What are derivatives and integrals?

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    Derivatives Integrals
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Discussion Overview

The discussion revolves around the concepts of derivatives and integrals in calculus, focusing on their definitions, interpretations, and implications. Participants explore both geometric interpretations and the nature of derivatives as functions, while also expressing personal experiences with learning calculus.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant requests a clear explanation of derivatives and integrals, noting that their textbook does not provide sufficient detail.
  • Another participant offers a geometric interpretation, explaining that the derivative represents the slope of a function at a given point, using the example of a linear function versus a quadratic function.
  • A follow-up question arises about whether there can be as many derivatives as there are points on a function, suggesting a curiosity about the nature of derivatives.
  • It is noted that different functions can be differentiated a varying number of times, indicating that not all functions have the same differentiability properties.
  • Further clarification is provided that the derivative of a function is itself a function of x, with specific examples illustrating how to calculate the slope at particular points on a curve.
  • A participant expresses appreciation for the explanations and shares a growing interest in learning calculus as a potential hobby.

Areas of Agreement / Disagreement

Participants generally agree on the basic interpretations of derivatives and integrals, but there are nuances regarding the number of derivatives a function can have and the nature of differentiability that remain open for discussion.

Contextual Notes

Some assumptions about the definitions of derivatives and integrals are not explicitly stated, and the discussion does not resolve the complexities of differentiability across different types of functions.

Who May Find This Useful

Individuals beginning to learn calculus, those seeking clarification on the concepts of derivatives and integrals, and anyone interested in the geometric interpretations of these mathematical concepts.

The_Z_Factor
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What are they? In my book I am studying limits and it has mentioned a few times before and in the current chapter Derivatives and Integrals, but hasnt explained them. Could anybody explain what these two things are, exactly?
 
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if your asking for a formal definition then goto www.wikipedia.com and search for derivative and separately integration.

Geometric calculus interpretation:

If f(x) is a line and can be represented by mx +b then the slope is m, but this is a line and the slope is the same throughout the real numbers. Now consider y= x^2, what is the slope? it changes at each point, the slope at any point is the derivative of the function evaluated at that point.

the integral of a real function gives you the area under the curve of the function.
 
SiddharthM said:
If f(x) is a line and can be represented by mx +b then the slope is m, but this is a line and the slope is the same throughout the real numbers. Now consider y= x^2, what is the slope? it changes at each point, the slope at any point is the derivative of the function evaluated at that point.

So does this mean that there can be as many derivatives as there are points?
 
The_Z_Factor said:
So does this mean that there can be as many derivatives as there are points?

Different functions can be differentiated a different number of times.
 
The_Z_Factor said:
So does this mean that there can be as many derivatives as there are points?

It means that in general the derivative of a function of x is itself a function of x. i.e., the slope of a function is different at each point on that function.

For example, the derivative of x^2 is 2x. This means that on the curve y = x^2, at the point x = 4, the slope of the curve at x = 4 (or, perhaps more precisely, the slope of the line tangent to the curve at x = 4) is 2*4 = 8. Similarly, the slope at the point x = -5 is -10.
 
Ah, thanks for clearing that up for me everybody, that explains it. I think as I'm beginning to learn more about simple calculus I'm beginning to like it more. Haha, it just might turn into a hobby once I learn enough.
 

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