Linearization of a function error viewed with differentials

In summary: Some of my comments here may be somewhat subtle, but the following example may be more clear.The Weierstrass function, no-where differentiable, is continuous and satisfies the intermediate value property, but is not monotone. However, it is not differentiable at any point, so the intermediate value property does not imply the existence of a derivative, like it does for a function that is differentiable at least at one point of an open interval. To see a less subtle example, take a smooth function like y=x^3 and erase the derivative at x=0, turning it into a sharp corner, but leave
  • #1
mcastillo356
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TL;DR Summary
Can't assume a premise of the reasoning
Hi, PF, want to know how can I go from a certain error formula for linearization I understand, to another I do not

Error formula for linearization I understand:

If ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x##, then there exists some point ##s## between ##a## and ##x## such that the error ##E(x)=f(x)-L(x)## in the linear approximation ##f(x)\approx{L(x)=f(a)+f'(a)(x-a)}## satisfies

##E(x)=\dfrac{f''(s)}{2}(x-a)^2##

(...)

Quote I don't understand:

The error in the linearization of ##f(x)## about ##x=a## can be interpreted in terms of differentials (...) as follows: if ##\Delta x=dx=x-a##, then the change in ##f(x)## as we pass from ##x=a## to ##x=a+\Delta x## is ##f(a+\Delta x)-f(a)=\Delta y##, and the corresponding change in the linearization ##L(x)## is ##f'(a)(x-a)=f'(a)dx##, which is just the value at ##x=a## of the differential ##dy=f'(x)dx##. Thus,

##E(x)=\Delta y-dy##

The error ##E(x)## is small compared with ##\Delta x## as ##\Delta x## approaches 0, as seen in Figure.

Attempt to understand ##E(x)=\Delta y-dy##:

In any approximation, the error is defined by

error=true value-approximate value

If the linearization of ##f## about ##a## is used to approximate ##f(x)## near ##x=a##, that is,

##f(x)\approx{L(x)=f(a)+f'(a)(x-a)}##

then the error ##E(x)## in this approximation is

##E(x)=f(x)-L(x)=f(x)-f(a)-f'(a)(x-a)##

It is the vertical distance at ##x## between the graph of ##f## and the tangent line to that graph at ##x=a##, as shown in Figure. Observe that if ##x## is "near" ##a##, then ##E(x)## is small compared to the horizontal distance between ##x## and ##a##

##\displaystyle\lim_{\Delta x \to{0}}{\dfrac{\Delta y -dy}{\Delta x}}=\displaystyle\lim_{\Delta x \to{0}}{\left({\dfrac{\Delta y}{\Delta x}-\dfrac{dy}{dx}}\right)}=\dfrac{dy}{dx}-\dfrac{dy}{dx}=0##

Well, actually is a fake attempt: either of the limits tend to zero when ##\Delta x\rightarrow{0}##, but I'm confused by the premise of reasoning:
##\Delta x=dx=x-a##. Is there any explanation for a dummy like me? I fear the answer might need non standard analysis. But the premise sounds like "if apple=pear=apple".

0.jpg
 
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  • #2
I am not sure I understand the question. That being said I believe your equivalence follows rom the Mean Value Therorem. ( There will always be a point s within the interval where the definitions coincide...is that the question?)
 
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  • #3
hutchphd said:
I am not sure I understand the question. That being said I believe your equivalence follows rom the Mean Value Therorem. ( There will always be a point s within the interval where the definitions coincide...is that the question?)
@hutchphd, I've intentionally put a lightbulb to your response; now, the fact is that past year I was an undergraduate in Physics. I chose UNED
https://en.wikipedia.org/wiki/National_University_of_Distance_Education
because it offered me the possibility of making my desire to learn compatible with the lack of time; to the point, and out of the pot: I want a good introductory book to non-standard analysis.
Sorry, that is the question I should have asked.
I prefer English, I manage better.
Love
 
  • #4
mcastillo356 said:
Summary:: Can't assume a premise of the reasoning

If exists for all in an interval containing and , then there exists some point between and such that the error in the linear approximation satisfies
Not quite true, you must also assume that f'' is continuous.
 
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  • #5
Svein said:
Not quite true, you must also assume that f'' is continuous.
@Svein, my personal opinion is that if ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x#, shoudn't be continuous in that interval, understanding it's a linearization?
 
  • #6
Try this example: Let f be given as [itex] f(x)=-x^{2}(x<0);x^{2}(x\geq 0) [/itex]. Then [itex]f''(x) [/itex] is discontinuous at x=0.
 
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  • #7
mcastillo356 said:
@Svein, my personal opinion is that if ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x#, shoudn't be continuous in that interval, understanding it's a linearization?
That is equivalent to saying that all derivatives are continuous.
 
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  • #8
hutchphd said:
I am not sure I understand the question. That being said I believe your equivalence follows rom the Mean Value Therorem. ( There will always be a point s within the interval where the definitions coincide...is that the question?)
@hutchphd, sorry, rom means right or maybe?

Svein said:
Not quite true, you must also assume that f'' is continuous.
@Svein, I've been wondering...Well, I'm talking nonsense for sure, but this is what I've done: take ##y=\dfrac{1}{x}##, which is discontinous at ##x=0##; compute the first and the second integral with an online calculator, and now I think: ##g=x(\ln{(|x|-1)}## is not linearizable?. It is. But the second derivative is not continuous in ##\Bbb{R}##. I can't tell I must assume it. I apologize in advance.

PeroK said:
That is equivalent to saying that all derivatives are continuous.
Wise remark. And true. I mean I'm stucked.

Love
 
  • #9
mcastillo356 said:
@hutchphd, sorry, rom means right or maybe?

My apologies its just a typo: from
 
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  • #10
mcastillo356 said:
compute the first and the second integral with an online calculator, and now I think: is not linearizable?. It is.
No. [itex] \ln(\vert x\vert )[/itex] does not exist at x=0 (in popular terms: it goes to -∞).
1648732815599.png
 
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  • #11
Could it be something like this?
Svein said:
Not quite true, you must also assume that f'' is continuous.
Yes.
To get to ##E(x)=\dfrac{f''(s)}{2}(x-a)^2##, which is an error formula for linearization if we know bounds for the second derivative of ##f##, it is needed to apply twice the Mean Value Theorem: "Suppose that the function ##f## is continuous on the closed, finite interval ##[a,b]## and it is differentiable on the open interval ##(a,b)##...Is this the right way?
 
  • #12
It seems the error term in post #1 is valid if only the second derivative exists, even if not continuous. See Courant, vol 1, p. 324, footnote, for example. This extends (and uses) the mean value theorem, where only the existence of the derivative is required. The continuity of the second derivative is apparently used to deduce the more precise integral form of the error term.

One consequence of this interesting property of the derivative is that it satisfies the intermediate value property even if not continuous. I.e. if the derivative of a function is negative at one point of an interval, and positive later, it implies that the function is not monotone, but changes direction, and a function that changes direction between two points must have a local extremum between them, hence the derivative, if it exists, must be zero there.

We often think intuitively of the intermediate value property as equivalent to continuity, but technically it is slightly weaker. This distinction is sometimes blurred in some less precise discussions, and in fact, continuity is (incorrectly to a mathematician) "defined" by this property on page 6 of the prelilminary section of volume one of Maxwell's Treatise on electricity and magnetism.
 
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  • #13
mathwonk said:
It seems the error term in post #1 is valid if only the second derivative exists, even if not continuous. See Courant, vol 1, p. 324, footnote, for example.
Sorry, I don't find the quote. Could you give me more details?
Your posts have been very educational, thanks!
 
  • #14
mcastillo356 said:
Sorry, I don't find the quote. Could you give me more details?
Your posts have been very educational, thanks!
Here you go

4A880394-4F69-479C-A877-C8B5C41A1FD1.jpeg
 
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  • #15
Here is another discussion of the proof in more detail:
https://gowers.wordpress.com/2014/02/11/taylors-theorem-with-the-lagrange-form-of-the-remainder/

See also the comment near the end of the comments section, where a simpler version of the proof is given, from a book used in Flanders, Belgium.

A similar proof appears on pages 494-495 of Calculus of one variable, by Joseph Kitchen, Addison Wesley 1968.

The "usual" proof, using Cauchy's mean value theorem which Tim Gower complains about in the link above to his blog, is on pages 345-7 of Spivak's famous Calculus book.
 
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Related to Linearization of a function error viewed with differentials

1. What is linearization of a function error?

Linearization of a function error is a method used in calculus to approximate the error in a function by using its tangent line at a specific point. It is useful in estimating the accuracy of a function's value, especially when the function is difficult to evaluate exactly.

2. What are differentials in linearization of a function error?

Differentials in linearization of a function error refer to the small changes in the independent and dependent variables of a function. These changes are used to calculate the slope of the tangent line and approximate the error in the function.

3. How is linearization of a function error useful in real-world applications?

Linearization of a function error is useful in real-world applications because it allows for the estimation of errors in measurements and calculations. This is important in fields such as engineering, physics, and economics, where accurate approximations are necessary for making predictions and decisions.

4. Can linearization of a function error be used for any type of function?

No, linearization of a function error is only applicable to functions that are differentiable at the point of interest. This means that the function must have a well-defined tangent line at that point.

5. What is the difference between linearization of a function error and linear approximation?

Linearization of a function error and linear approximation are closely related but have different purposes. Linearization of a function error is used to estimate the error in a function, while linear approximation is used to approximate the value of a function at a specific point. Linear approximation is a special case of linearization of a function error, where the error is assumed to be zero.

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