What are derivatives and integrals?

[The_Z_Factor]

What are they? In my book I am studying limits and it has mentioned a few times before and in the current chapter Derivatives and Integrals, but hasnt explained them. Could anybody explain what these two things are, exactly?

 

[SiddharthM]

if your asking for a formal definition then goto www.wikipedia.com and search for derivative and separately integration.

Geometric calculus interpretation:

If f(x) is a line and can be represented by mx +b then the slope is m, but this is a line and the slope is the same throughout the real numbers. Now consider y= x^2, what is the slope? it changes at each point, the slope at any point is the derivative of the function evaluated at that point.

the integral of a real function gives you the area under the curve of the function.

 

[The_Z_Factor]

If f(x) is a line and can be represented by mx +b then the slope is m, but this is a line and the slope is the same throughout the real numbers. Now consider y= x^2, what is the slope? it changes at each point, the slope at any point is the derivative of the function evaluated at that point.

So does this mean that there can be as many derivatives as there are points?

 

[Gib Z]

So does this mean that there can be as many derivatives as there are points?

Different functions can be differentiated a different number of times.

 

[Mute]

So does this mean that there can be as many derivatives as there are points?

It means that in general the derivative of a function of x is itself a function of x. i.e., the slope of a function is different at each point on that function.

For example, the derivative of x^2 is 2x. This means that on the curve y = x^2, at the point x = 4, the slope of the curve at x = 4 (or, perhaps more precisely, the slope of the line tangent to the curve at x = 4) is 2*4 = 8. Similarly, the slope at the point x = -5 is -10.

 

[The_Z_Factor]

Ah, thanks for clearing that up for me everybody, that explains it. I think as I'm beginning to learn more about simple calculus I'm beginning to like it more. Haha, it just might turn into a hobby once I learn enough.