Graphical link between function and derivate

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Discussion Overview

The discussion revolves around understanding the graphical relationship between a function and its derivative, specifically using the example of the function y = x^2 and its derivative y = 2x. Participants explore how to link the graphs of a function to its derivative and clarify the implications of positive and negative slopes.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about linking the graph of a function to its derivative, noting that while the slope of y = x^2 is negative for x < 0, the derivative y = 2x has a positive slope.
  • Another participant explains that the gradient function y = 2x is negative for x < 0, but emphasizes that the value of the derivative at a specific point (e.g., x = -2) reflects the slope of the original function at that point.
  • It is proposed that when a function is increasing, its derivative is positive, and when it is decreasing, the derivative is negative. The point where the function changes direction corresponds to the derivative being zero.
  • A participant highlights that although the slope of the derivative function is positive, the actual value of the function can still be negative, suggesting that the focus should be on the function's value rather than just the slope of the derivative.

Areas of Agreement / Disagreement

Participants express differing views on how to interpret the relationship between a function and its derivative, particularly regarding the implications of positive and negative slopes. The discussion remains unresolved as participants explore these concepts without reaching a consensus.

Contextual Notes

Some assumptions about the behavior of functions and their derivatives are not fully articulated, and there may be dependencies on specific definitions of increasing and decreasing functions that are not clarified.

mad
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Hello all,
first, excuse my english I don't speak it very well

I have a problem. We have two sheets. One are graphics of functions, and the other are graphics of the derivate of those function. Now my problem is I don't know how link a graphic of a function to the graphic of its derivate. I know that, for example, y = x^2, for ]-oo, 0[ , that the slope (sp?) will be negative. So why , on the graphic of the derivate which is y=2x, is the slope positive? How can I associate a function to its derivate?
thanks a lot
 
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If you don't understand what I'm asking, here is an exercice exactly like the one I'm talking about.
http://gmca.eis.uva.es/wims/wims.cgi?lang=es&+module=U1%2Fanalysis%2Fderdraw.en

Choose degree 3 or more and it asks to draw its derivate graphic. But I don't know how
 
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The gradient function of y = x^2, that is: y = 2x, is negative for values of x less than zero. So, although the slope of the gradient function is positive, you can see that value of the gradient function at say, x = -2 still gives the slope of y = x^2 at x = -2.

The slope of the gradient function would only be negative if the original function was y = -x^2.

It helps to plot the two graphs y = x^2 and y = 2x above and below each other, and matching respective x values on both, to get a feel for what's happening in the gradient function.
 
The relationship is:

When the function is increasing, the derivative is positive

When the function is decreasing, the derivative is negative

When the function is changing direction, the derivative is zero

So, in the case of [tex]f(x)=x^2[/tex] from [tex]-\infty\rightarrow 0[/tex] the function is decreasing and the derivative is negative. At the point (0,0) the function changes direction, so the derivative is zero, and from [tex]0\rightarrow\infty[/tex] the function is increasing so the derivative is positive.
 
Although the slope of 2x is positive, the value of the function is negative.

It is the value of the function which you must be concerned with.
 

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