Question about limits and derivatives.

[MarcL]

Hi to all of Physics forums community :),


I'm not sure whether or not I am in the right section. If not, I apologize for my mistake. Also, I am new to Calculus (I am in Cegep --> Quebec schooling system...) and I am taking Calculus 1 single variable. My teacher just started our class by giving us a bunch of formula but really, I don't understand the logic behind it. I found this forum by trying to find an explanation that I would hopefully understand.

First, I asked my teacher about limits and their role. I got a totally evasive answer. I know how to solve certain problems using limits. However, (and I am probably repeating myself) I feel like a robot just solving a problem... I was told that a function can be discontinuous, continuous or does not exist at a certain point. My question for this is, how can it be discontinuous but still exist and how can it just not exist ( I would assume if the denominator is 0?)? Also, what are limits in general?

For Derivatives, when asked, my teacher said to take it as a "game". I honestly do not know what he meant by that ( and I'm not looking to an answer to the statement above). Though, I thought that the derivative was simply the tangent of a function at a certain point. However, isn't the definition lim _h-->0f(x+h)-f(x) /h. So, does it only stands for when h approaches 0? and I am sorry for sounding stupid but what does the h stands for..?

I am sorry if my questions are all over the place or you guys don't understand what I am asking. I am just hopelessly trying to understand what I am doing. I find math wonderful when you understand but when you don't... Its just another like trying to understand a language you don't speak.

 

[mathman]

I suggest you look up "calculus basics" using google or bing.

 

[Bacle]

I agree with the previous suggestion. Still, hope this will help:

h is a change of values in the domain of the function. The idea is that the limit may

or may not exist as h→0 ; if the limit does exist, then we say f is differentiable at x,
and we call its derivative at x f'(x).

The derivative is an

effort to "linearize" the change of the function f locally; by linearizing we mean

describing the local change of the values of the function by a linear (affine, actually)
function f'(x). Linear approximations are useful, because linear functions are easier
to work with than non-linear ones in most respects.

If your function f satisfies certain conditions, this linear approximation exists. For
example, the local change of the function x² is described by the linear
function f'(x)=2x , so, in a small region near a point x, the change in the values of
x² can be approximated to any level of precision, in a precise ε -δ, sense.

Hope this helps you get started; let us know if you have a followup.

 

[lurflurf]

Limits are a way of guessing the value of a function f(x) based on the values of f(x+h) for various small h. The use of doing this is precisely because a function can be discontinuous, continuous or does not exist at a certain point. When a function is discontinuous we an find the function and the limit but they do not agree. When a function is discontinuous we an find the function and the limit they do agree. When a function does not exist at a certain point the limit gives us information near that point. A problem some have when new to calculus is that they do not have much imagination for functions and many calculus books do not address this. If you say hey new to calculus person name a few function they will say 2x or sin(x). The use of limits is not made clear by such simple functions. You might imagine various situations of each type. The definition lim h-->0 (f(x+h)-f(x) )/h is a good example that is often not understood. We cannot evaluate this for h=0 so we figure out what it would be if we could. Common examples lose this. we might let f(x)=x^2 then (f(x+h)-f(x) )/h=(2x h+h^2)/h=2x+h and say we cannot take h=0, but what would happen if h gets closer and closer to 0, we get 2x. Students think this is obvious, stupid, or confusing; they are not impressed.

 

[pwsnafu]

I'm not sure whether or not I am in the right section. If not, I apologize for my mistake. Also, I am new to Calculus (I am in Cegep --> Quebec schooling system...) and I am taking Calculus 1 single variable.

It's the right place.

My teacher just started our class by giving us a bunch of formula but really, I don't understand the logic behind it. I found this forum by trying to find an explanation that I would hopefully understand.

Unfortunately, this is very common practice. Let's compare with trigonometry. Trig was discovered by the ancient Greeks, and they pretty much worked the majority of it out during their studies. What you learned in high school is just following their footsteps.

But they didn't understand calc. They had heuristics sure. But they didn't have a universal theory. It took Newton's era for that, and even he only had ad hoc methods. It took until Cauchy and his contemporaries to lock down. Calculus is easy to use but very hard to understand. This is why your teacher is teaching it this way.

how can it be discontinuous but still exist

The function $\chi : \mathbb{R}\rightarrow\mathbb{R}$ defined as $\chi(x) = 1$ when x is rational and $\chi(x)=0$ when x is irrational is defined for all real numbers but is discontinuous everywhere.

and how can it just not exist ( I would assume if the denominator is 0?)?

An example of a function which doesn't have a limit at a point but defined there is simply f(0)=1/2 and f(x)=0 for all x < 0 and f(x)=1 for all x > 0. The left and right limits are not the same.

An example of a function which is not defined at a point but none the less has a limit there is $\frac{\sin(x)}{x}$ when x = 0.

Also, what are limits in general?

Do you want a mathematical definition or a heuristic?

I thought that the derivative was simply the tangent of a function at a certain point.

This is good enough for one variable case. To be technical it's the slope of the tangent.

PS: There are also functions which are continuous everywhere differentiable nowhere. In fact most continuous functions are not differentiable, but you will not see them in high school. Also continuity is a strong property, and is the most critical object in topology. But again you will be unlikely to appreciate this until university/college.

 

[paulfr]

PS: There are also functions which are continuous everywhere differentiable nowhere. In fact most continuous functions are not differentiable, but you will not see them in high school.
===============
I am curious
Could you elaborate on this with some examples ?
Thanks

 

[pwsnafu]

I am curious
Could you elaborate on this with some examples ?
Thanks

It's easy to find a function which is continuous but not differentiable a point, namely, the absolute value. Similarly, if you have points x₁, x₂... the function
$\sum_{k=1}^\infty \frac{|x-x_k|}{3^k}$
is not differentiable at a countable number of points.

The first published example of a function which is not differentiable anywhere is the http://en.wikipedia.org/wiki/Weierstrass_function" . For

1. An odd natural number, a,
2. 0 < b < 1,
3. and $ab > 1 + 3\pi / 2,$

the function is
$W(x) = \sum_{k=0}^\infty b^k \cos(a^k \pi x).$
The proof is elementary and only about a page, but it is terse. If people want, I'll post the full proof tomorrow.

If you want a simple example that a high school student will understand the van der Waerden's function is an elegant example. Define f₀(x) to be the distance from x to the nearest integer. Clearly this is periodic, zero on the integers and equal to a half for all half-integers. The graph looks like a sawtooth.

Define
$f(x) = \sum_{k=0}^\infty \frac{f_0 (10^k x)}{10^k}.$
For a given x, we will construct a sequence for h so that the difference quotient diverges.
Because f is periodic, assume 0 ≤ x < 1.
Using the decimal expansion
$x = 0.x_1 x_2 x_3 \ldots$
we define
$h_n = -10^{-n}$ when x_n is equal to 4 or 9; and
$h_n = 10^{-n}$ otherwise.

Example: 0.1245...
h₁ = 0.1
h₂ = 0.01
h₃ = -0.001
h₄ = 0.0001
and so on.

Our difference quotient is
$\frac{1}{h_n} \sum_{k=0}^\infty \frac{f_0 (10^k x + 10^k h_n) - f_0(10^k x)}{10^k}$
Using our example:
when n = 1, the first term of the sum $f_0 ( x + h_1) - f_0 (x) = f_0 (0.2245\ldots)-f_0(0.1245\ldots)=0.1$, whence dividing through we have +1. The remaining terms vanish due to periodicity of $f_0$.

When n=2, only two terms remain giving +1+1 = +2.
When n=3, only three terms remain giving -1-1-1 = -3

and so on. QED.

You could turn this into a worksheet to give an enrichment class.

 

[MarcL]

First, thank you all for your answers, they are very appreciated (and helpful I must say) However I don't know if the last post was addressed to my question because if it was, I must say I'm stupid because I did not understand :P.

Anyway, thank you for you're answers!
And pwsnafu, I was looking for a mathematical answers. Or just how it came to be found. I mean if I look at chemistry, it does make sense how plank discovered quantum theory with his blackbody experiment. But when it comes to math, it is harder to just do an experiment and demonstrate the result. I don't know if I actually make sense by saying this and if it doesn't I'll try to re-phrase it :)

 

[pwsnafu]

However I don't know if the last post was addressed to my question because if it was, I must say I'm stupid because I did not understand :P.

If you are talking about my post #7, it was in reply to paulfr's post, which in turn was a reply to the postscript in my post #5: "there exists a function which is continuous everywhere differentiable nowhere".

You see when I was in high school, the teachers would say this is the definition of continuity and this is differentiability. But the latter was the important bit. Continuity never came up. Sure there would be a problem on the exam. "Show this function is continuous". But that function happened to differentiable too! Why should I care about continuity? It took some time at uni before I was able to appreciate that continuity is actually useful.

Oh, if you want I'll walk you through post #7, bit by bit.

And pwsnafu, I was looking for a mathematical answers. Or just how it came to be found.

The problem is, you wrote "what is a limit in general". The most general concept of a limit requires topology, and I doubt you have a necessary mathematical background for that.

This is how I was taught: start with the concept of http://en.wikipedia.org/wiki/Limit_of_a_sequence" .

For now have a quick read, confuse yourself (yes its a good thing), then come back and ask questions.

 

[MarcL]

I will go confuse myself after my last minute studying :) haha and I will certainly ask my question. And before you walk me through your post I believe I should gain the needed knowledge. If I had it, I would understand :)