# Can anyone clarify the following points regarding calculus. Please

1. Jul 4, 2008

### superstonerman

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

2. Jul 4, 2008

### matt grime

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.

3. Jul 4, 2008

### kkrizka

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.

4. Jul 4, 2008

### HallsofIvy

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".

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.

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".

Okay. I started to disagree until I saw your second sentence!