs and little intersections a highway only stops at major intersections. By making fewer stops you get further faster... effeciency. It's been said that a person with a shallow broad understanding will fair better than a person with a deep narrow one.

It sucks around here because people are just dumb and slow. People are also lazy, which is part of our society. Changing the school system won't do anything.

 

[master_coda]

It's the same was a street goes through all the stop signs and little intersections a highway only stops at major intersections. By making fewer stops you get further faster... effeciency. It's been said that a person with a shallow broad understanding will fair better than a person with a deep narrow one.

In general, a shallow broad understanding of a subject is useless for doing anything. Including learning. In fact, back when I took calculus 1 & 2 in my first year at university, students who had gained a "shallow" understanding of calculus from high school had an extremely hard time adjusting to doing real calculus; a lot of people had a hard time letting go of the fast and easy (and wrong) way they were first taught to do things.

Teaching people wrong answers just because it's faster and easier is not a good thing.


And test scores aren't a good indicator of anything. Trying to compare the education in different countries via standardized tests is like trying to test for kids who are gifted at math by testing how well they've memorized multiplication tables (which is something that I am sad to say I have seen).

 

[Robokapp]

And just what is an integral?

It's the area under x-axis of a given function.

It's the opposite of a deivative.

If you know a car is going 50 miles per hour for 3 hours, how far does it go? Well...that's easy one, but basically the distance under the x-axis is what integral does.

Let's say you have a car accelelrating at a rate of 1 km/s^2 starting from rest. How fast does it go after 10 seconds? how far did it travel?

That knid of stuff is what Integrals do. I'm guessing you scheemed trough the book, because I'm in capter 5 (Integrals) right now.

 

[Line]

In general, a shallow broad understanding of a subject is useless for doing anything. Including learning. In fact, back when I took calculus 1 & 2 in my first year at university, students who had gained a "shallow" understanding of calculus from high school had an extremely hard time adjusting to doing real calculus; a lot of people had a hard time letting go of the fast and easy (and wrong) way they were first taught to do things.
Teaching people wrong answers just because it's faster and easier is not a good thing.
And test scores aren't a good indicator of anything. Trying to compare the education in different countries via standardized tests is like trying to test for kids who are gifted at math by testing how well they've memorized multiplication tables (which is something that I am sad to say I have seen).

I'm afraid you're wrong, learning a little something about everything will take you a long way. When you start off in lementary school they teach you a little something about each of the subjects. as you progress further and further they add on. And if you want you can expand it more in college. There noway I'd teach a first grader just all hard math.

And as for Norway, do they have a worse school system?

 

[Integral]

Closed for lack of math disscussion.