Visual Derivative: Understanding Slope of 3-D Functions

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Discussion Overview

The discussion revolves around the visual representation of derivatives in the context of 3-D functions, particularly focusing on the derivative of the function y = x^2. Participants explore the concept of limits and how they relate to graphical representations of derivatives, while grappling with the implications of indeterminate forms.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses a belief that a derivative can be visualized as a "gap" in a 3-D function, specifically questioning the graphical outcome of the derivative of x^2 when represented in three dimensions.
  • Another participant challenges the initial definition of a derivative provided by the first participant, suggesting a more standard limit-based definition.
  • There is a discussion about the expression for the derivative, with one participant attempting to reformulate it using y instead of dx, leading to confusion about the resulting graph.
  • One participant later claims to have resolved their confusion regarding the graphing of the derivative, attributing the issue to a mistake in inputting data into a graphing calculator.
  • Another participant asserts that the expression discussed does not represent a plane, while a different viewpoint claims that it does, highlighting the ongoing debate over the graphical representation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the graphical representation of the derivative in 3-D space, with multiple competing views and interpretations of the expressions involved. The discussion remains unresolved regarding the nature of the graph and the validity of the proposed representations.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the definitions of derivatives and the conditions under which the expressions are evaluated, particularly concerning indeterminate forms.

aquaregia
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I am learning calculus and I liked the idea of how limits are represented visually, as a removable discontinuity (a single point gap) in a function of x. When we started studying derivatives I immediately came to the conclusion that a derivative is like a single curve "gap" in a function of x and y, however after doing some searching I have not found a visual representation of the derivative being like that, and after graphing some 3-d functions they did not turn out like I expected.

For example the derivative of x^2 is: (x + dx)^2 - (x^2) / dx. Now replace the dx with y and you have (x + y)^2 - (x^2) / y, this should be a 3-d surface, right? And it should basically look like a plane with a slope of 2 with gap at y=0 since that is indeterminate because it would be 0/0, however the graph is a strange 3-d curve. Does anyone know if my reasoning is wrong?
 
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aquaregia said:
For example the derivative of x^2 is: (x + dx)^2 - (x^2) / dx. Now replace the dx with y and you have (x + y)^2 - (x^2) / y,

What is your definition of a derivative?!?

y = x^2

then dy/dx = lim (as h approaches 0) ((f(x+h) - f(x))/h)
 
aquaregia said:
I am learning calculus and I liked the idea of how limits are represented visually, as a removable discontinuity (a single point gap) in a function of x. When we started studying derivatives I immediately came to the conclusion that a derivative is like a single curve "gap" in a function of x and y, however after doing some searching I have not found a visual representation of the derivative being like that, and after graphing some 3-d functions they did not turn out like I expected.

For example the derivative of x^2 is: (x + dx)^2 - (x^2) / dx. Now replace the dx with y and you have (x + y)^2 - (x^2) / y, this should be a 3-d surface, right? And it should basically look like a plane with a slope of 2 with gap at y=0 since that is indeterminate because it would be 0/0, however the graph is a strange 3-d curve. Does anyone know if my reasoning is wrong?

You mean z= [(x+y)^2- x^2]/y? No, that is not a plane.
 
Whatever, I figured it out. I just typed it into the graphing calculator wrong, but the basic idea is true.
 
HallsofIvy said:
You mean z= [(x+y)^2- x^2]/y? No, that is not a plane.

Are you sure? I just graphed it, and it was, actually I graphed ((x+y)^2- x^2)/y, but I don't think switching the parenhesis style makes a difference. Of course its indeterminate at y = 0 because that is the deraritive.
 
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