# What's this symbol stand for?

1. Dec 13, 2005

### 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?

2. Dec 13, 2005

### 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)

3. Dec 13, 2005

### Line

And just what is an integral?

4. Dec 13, 2005

### Integral

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

5. Dec 13, 2005

### Tide

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

6. Dec 13, 2005

### El Hombre Invisible

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 travelled ($$\Delta x$$) 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:

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

Last edited: Dec 13, 2005
7. Dec 13, 2005

### Line

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

8. Dec 13, 2005

### HallsofIvy

As opposed to short, simple, wrong answers?

9. Dec 13, 2005

### HallsofIvy

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

10. Dec 13, 2005

### 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.

11. Dec 13, 2005

### JasonRox

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

Personally, I always thought Americans gave short lazy answers.

12. Dec 13, 2005

### 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 extremly 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.

13. Dec 13, 2005

### Math Is Hard

Staff Emeritus
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.

Last edited: Dec 13, 2005
14. Dec 13, 2005

### cronxeh

Actually thats 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 :surprised expression on your face

15. Dec 13, 2005

### -Job-

Most of the time you can find alot 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.

16. Dec 14, 2005

### 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.

17. Dec 14, 2005

### Math Is Hard

Staff Emeritus
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!

18. Dec 14, 2005

### JasonRox

I wouldn't be so sure about that.

19. Dec 14, 2005

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.

20. Dec 14, 2005

### Line

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.

Last edited: Dec 14, 2005
21. Dec 14, 2005

### arildno

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

22. Dec 14, 2005

### JasonRox

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.

23. Dec 14, 2005

### master_coda

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).

24. Dec 14, 2005

### Robokapp

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.

25. Dec 16, 2005

### Line

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?