Help with Derivatives and L'Hospital's Rule - K.

[_DJ_british_?]

Oh hai, didn't know you were there!

I was just wonderin', when the derivative of a function is equal to zero, is the the derivative positive, negative or neither? My teacher says "Oh, it's like you want it to be", which seems to be a pretty dumb response (for a otherwise smart professor, but anyway). For exemple, if we take x^2, he seems to say that the decreasing interval can be ]-infinity,0] AND/OR ]-infinity,0[ (that's our choice!). Fact is, he doesn't give a damn in tests if responses like this differs (I've checked with friends). Is that real? Just wanted some clarifying!

P.S. Oh and another question! In an exam, I used L'Hospital's Rule to solve an 0/0 limit, but my teacher give me wrong because we didn't cover that and I wasn't suppose to use this rule yet. Is that ok?

P.P.S Sorry for the grammar, I'm french-canadian.

Thanks in advance!

K.

 

[NoMoreExams]

What makes you think it has to be one or the other? A real number can either be positive, negative or neither (i.e. 0). As far as your other question, yeah he doesn't have to mark it correct especially if he specified that you should use the methods he taught. Now if you can prove L'Hopital's on your own, then yes you should be able to use it. The place where the derivative is 0 is where you can say it vanishes which among other things that you have an extrema point.

 

[_DJ_british_?]

What makes you think it has to be one or the other? A real number can either be positive, negative or neither (i.e. 0).

Yeah, I know that, but I'm talking about rates of change. Can a derivative be positive and negative at the same time (the magnitude of -1 is equal to one, but I ain't taling about absolute value. Or equal to zero at the same time? Sorry if I don't understand, I'm just doin' differential calculus so I don't know much much about maths, but to me, it seems kinda weird!

As far as your other question, yeah he doesn't have to mark it correct especially if he specified that you should use the methods he taught. Now if you can prove L'Hopital's on your own, then yes you should be able to use it. The place where the derivative is 0 is where you can say it vanishes which among other things that you have an extrema point.

Yeah, you're right about that, it's just that L'Hospital's Rule is pretty useful and I didn't think mind teacher would mind me using it.

 

[Fredrik]

The function f defined by f(x)=x² for all real numbers x is neither increasing nor decreasing at 0. I have no idea what he would say that it's both.

 

[adriank]

A positive number is defined to be anything greater than zero. A negative number is defined to be anything less than zero. Thus, a number is either positive, negative, or zero; in particular, it can't be both positive and negative.

My analysis text defines a function f to be increasing on a set S if given x and y in S, x < y implies f(x) ≤ f(y), and decreasing if x < y implies f(x) ≥ f(y); this makes no reference to the derivative, although if a function is differentiable then it is decreasing if and only if the derivative is everywhere negative or zero. Thus, f(x) = x² is decreasing on (∞, 0].

A consequence of this definition (I'm looking at you, Fredrik) is that it doesn't make much sense to talk about whether a function is increasing or decreasing at a single point. (If you apply the definition strictly, then any function is both increasing and decreasing on a set containing a single point.)

For your limit question, what was the limit to be evaluated? Do you know how to evaluate it without using l'Hôpital's rule?

 

[Fredrik]

My analysis text defines a function f to be increasing on a set S if given x and y in S, x < y implies f(x) ≤ f(y), and decreasing if x < y implies f(x) ≥ f(y);
...
A consequence of this definition (I'm looking at you, Fredrik) is that it doesn't make much sense to talk about whether a function is increasing or decreasing at a single point. (If you apply the definition strictly, then any function is both increasing and decreasing on a set containing a single point.)

Good post, and good point.

 

[NoMoreExams]

Looks like a lot has been said since I last posted and most of it has answered your confusion I hope. I will say this about your question whether your teacher should've minded you using L'Hopital's. Unless you guys studied it and he said you can use it OR you can prove it on your own, I can definitely see why he would mark it wrong or at best give you partial credit. There's probably a reason you learn things in order and you probably had to apply the limit definition of derivatives to evaluate them at first. That stuff doesn't just disappear after Calc. 1. If I were you, I'd make sure you understand the material being taught and use the more advanced material that you know to check your work. Just my .02

 

[Tac-Tics]

Yeah, I know that, but I'm talking about rates of change.

A derivative of a function at a particular point is a real number. Just a single, boring real number. It can be positive. It can be negative. It can be zero. It can't really be infinity (in such cases, the derivative is not actually defined at that point).

In fact, when a function's derivative is zero, it has special physical significance. If f is a function, and for some point c, f'(c) = 0, we call c a critical point. If a function has any maximums or minimums, they will always appear at critical points.

 

[adriank]

Or at boundary points, or at points where the function is not differentiable. :)