integrating and differentiating

[baric]

That is precisely the point.

 

 

Sorry, I lost track of who derailed this mess due to all of the math peens flying around.

[between3and26characterslon]

You would not be splitting hairs in the context of a formal math classroom. I was aware of the correction Dr. Rocket posted, but given that this thread is for the most elementary level of knowledge transfer my explanation was sufficient and I will stand by it. The original poster doesn't even know what continuity and jump discontinuities are probably, no insult to him.

 

If you wish to criticize me on my lack of rigor then you can, but for what?

 

No insult taken, I havent a clue what you're talking aout.

 

Nor do I have any clue why this arguement about FTC ensued. Sorry Zone Ranger but your comment was irrelevant here because it meant nothing to me and didn't help in any way to answer my original question.

[baxtrom]

Nor do I have any clue why this arguement about FTC ensued.

 

Just relax and enjoy the squabble. I'd put my money on Dr Rocket, he may be a bit grumpy from time to time but he knows his math..

[Hal.]

The Fundamental Theorem of Calculus argument ensued over a difference of opinion as to what should be said in this thread as being the Fundamental Theorem of Calculus . If the Fundamental Theorem of Calculus is going to be altered for the purpose of explanation then the explanation may not be describing what the person says they are describing .

[baric]

No insult taken, I havent a clue what you're talking aout.

 

Nor do I have any clue why this arguement about FTC ensued. Sorry Zone Ranger but your comment was irrelevant here because it meant nothing to me and didn't help in any way to answer my original question.

 

Discontinuities are easy! They are simply points on a graph where the line is not connected. There are a few different types, such as a "hole" on the line or where the line suddenly breaks. The second type is a "jump" discontinuity because the line "jumps" to a new value. Here is an example:

 

Edited June 15, 2011 by baric

[baxtrom]

Discontinuities are easy!

 

Using the plumber methaphor, you don't want discontinous pipes, especially not sewer..

[DrRocket]

No insult taken, I havent a clue what you're talking aout.

 

Nor do I have any clue why this arguement about FTC ensued. Sorry Zone Ranger but your comment was irrelevant here because it meant nothing to me and didn't help in any way to answer my original question.

 

OK, if you have not head much about discontinuities, then you have not seriously studied limits and continuous functions either. Without that background you cannot understand derivatives or integrals beyond my earlier post on tangent lines and areas. If that is all that you need, then there is not much more to be said.

 

If you need something more, then you need to take a real calculus class, preferably from someone who undersatnds the subject. So, my suggestion is to take a class at the university nearest to you. I don't know of any text suitable for self-study at your level.

[baric]

I don't know of any text suitable for self-study at your level.

 

This is a good start:

 

http://www.amazon.com/Complete-Idiots-Guide-Calculus/dp/0028643658

[imatfaal]

I don't know of any text suitable for self-study at your level.

 

Have any of the maths gurus looked at the MIT opencourseware maths courses? I realise that the areas they cover are way back in your undergraduate past - but bearing in mind the discussions in another place on the quality of current teaching - I wondered what opinions were.

 

They are basic first year (I think) undergrad courses - I haven't had the chance to watch any of them apart from Golbert Strang's Linear Algebra yet. Whilst they would not provide any deep understanding, for those of us who last formally studied maths in the 20th Century they are great at "reformalising" (is that a word?) ideas that have become a bit woolly.

[DrRocket]

This is a good start:

 

http://www.amazon.co...s/dp/0028643658

 

 

Have any of the maths gurus looked at the MIT opencourseware maths courses? I realise that the areas they cover are way back in your undergraduate past - but bearing in mind the discussions in another place on the quality of current teaching - I wondered what opinions were.

 

They are basic first year (I think) undergrad courses - I haven't had the chance to watch any of them apart from Golbert Strang's Linear Algebra yet. Whilst they would not provide any deep understanding, for those of us who last formally studied maths in the 20th Century they are great at "reformalising" (is that a word?) ideas that have become a bit woolly.

 

Either of thse likely have merit.

 

However, my take is this: Calculus is a huge philosophical change from elementary algebra. it is no longer "solve the equation and find the number". Rather, there are entirely new concepts, based on estimates and refined estimates. Inequalities are as important or more important than equalities. Limits are a couple of levels of sophistication removed from high school algebra. Calculus requires a change in mind set.

 

The change in mind set is greatly facilitated by classroom discussions and talks with fellow students. This is not available in a self-study environment. Average students (and I suspect that neither of you are in this category) probably cannot really grasp the material without the intangibles that come from these talks and the opportunity to have questions answered and misconceptions corrected in real time.

 

Add to this my prejudice that the (standard) textx from which I have taught the subject are generally poor, and my position that to learn calculus for the first time most students are best served by a traditional class. Books, videos etc. may be useful supplements, but not substitutes for the class.

Edited June 15, 2011 by DrRocket

[between3and26characterslon]

ok, here you go.

 

Think of a curve on a graph... maybe a curve representing the square of a number

 

 

For example, at x=1, then y =1. at x=2, y=4. at x=3, y=9.

 

Got that?

 

Notice how the value for y goes up faster than the value of x? That means that the SLOPE of the curve is increasing as x increases.

 

What differentiating does is give us a new equation that tells us the particular slope of the line at any x value.

 

In the real world, this slope is often called the rate of change.

 

 

 

Integration goes the other way. Imagine that same curved line as before, but also visualize the area underneath it. What integration tells us is how much area is under the curve between two x points.

 

In the real world, this can give you specific values at points in time for something that is changing. For example, the integration of a velocity equation will tell us how far something has traveled in a specified amount of time.

 

 

These two concepts are the fundamental basis of calculus and have a tremendous amount of use in the natural world. What calculus boils down to is understanding how to differentiate or integrate the various COMPLICATED equations that are used to describe things in the real world.

 

 

This is a good explanation for me (if you've read any of my other posts you will notice a distinct absense of maths, I am more comfortable with concepts).

 

Can I ask, if you plotted a graph of 'gravitational force of attraction against distance from the centre' following the 1/r² rule you would get a curved graph. Could you differentiate and integrate this graph and if so how would you and what would it tell you?

 

DrRocket, all those squiggly lines you keep posting mean absolutely nothing to me just yet, you may as well be speaking Klingon (which you probably do... phnarrr! no offence) thanx for your input, and sorry to single you out, but you're way over my head. Perhaps you're not up to the challenge of coming down to my level....?

[baric]

This is a good explanation for me (if you've read any of my other posts you will notice a distinct absense of maths, I am more comfortable with concepts).

 

Can I ask, if you plotted a graph of 'gravitational force of attraction against distance from the centre' following the 1/r² rule you would get a curved graph. Could you differentiate and integrate this graph and if so how would you and what would it tell you?

 

Gravity is a force of acceleration. Acceleration is the differential of velocity over time, which itself is the differential of distance over time.

 

For example, if you knew a rate of acceleration (from gravity), then integrating that into a velocity equation will tell you how fast something is falling after a specified amount of time. You could then integrate that velocity equation into a distance equation to tell you how far something has fallen. You can obviously imagine how incredibly useful calculus is in science.

 

You could also differentiate the acceleration equation, which would give you an equation defining the rate of change in acceleration. This is known as "jerk" or "jolt" and is best felt as the sensation you get when you press further down on the gas pedal in your car while it is still speeding up.

Edited June 18, 2011 by baric

[DrRocket]

DrRocket, all those squiggly lines you keep posting mean absolutely nothing to me just yet, you may as well be speaking Klingon (which you probably do... phnarrr! no offence) thanx for your input, and sorry to single you out, but you're way over my head. Perhaps you're not up to the challenge of coming down to my level....?

 

 

My mistake. I thought that your objective was to raise your level.

Edited June 18, 2011 by DrRocket

[between3and26characterslon]

My mistake. I thought that your objective was to raise your level.

 

 

Then you are mistaken.

 

My objective is not to learn more things about this subject but rather to understand one thing. I wasn't seeking to broaden my knowledge, I wanted to deepen my understanding. If someone asked me what differentiating and integrating were I'd still have to answer "I don't know"

 

Gravity is a force of acceleration. Acceleration is the differential of velocity over time, which itself is the differential of distance over time.

 

For example, if you knew a rate of acceleration (from gravity), then integrating that into a velocity equation will tell you how fast something is falling after a specified amount of time. You could then integrate that velocity equation into a distance equation to tell you how far something has fallen. You can obviously imagine how incredibly useful calculus is in science.

 

You could also differentiate the acceleration equation, which would give you an equation defining the rate of change in acceleration. This is known as "jerk" or "jolt" and is best felt as the sensation you get when you press further down on the gas pedal in your car while it is still speeding up.

 

Ok so the bits in bold are where I'm struggling, would you be so kind as to put some simple equations in so I can see what is going on.

 

You can also find what I hope is an intuitive explanation of the meaning of differentiation here:

 

http://www.sciencefo...ifferentiation/

 

I got distracted by the rest of the posts here but now I've given this a proper read it is starting to make sense. Thanks Cap'n

[baric]

Ok so the bits in bold are where I'm struggling, would you be so kind as to put some simple equations in so I can see what is going on.

 

ah, but that is the crux of calculus! Simple equations are easily differentiated and integrated using the "power rule", so those would be a good illustrative example. However, proficiency of calculus involves honing your differentiating and integrating skills to handle complex equations.

 

However, it's a quick example of the power rule.

 

Look at the following series for [math]y=x^2[/math]

 

0, 1, 4, 9, 16, 25, 36, 49 ....

 

We can easily see the rate of change by manually calculating the difference between each value:

 

1, 3, 5, 7, 9, 11, 13

 

Notice how it goes up by 2 each time.

 

Now look at the "power rule" for differentiation:

 

[math](x^n)' = nx^{n-1}[/math]

 

This is interpreted as "the differential of [math]x^n[/math] is n times [math]x^{n-1}[/math]

 

 

So what is the differential of our original equation, [math]y=x^2[/math]?

 

It would be 2 times [math]x^1[/math] or, more simply, [math]2x[/math]. In other words the value of y will increase by 2 with each value of x. And this is exactly what we see when we manually calculate the rate of change for [math]y=x^2[/math]. Does this make sense?

 

 

Integrating with the power rule is just the exact opposite. The integral of [math]x^n[/math] is [math]x^{n+1}/(n+1)[/math]. You can see that we increased the exponent by 1 and then divided by the new exponent. This is the reverse process of differentiation.

 

Here are some more basics on differentiation if you want to get a feel for the different techniques that can be used: http://www.math.ucda...pts/node17.html

Edited June 19, 2011 by baric

[khaled]

I'm not really good in mathematics .. but, differentiation and integration, sounds like cracking and collecting, to me it seem more like degrade and upgrade to a function f(x) ...