Can it be argued that derivatives should be undefined?

[CuriousBanker]

I understand derivatives and I am not trying to be like a stickler or anything, but before manipulating the equation to arrive at a form where we can find a real answer for a derivative, we are left with [f(x+h)-f(x)]/h (where h is delta x I guess as most people write it). Before evaluating it further, if we are taking the limit as h goes to zero...then wouldn't the equation, which once manipulated gives us a reasonable answer, be equal to (0-0)/0 for all derivatives? Why is it that we are allowed to ignore this form and use the manipulated form?

I hope my question was clear, thanks in advance.

 

[Borek]

Just because the division is undefined doesn't mean the limit is undefined. These are two different things.

 

[HallsofIvy]

Have you considered looking at the graph of such a "difference quotient"? For example, if [itex]f(x)= x^2[/itex] then [itex]\frac{f(2+ h)- f(2)}{h}= \frac{(2+h)^2- 4}{h}= \frac{h^2+ 4h+ 4- 4}{h}= \frac{h^2+ 4h}{h}[/itex].

Now, as long as h is NOT 0, we can write that as [tex]\frac{h(h+ 4)}{h}= h+4[/tex] and its graph would be a straight line, with slope 1. The actual graph is that straight line with the point (0, 4) removed. That is, the value of the difference quotient is not defined at h= 0 but the limit, as h goes to 0, is 4. That is why we have to define the derivative in terms of the limit.o

(I have no doubt that you are good at manipulating the formulas of Calculus, but until you understand exactly why the limit is so important in Calculus, you do not really "understand Calculus". Actually, most students don't really "understand Calculus" until they have taken "Mathematical Analysis".)

 

[CuriousBanker]

I know, but in the power rule, we assume h goes to zero when we have 2x + h as our answer...so if we are assuming it goes to zero then, why not assume it goes to zero when in the denominator?

 

[CuriousBanker]

Yeah, I get it but doesn't that mean that the derivative must have an infinitesimally small error at all times?

 

[pwsnafu]

I know, but in the power rule, we assume h goes to zero when we have 2x + h as our answer...so if we are assuming it goes to zero then, why not assume it goes to zero when in the denominator?

If you look at the numerator separately from the denominator, then you are right. The point is that the numerator goes to zero faster than the denominator, so the ratio goes to a finite number.

Yeah, I get it but doesn't that mean that the derivative must have an infinitesimally small error at all times?

The derivative has no error.
Edit: Maybe I'm misunderstanding. Error respect to what?

 

[WannabeNewton]

Look at the ##\epsilon - \delta## definition of a limit in ##\mathbb{R}##. This will surely get rid of any misconceptions you have about the limit, especially the part about the "infinitesimally small error".

 

[Mandelbroth]

Yeah, I get it but doesn't that mean that the derivative must have an infinitesimally small error at all times?

No. Especially not if you use a more complicated definition of a derivative, but that's a bit advanced for right now.

The idea of a real limit is this: Say we have a number ##\epsilon>0##. Then, we can say that ##\displaystyle \lim_{x\rightarrow\alpha}f(x) = \mathfrak{L}## if and only if, for all ##\epsilon>0##, there exists another number ##\delta>0## such that, for all x, the inequality ##0<\left|x-\alpha\right|<\delta## implies the inequality ##\left|f(x)-\mathfrak{L}\right|<\epsilon##.

What the heck does this mean, you ask? It means that we can make ##\epsilon## as close to 0 as we want. In fact, it means we can make ##\epsilon## ARBITRARILY small. We might even pretend that we can make it SO small that we can imagine that there is no real number between ##\epsilon## and 0. In simpler terms, this basically means that f approaches (and gets infinitesimally close to) ##\mathfrak{L}## as x approaches ##\alpha##. This does NOT mean that ##\displaystyle f(\alpha)=\lim_{x\rightarrow\alpha}f(x)##.

For a rather inflated and sugar-driven example, consider the function

$$f:\mathbb{R}\rightarrow\mathbb{R}\cup\left\{TOOTSIEPOP\right\} \\ f(x) = \left\{\begin{matrix} 2, & x\neq 2 \\ TOOTSIEPOP, & x=2 \end{matrix}\right. .$$

For all values of x, other than 2, f(x)=2. In fact, if we get infinitesimally close to 2, we are still approaching a value of 2 for f(x) at x=2. However, as stated, f(2)=TOOTSIEPOP. Clearly, we aren't going to approach TOOTSIEPOP in the real numbers. The limit as x approaches 2 is, thus, 2.

Edit:
WannabeNewton post above basically suggests the same. For more info, you might start here.

 

[CuriousBanker]

Yeah, I am familiar with epsilon-delta. I see now...I was just confused how we could assume delta x = 0, but when it was in fraction form we could not (otherwise we would have it undefined). I get it though. Thanks for the examples, all

 

[lavinia]

.then wouldn't the equation, which once manipulated gives us a reasonable answer, be equal to (0-0)/0 for all derivatives? Why is it that we are allowed to ignore this form and use the manipulated form?

.

Think about this example. the ratio 2x/x is 2 for any x except 0. What is its limit as x approaches zero?

 

[lurflurf]

In a derivative we have [f(x+h)-f(x)]/h, but h is not zero so there is no problem. Assume f(x) is a polynomial. We may define the derivatives of f by

$$\mathop{f}(x+h) = \sum_{k=0}^\infty \frac{h^k}{k!} {\mathop{f}}^{(k)}(x)$$

We do not worry as this does not remind us of dividing by zero.

$$\lim_{h \rightarrow0} \frac{\mathop{f}(x+h)-\mathop{f}(x))}{h}$$

Reminds us of dividing by zero, but does not actually involve it. The derivative is what it is. For example to find the derivative of x^2 we have

$$\lim_{h \rightarrow 0} \frac{2 \, x \, h}{h}$$

Since h is not 0 we might just as well have
$$\lim_{h \rightarrow 0} 2x $$

$$\frac{2 \, x \, h}{h}=2x $$

When h is not zero, 2x does not remind us of dividing by zero.

 

[CuriousBanker]

Ah, I understand now. The one thing though, is if you have 2 + h, shouldn't the derivative also be 2+h? The reason it is always shown to be 2 as that as h goes to zero, 2 + 0 = 2. But if h is not zero, should it not be 2+dx? If h is never zero how come 2 + h = 2?

 

[HallsofIvy]

The problem appears to be that you still do not understand the concept of a limit.

 

[WannabeNewton]

The whole point of the limit is that you take ever decreasingly small open balls around the limit, when it exists, so that the values converging to the limit become arbitrarily close to the limit via containment in the open balls.

 

[CuriousBanker]

I see now...I don't know what I was missing before

 

[CuriousBanker]

Thanks all

 

[micromass]

Why is ##\lim_{h\rightarrow 0} 2+h## evaluated by simply substituting ##h=0## into the function, and why is ##\lim_{h\rightarrow 0} \frac{h}{h}## not evaluated like that?

Well, the answer is the notion of a continuous function. Basically, if ##f:D\rightarrow \mathbb{R}## is a continuous function that is defined in a point ##c## (this means that ##c\in D##), then [tex]\lim_{h\rightarrow c} f(h) = f(c)[/tex]is true. This is a theorem that one can prove (or you can accept it as definition of continuity). For example, consider ##f:\mathbb{R}\rightarrow \mathbb{R}:h\rightarrow 2+h##. This function is certainly defined in ##0## and is continuous there. So we can write [tex]\lim_{h\rightarrow 0} 2+h = \lim_{h\rightarrow 0} f(h) = f(0) = 2+0 = 2[/tex]

Now, why does this not work for [tex]\lim_{h\rightarrow 0} \frac{h}{h}[/tex] Well, the simple reason is that the function ##f(h) = h/h## is not defined at ##0##. So the above does not apply. In other words, we can write ##f:\mathbb{R}\setminus \{0\}\rightarrow \mathbb{R}:h\rightarrow h/h## and ##0## is not in the domain.

How do we handle this situation then? Well, we have the following theorem:

Let ##D## be a set such that ##c\notin D##. Given two functions ##f:D\rightarrow \mathbb{R}## and ##g:D\cup \{c\}\rightarrow \mathbb{R}##. If for all ##x\in D## holds that ##f(x) = g(x)##, then ##\lim_{h\rightarrow c} f(h)=\lim_{h\rightarrow c} g(h)##.

So this theorem tells us that two functions have the same limit if they agree everywhere except possibly in ##c##.

Now, let ##f:\mathbb{R}\setminus \{0\}\rightarrow \mathbb{R}:h\rightarrow h/h## and ##g:\mathbb{R}\rightarrow \mathbb{R}:h\rightarrow 1##. Then ##f(h)= g(h)## for all ##h\in \mathbb{R}\setminus \{0\}##, so the theorem implies that the limits agree. So [tex]\lim_{h\rightarrow 0} f(h) = \lim_{h\rightarrow 0} g(h)[/tex] Thus [tex]\lim_{h\rightarrow 0}\frac{h}{h} = \lim_{h\rightarrow 0}1[/tex] Now, the right-hand side is a continuous function and it is defined at ##0##, so we can apply the previous result: [tex]\lim_{h\rightarrow 0} 1 = \lim_{h\rightarrow 0} g(h) = g(0) = 1[/tex] This is what actually is going on when evaluating limits.

 

[CuriousBanker]

I have another unrelated question but don't want to start a new thread and clog up the forums, so answer it if you wish and I will be happy.

For a definite integral I have read that ∫a-b f(x)dx = F(b)-F(a). Easy enough to plug numbers in and compute, but there was no explanation as to why that is the formuia. Why is the area under a curve equal to the antiderivative at the end point minus the origin? WHat about all the points in between?

 

[CuriousBanker]

Also how long should it take me to teach myself calc 1-3 (not analysis) if I am studying 15 hours a week? I know everybody is different, but so far I have put in about 30 hours and I am freaking out that I don't fully get it yet, but so far I have learned limits, derivative chain rule and power rule, antiderivatives, continuous functions, derivatives of common functions. Seems like by now I should fully get integrals but maybe I am just trying to rush things

 

[pwsnafu]

I have another unrelated question but don't want to start a new thread and clog up the forums, so answer it if you wish and I will be happy.

For a definite integral I have read that ∫a-b f(x)dx = F(b)-F(a). Easy enough to plug numbers in and compute, but there was no explanation as to why that is the formuia. Why is the area under a curve equal to the antiderivative at the end point minus the origin? WHat about all the points in between?

It's called the Fundamental Theorem of Calculus.

 

[lurflurf]

Also how long should it take me to teach myself calc 1-3 (not analysis) if I am studying 15 hours a week?

Well 15 hours a week is about the time a person in a class would spend and it would take a year for calculus 1-3. Self study might be slightly faster or slower depending on the person. Knowing calculus 1-3 means different things different places depending on differences in difficulty, amount of material, number of sessions, rigor, topics covered, applications, computational skill, and other things that very class to class.

To put the pacing in perspective a class would spend 1-3 weeks on many of these topics
Calculus 1
01-Introduction
02-Limits
03-Elementary functions
04-Derivatives
05-Appliations of derivatives
06-Integrals
07-Applications of Integrals
08-More about Elementary functions
09-Techniques of integration
10-Assorted loose ends
11-Seqences and series
12-Convergence
13-Coordinate systems
14-Analytic geometry
15-Differential equations
16-Vector, matrices, determinants, and complex Numbers
17-Derivatives in several variables
18-Applications of derivatives in several variables
19-Integrals in several variables
20-Application of integrals in several variables
21-introduction to vector calculus
22-differential vector calculus
23-integral vector calculus
24-More on Coordinate systems
25-Vector Calculus in Space
26-Vector Calculus on Surfaces
27-Transport
28-Differential forms/Stokes theorem
29-Applications of vector Calculus
30-Topics in vector Calculus

A danger in self study is in going much faster than a class (which is good) you might not learn as
thoroughly as the class (which is bad). Watch out for this.

 

[lurflurf]

The Fundamental Theorem of Calculus can be understood a few ways, here is one.

An integrable function f can be approximated by a constant C and the constant can be given by a value of the function.
∫f(x)dx ~f(c) h if x~c and h~0

A difference of a differentiable function F can be approximated by a constant C and the constant can be given by a value of the derivative.
F(x+h)-F(x)~h F'(x) if x~c and h~0

If F'=f we can combine the above
∫f(x)dx~F(x+h)-F(x) if x~c and h~0

This hold only for small intervals, but many small intervals can be combined
∫f(x)dx~Ʃ(F(x+h)-F(x))

The sum collapses and we get the theorem
∫f(x)dx~F(b)-F(a)

 

[Mandelbroth]

Also how long should it take me to teach myself calc 1-3 (not analysis) if I am studying 15 hours a week? I know everybody is different, but so far I have put in about 30 hours and I am freaking out that I don't fully get it yet, but so far I have learned limits, derivative chain rule and power rule, antiderivatives, continuous functions, derivatives of common functions. Seems like by now I should fully get integrals but maybe I am just trying to rush things

It may seem like a weird idea, but read Wikipedia. The math pages are generally reliable, and bouncing around different areas to understand all the concepts in depth can really help you learn and make connections before you start fully applying the material. Start with something math related that you enjoy, and then open somewhere in the neighborhood of 15 to 20 new tabs.

To give you an idea, the first integral I truly tried to evaluate on my own was ##\displaystyle \int \frac{e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}}{\sigma\sqrt{2\pi}}\, dx##, an integral that has notable use in statistics. After a somewhat pathetic effort, I finally got to my answer of ##\displaystyle \frac{1}{\sqrt{\pi}}\int_{0}^{\frac{x-\mu}{\sigma\sqrt{2}}}e^{-\xi^2} \, d\xi - C##. At that point, I was fairly certain that I understood most of the formulas.

So, anecdotes and stories aside, my suggestion is to read Wikipedia pages and, every once in a while, do stupid stuff with what you learn. That's how I learned math, so hopefully it works for you too.

 

[CuriousBanker]

Thanks for all the tips and help!