Derivative: Difference between revisions

From Objective Mathematics
Jump to navigation Jump to search
VisualWikitext
mNo edit summary
Line 4: Line 4:


== Criticism ==
== Criticism ==
This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose that <math>f(x) = x^3</math> so <math>(\Delta_\epsilon f)(x) = 3x^2 + 3\epsilon x + \epsilon^2 </math> and <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 3 (\epsilon - \epsilon' )x + ( \epsilon^2 - \epsilon'^2)</math>. And let's say that we are in a context where anything with absolute value below 0.1 is nill. Let <math>\epsilon  = 0.091 </math> and <math>\epsilon' = 0.001</math>. Then <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 0.27 x + 0.00828 </math>. This quantity is greater than 0.1, and thus non-nill, when <math>x > 0.339\overline{703}</math>, so we reach the seemingly absurd conclusion that <math>f(x) = x^3</math> is not differentiable where <math>x > 0.339\overline{703}</math>. 
This definition makes things way more complicated. I will demonstrate this with the following example.  


This complication is not present in the standard definition of derivative, where the remainder terms go away in the limit and we are just left with <math>f'(x) = 3 x^2 </math>. It's not completely clear whether or not this complication is a problem. Those pesky terms are measuring something real, which calculus is ignoring. They are measuring the difference between two different methods of finding the slope of the tangent line to a real curve.
Let's suppose that <math>f(x) = x^3</math> so <math>(\Delta_\epsilon f)(x) = 3x^2 + 3\epsilon x + \epsilon^2 </math> and <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 3 (\epsilon - \epsilon' )x + ( \epsilon^2 - \epsilon'^2)</math>. And let's say that we are in a context where anything with absolute value below 0.1 is nill. Let <math>\epsilon  = 0.091 </math> and <math>\epsilon' = 0.001</math>. Then <math>(\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 0.27 x + 0.00828 </math>. This quantity is greater than 0.1, and thus non-nill, when <math>x > 0.339\overline{703}</math>, so we reach the seemingly absurd conclusion that <math>f(x) = x^3</math> is not differentiable where <math>x > 0.339\overline{703}</math>. 
 
More generally, if <math>\nu : \mathbb{Q}_{>0}</math> is the nill cutoff (where we also assume <math>\nu \leq 1</math>), then <math>f(x) = x^3</math> is only "differentiable" in the region <math display="block">-\frac{1}{3}(1 + \nu) < x < \frac{1}{3}(1 - \nu)</math>This seems like an absurd conclusion.

Revision as of 22:58, 28 January 2024

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} be a function, and let . I define byFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f) (x) := \frac{f(x + \epsilon) - f(x)}{\epsilon}, \quad x : \mathbb{Q}. } I say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} is differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x:\mathbb{Q}} , if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f)(x) - (\Delta_{\epsilon'} f)(x) } is nill whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon, \epsilon' : \mathbb{Q}_{>0} } are both nill.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{Q} \rightarrow \mathbb{Q}} is differentiable at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x:\mathbb{Q}} , then I define the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) := (\Delta_\epsilon f)(x) } , for some nill Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon : \mathbb{Q}_{>0}} .

Criticism

This definition makes things way more complicated. I will demonstrate this with the following example.

Let's suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^3} so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f)(x) = 3x^2 + 3\epsilon x + \epsilon^2 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 3 (\epsilon - \epsilon' )x + ( \epsilon^2 - \epsilon'^2)} . And let's say that we are in a context where anything with absolute value below 0.1 is nill. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon = 0.091 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon' = 0.001} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta_\epsilon f - \Delta_{\epsilon'} f)(x) = 0.27 x + 0.00828 } . This quantity is greater than 0.1, and thus non-nill, when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 0.339\overline{703}} , so we reach the seemingly absurd conclusion that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^3} is not differentiable where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 0.339\overline{703}} .

More generally, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu : \mathbb{Q}_{>0}} is the nill cutoff (where we also assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu \leq 1} ), then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^3} is only "differentiable" in the region Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{3}(1 + \nu) < x < \frac{1}{3}(1 - \nu)} This seems like an absurd conclusion.

Retrieved from "http://64.23.165.198:80/index.php?title=Derivative&oldid=233"