Relation of the graph of a function with the graph of it's derivative

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Discussion Overview

The discussion revolves around the relationship between the graph of a function and the graph of its derivative, specifically using the example of the function x^2 and its derivative 2x. Participants explore what can be inferred about the original function from the derivative's graph and vice versa, focusing on aspects such as increasing/decreasing behavior, concavity, and points of interest like maxima and minima.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the relationship between the graphs of a function and its derivative, asking if conclusions can be drawn about x^2 from the graph of 2x or about 2x from the graph of x^2.
  • Another participant suggests that the sign of the first derivative can indicate where the function is increasing or decreasing, as well as where it has maxima, minima, or inflection points.
  • A participant clarifies that they are interested in understanding what can be inferred about x^2 from the equation of its derivative, 2x, without substituting specific values.
  • There are repeated affirmations of earlier points, indicating a lack of new contributions or clarification on the topic.

Areas of Agreement / Disagreement

Participants express similar views regarding the potential insights gained from the derivative's graph, but there is no consensus on the specific conclusions that can be drawn without substituting values. The discussion remains somewhat unresolved as participants reiterate points without introducing new perspectives.

Contextual Notes

Participants do not fully explore the implications of their claims, leaving some assumptions and dependencies on definitions unaddressed. The discussion does not resolve the nuances of how much can be inferred from the derivative's graph without specific value substitution.

Juwane
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What is the relation between the graph of a function and the graph of it's derivative?

Suppose that the function is x^2. It's derivative is 2x. If we graph both x^2 and 2x on the same coordinate axes, can we conclude anything about x^2 by looking at the graph of 2x, or vice versa (i.e. can we conclude anything about 2x by looking at the graph of x^2)?

For example, can we tell what will be the slope of the tangent line at a curve at x^2 by looking at the graph of it's derivative 2x?

If there is no relation between the two, then what is the use of graphing the derivative of a function?
 
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You ought to be able to tell where it it is increasing, decreasing, concave up, concave down, where it has maxima or minima or inflection points just by looking at the sign of the first derivative and where it's increasing or decreasing.
 
rochfor1 said:
You ought to be able to tell where it it is increasing, decreasing, concave up, concave down, where it has maxima or minima or inflection points just by looking at the sign of the first derivative and where it's increasing or decreasing.

I'm not talking about when you find the slope of the tangent at some point by substituting the point into the the equation of the derivative. I'm talking about the equation of the derivative itself (such as y=2x for y=x^2).

I mean: what can we find out about x^2 just from the equation 2x (or it's graph) (without substituting any values)?
 
Exactly what I said.
 
Oops.. I think I'm asking the question the wrong way. My question is that by looking at the plot of 2x (which is a straight line), what can we say about the x^2 (which is a curve), without knowing that the straight line's equation is 2x and the curve's equation is x^2?
 
Again, exactly what I said.
 

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