Thread: limit definition

Hi,

i'm just starting calculus and have been trying to figure out the derivative defined as a limit for what already seems like far too long. I've heard people say that the formal definition is hardly ever used in real world situations, but as i'm just starting out i feel i need some insight into how much time to spend on this topic at this point.

What level of understanding of the formal definition of the derivative is actually neccesary in physics and engineering? (BTW the context of my questions is that i hope to study engineering in the future, which i imagine will involve calculus based physics)

My book goes into excruciating detail to describe it, but is the reason for this simply to show me the logic, or is it because i will have to think conceptually(?) in terms of the formal definition later in maths and physics?

In other words, can i get away with just learning it superficially for now? I don't see how i can hope to understand it properly without first studying inductive logic and maybe predicate logic anyway.

Should i just move on?

Thanks in advance for any help with this.

The limit definition of the derivative really isn't used in application. It's not that hard to understand really, if you think about it a little you'll realize that the portion on top is the change in the -value, and then the (I don't know what your book uses...it could be but it's the same thing) on the bottom is the change in the -value, giving you which equals the slope of the tangent line at , or the instantaneous rate of change, which is what the value of the derivative is.

So, it's definitely good to understand, but it's really not needed in application because all the differentiation rules make taking derivatives much easier.

Originally Posted by CMR80606

Hi,

i'm just starting calculus and have been trying to figure out the derivative defined as a limit for what already seems like far too long. I've heard people say that the formal definition is hardly ever used in real world situations, but as i'm just starting out i feel i need some insight into how much time to spend on this topic at this point.

What level of understanding of the formal definition of the derivative is actually neccesary in physics and engineering? (BTW the context of my questions is that i hope to study engineering in the future, which i imagine will involve calculus based physics)

My book goes into excruciating detail to describe it, but is the reason for this simply to show me the logic, or is it because i will have to think conceptually(?) in terms of the formal definition later in maths and physics?

In other words, can i get away with just learning it superficially for now? I don't see how i can hope to understand it properly without first studying inductive logic and maybe predicate logic anyway.

Should i just move on?

Thanks in advance for any help with this.

It is actually more important to understand what the derivative is, a theoretical issue, than to simply to learn symbols -- though you will be told differently by engineering students and even some professors. They are simply wrong.

On the other hand if you are just learning calculus it is very likely that you are seeiing formal definitions that obcure what is really goiing one. It is important to understand the formal definition, but it is equally important to understand the purpose of that formal definition and the intuition behind it.

So, here is the basic idea.

1). Fix a point . A neighborhood of is a set that contains an open interval that contains . That open interval can be taken to be of the form for some . So a "neighborhood" of is a set that contains all points "sufficiently close" to .

2) To say that for some function means that is "close" to whenver is "close" to . More precisely this means that given any neighborhood B of there is a neighborhood A of such that whenever . This gets translated to: Given there is a such that whenever

3) The idea of a derivative is to approxiimate, as closely as possible, a rather arbitgary function by a linear function. More precisely one want to approximate perturbations of a function near a fixed point with a linear function. In the case of a real valued function of a single variable a linear function is just multiplication by some constant, call it A. So near a fixed we are looking for a number A such that . We want the "approximately" to mean that the error is much smaller than . This translates to . Geometrically the number A is the slope of a line tangent to the graph of at

Why do we want to approximate the function by a straight line ? Because we know a lot about straight liines, and because it turns out that this definition of a "derivative' allows us to use that knowledge of straight lines to learn a lot about more general functions. It is a way to study the complicated by studying a siimple approximation.

So the reason the error need not be specified is that if the limit exists, the limit definition proves that x can always be held close enough to x0 such that the outputs of f(x) will fall within any prescribed error bounds.

Is it reasonable to say that if the purpose of the derivative is to approximate the rate of change of a function, then the purpose of the limit definition is to remove as much of the assumption in that approximation as possible? so is the limit definition true by induction?
thanks

Originally Posted by CMR80606

So the reason the error need not be specified is that if the limit exists, the limit definition proves that x can always be held close enough to x0 such that the outputs of f(x) will fall within any prescribed error bounds.

more or less.

Originally Posted by CMR80606

Is it reasonable to say that if the purpose of the derivative is to approximate the rate of change of a function, then the purpose of the limit definition is to remove as much of the assumption in that approximation as possible? so is the limit definition true by induction?
thanks

No.

The purpose of any mathematical definition is to make the associated concept precise.

Induction has nothing to do with limits.

Definitions are just that, definitions. There is no question of "truth" and definitions do not require proof.

Originally Posted by DrRocket

Definitions are just that, definitions. There is no question of "truth" and definitions do not require proof.

ok, apparently i need to learn how to read . I think difficulty was that i was labouring under the faulty assumption that maths should be empirical. ho hum.

Originally Posted by cmetc

In other words, can i get away with just learning it superficially for now?

That would be a shame.

It can take a very long time for the idea of a ratio of rates of approach coming to a limit to sink in, for the significance of a linear approximation to make itself clear, for the implications of the derivative to become apparent. And they would ease and organize your comprehension of many otherwise arbitrary seeming and endlessly complicated tricks of the engineering trade.