Can anyone clarify the following points regarding calculus. Please

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Discussion Overview

The discussion revolves around clarifying fundamental concepts in calculus, including definitions of functions, derivatives, and integrals. Participants seek to ensure their understanding of these concepts is accurate and address potential misconceptions.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant defines a function as the relationship between two variables, x and y, while another suggests it should be described as a unique relationship where each x corresponds to only one y.
  • There is a claim that a differential indicates the rate of change between dependent and independent variables, with an example provided (y' = 2x for y = x²). However, another participant notes that a continuous function need not be differentiable, questioning the justification for this claim.
  • A participant emphasizes the distinction between "derivative" and "differential," indicating that the term "derivative" is more appropriate in the context of discussing the slope of the tangent to a function.
  • Integration is described as finding the area under a curve, with acknowledgment that it has other uses. There is a discussion about integration being the opposite of differentiation, but this point remains under scrutiny.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and relationships between functions, derivatives, and integrals. There is no consensus on several points, particularly regarding the nature of differentiability and the terminology used.

Contextual Notes

Some claims are challenged or refined by participants, indicating a need for careful consideration of definitions and conditions, such as differentiability and the distinction between derivatives and differentials. The discussion reflects a range of interpretations and understandings of calculus concepts.

superstonerman
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Wassup peeps. Can anyone clarify the following points to make sure what I know about calculus is correct. Anyway:

1. A function is defined as the relationship between two variables usually x and y.

2. A differential of a function states the rate of change between these two variables. How the dependent variable changes in regard to changes in the independent variable. Therefore given a continuous function it is possible to differentiate this function to obtain a statement that describes the rate of change between the variables x and y. For example the differential of y=x^2 is y'=2x. Therefore 2x describes how the dependent variable y changes with regards to how the independent variable x changes.

3. This differential can also be seen as the slope of the tangent to a function.

5. The integral and therefore the process of integration is about finding the area under a curve. It has many other uses however this is how an integral can be defined.

3. Integration takes this rate of change, the derivative of the function and outlines a statement that describes the relationship between two variables, therefore a function. It can seen from this integration is the opposite of differentiation and vice-versa

4. Which is shown in the fundamental laws of calculus.

Please correct and explain anything you feel is incorrect. Thanks for your help
 
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2. A continuous function need not be differentiable.3 (The second 3). How can it be seen from this? I don't see a justification there.
 
1. A function is defined as the relationship between two variables usually x and y.
I would write that as:
1. A function is defined as the unique relationship between two variables usually x and y.

In other words, for each x there is only one y.
 
superstonerman said:
Wassup peeps. Can anyone clarify the following points to make sure what I know about calculus is correct. Anyway:

1. A function is defined as the relationship between two variables usually x and y.
A function is defined as a relationship between two quatitites so that if a is related to b, a is NOT related to any thing other than be. For example in y= x2 , y is a "function of x" but x is not a "function of y".

2. A differential of a function states the rate of change between these two variables. How the dependent variable changes in regard to changes in the independent variable. Therefore given a continuous function it is possible to differentiate this function to obtain a statement that describes the rate of change between the variables x and y. For example the differential of y=x^2 is y'=2x. Therefore 2x describes how the dependent variable y changes with regards to how the independent variable x changes.
Given a differentiable function. A continuous function is not necessarily differentiable. For the example y= x2, saying "2x describes how the dependent variable y changes with regards to how the independent variable x changes" is too general. The rate at which y changes is specficially 2x times the rate at which x changes.

3. This differential can also be seen as the slope of the tangent to a function.
Be careful to distinguish between "derivative" and "differential". They are quite different and what you are talking about here is the "derivative", not the "differential".

5. The integral and therefore the process of integration is about finding the area under a curve. It has many other uses however this is how an integral can be defined.
Okay. I started to disagree until I saw your second sentence!

3. Integration takes this rate of change, the derivative of the function and outlines a statement that describes the relationship between two variables, therefore a function. It can seen from this integration is the opposite of differentiation and vice-versa

4. Which is shown in the fundamental laws of calculus.

Please correct and explain anything you feel is incorrect. Thanks for your help
 

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