I think I made a mathematical discovery Check it out

[Skhandelwal]

Don't laugh if this basic stuff and has already been discovered centuries ago but I am just in basic Calculus. Here is what I have came up w/.

In y=x^odd number(just pick any) graph, the greater the coeffecient, the closer it is to the graph x=0. But if we were to imagine what the graph of y=infinityx^odd number(just pick any) look like, I believe it will look like a vertical line at x=0.(you can check this theory by finding greater and greater secant line from the origin of this graph to a farther and farther point, the slope keeps increasing.)

Same thing happens w/ the graph of y=x^infinity. Just at an exponentially faster rate but as long as you have a vertical line, I don't think slower and faster matters.

So I think that y=infinityx^odd number(just pick any) is equivalent to y=x^infinity which is equivalent to the graph x=0.

Now, y=smallestpositive#x^2 is equal to y=0 graph. And same w/ exponent but I think we all know that already; y=x^0. Which equals to y=1 graph.

I am like an infinitologist, always trying to understand infinity in other ways, I have read so many books and articles over infinity that you won't even believe it. I think if we can understand infinity, we well get answers to all our questions.

What do you think?

 

[jpr0]

I think your function would look something like this:

$$y(x) = 0,\,\,\,\,-1<x<1,$$
$$y(x) = \pm 1, \,\,\,\, x=\pm 1,$$
$$y(x) = -\infty, \,\,\,\, x<-1,$$
$$y(x)= \infty, \,\,\,\, x>1.$$

If you take a number less than one, and multiply it by itself, you get a number small than the one you started with. If you then mulitply the result by the same number you get an even smaller number. So in the limit of doing this infinitely many times you will get 0. Similarly, if you multiply a number greater than 1 (or less than -1) by itself inifinitely many times the result will diverge. If you multiply 1 by itself infinitely many times the result will always be 1.
(Sorry for the messy tex but I can't get arrays working).

 

[Skhandelwal]

So is this something new or common sense?(I am a senior in high school)

 

[matt grime]

I am like an infinitologist, always trying to understand infinity in other ways, I have read so many books and articles over infinity that you won't even believe it. I think if we can understand infinity, we well get answers to all our questions.

What do you think?

What makes you think that 'infinity', whatever you may mean by it, is not understood mathematically? We seem to have a prefect grasp of it. Writing things l ike y=infinityx^odd power doesn't mean anything. Just put limits in and it all drops out properly, which is after all what infinity in the sense you seem to want to use it means.

 

[matt grime]

In y=x^odd number(just pick any) graph, the greater the coeffecient

what coefficient?

Just take time to write things out properly and remember to treat things like infinityx as inherently meaningless at the moment (It is unless you chose to adopt some meaning for it).

 

[jpr0]

By the way that was for the function $$y(x)=\lim_{n\to\infty}x^{2n+1}$$. This kind of thing isn't new I suppose. In physics you sometimes use something called the Fermi distribution function, which is a smooth function of x say, except when you take one of it's parameters to 0 then it becomes step-like, in much the same way as your function does. For $$0\leq x < x^{\prime}$$ the Fermi function is $$1$$, at $$x=x^{\prime}$$ it is $$\frac{1}{2}$$, and for $$x>x^{\prime}$$ it is 0.

I may as well write it here:

$$f(E) = \frac{1}{\exp((E-E_f)/kT)+1}$$

When you take the limit of $$T\to 0$$ for this function you get the above step like behaviour.

 

[jpr0]

what coefficient?

Just take time to write things out properly and remember to treat things like infinityx as inherently meaningless at the moment (It is unless you chose to adopt some meaning for it).

I think he means the exponent

 

[Data]

I am like an infinitologist, always trying to understand infinity in other ways, I have read so many books and articles over infinity that you won't even believe it. I think if we can understand infinity, we well get answers to all our questions.

I think it'll probably lose its enigma when you learn something about the various notions of infinity in mathematics. They're pretty well-defined :tongue2:

 

[Skhandelwal]

since this topic has pretty much ended, I was just wondering, if I take the antiderivitive of a position graph, what do I get? Anything useful?

 

[Skhandelwal]

Nvm that, btw, what does an increasing curve(like a half of the positive quadratic) in an acceleration graph indicate?(I know technically but hoping you guys could give me a real life example.

Also, there is acceleration at linear rate, geometric rate, exponential rate, what comes after that?

One more thing, for 1,2,3... there is first, primary, single, and I can't remember, there is one more, but do you guys know more of these? I would really appreciate if you guys could tell me.(don't need to list all of them, just tell me the starting point.)

 

[Swapnil]

Nvm that, what does an increasing curve in an acceleration graph indicate?(I know technically but hoping you guys could give me a real life example.

You mean in an acceleration-time graph? Its just one of those things which are harder to visualize because its hard to distinguish constant acceleration from a change in acceleration just through experience. (Even when you are falling you are just going through a phase of constant acceleration).

Let me give it a try. Say you are in an intergallactic spaceship which is taveling at a constant velocity (i.e. a horizontal line [y=0] on your acceleration-time graph). How do you know this? Because you are experiencing no force. Now say the ship started accelerating at a constant rate (i.e. a flat horizontal line [y=constant] in the accleration-time graph). How do you detect this? Because you are feeling a constant force (maybe a throbbing pain in your chest). Now say the ship is going through a change in acceleration (i.e. a curve in the acceleration time graph). How do you know? Because each instant of time, you would be experiencing more and more force! (i.e. the pain in your chest would throbb more and more every second!).

That's basically how you can visualize the three aspects of acceleration: 1) no acceleration; 2) constant acceleration and 3) increasing acceleration.

 

[Integral]

A time changing acceleration is called JERK. You have experienced this in every day life. A good example is when shifting gears with a manual transmission. Very frequently when shifting to a different gear your head will jerk. That is because your rate of acceleration has changed. This will happen when going from constant acceleration to constant velocity or from constant velocity to an acceleration.

 

[Skhandelwal]

Here is what I knew,
1) no accel, empty graph
2) constant accel., horizontal line
3) increasing acceleration, a line w/ positive slope
4) But what about curved line in an accel?

You see, when you were explaning, you skipped 3). Increasing accel(linear rate) has a line w/ a positive slope, not a curved line. I think curve line is where accel is inc. at a geometric rate. Then what kind of line would indicate an accel. increasing at an exponential rate?

Btw, I know about jerk too, that wasn't my question.

 

[CRGreathouse]

Here is what I knew,
1) no accel, empty graph
2) constant accel., horizontal line
3) increasing acceleration, a line w/ positive slope
4) But what about curved line in an accel?

You see, when you were explaning, you skipped 3). Increasing accel(linear rate) has a line w/ a positive slope, not a curved line. I think curve line is where accel is inc. at a geometric rate. Then what kind of line would indicate an accel. increasing at an exponential rate?

Btw, I know about jerk too, that wasn't my question.

1) is actually a horizontal line at 0. You have 2) and 3) correct. A curve (that isn't a line) in the acceleration-time graph simply represents a non-constant derivative of acceleration wrt time. Any non-zero jolt* will cause the graph to be 'curved'. If jolt is constant, acceleration is linear; if jolt is linear, acceleration is quadratic. But you know this since it's just basic calculus (calc I or pre-calc, I can't remember).

Again, with basic calculus, you know that if any of position, velocity, acceleration, jolt, www.bme.jhu.edu/~reza/book/minimumjerk.pdf[/URL], ... are exponential they all are, and you know from high school algebra how to graph exponentials.

* Oddly, I have always known of the derivative of acceleration wrt time as jolt rather than jerk.

 

[CRGreathouse]

Also, there is acceleration at linear rate, geometric rate, exponential rate, what comes after that?

I don't see the connection between these. They are, generically:

$$y=ax+b$$

$$y=a\cdot b^x$$

$$y=a\cdot b^x$$

If you're loking for a function that increases faster than a geometric/exponential function as it tends toward infinity, you could consider factorials, double exponentials y=e^(e^x), tetration, or some version of the Ackerman function.

 

[Robokapp]

I have to say this: a lot of graphs can look in a lot of ways if you stretch the windows in ways to fit those...

I understand what you're saying...$$y=x^{10^{9999}-1}$$ does look almoust like y=0 but...I don't understand why that's in any way a 'discovery".

I mean if you're trying to graph a vertical asymptote as a function...just like squaring a circle, yuo'll never get it perfectly...am I right?

 

[Skhandelwal]

I apologize for not making my question clear b/c this isn't even my question. I was wondering if there is a curve on an acceleration graph, can you give me a real life situation that would represent that? I mean none of my teachers could come up w/ one.

 

[matt grime]

Perhaps no examples were forthcoming because the phrase 'a curve on an acceleration graph' doesn't mean much. UNdoubtedly there is some idea you have, but I don't think you're communicating it at all clearly.

 

[Skhandelwal]

All I am saying is that you have to imagine a quadratic graph on the acceleration graph.(but only the 1st quadrant of it) Now just like peddling down the accelerator in a car can be described by a line w/ some slope on the acceleration graph, how would you describe a real situation which matches w/ my description of the acceleration graph?

 

[Data]

You experience what would correspond to a curved acceleration graph all the time.

Suppose you're driving in a car and you floor the accelerator. At first, your car accelerates very quickly. After a while, friction and air resistance start limiting your acceleration, and it decreases, eventually getting close to zero, as the forces balance.

So your acceleration is decreasing. But it's not a linear decrease: the rate of decrease depends on how fast you're already accelerating.

Once the forces have "balanced," you'll still be experiencing a curved acceleration graph: wind will change, maybe the surface of the road will change, and you'll slow down and speed up in an unpredictable pattern (keeping the pedal floored).

 

[matt grime]

All I am saying is that you have to imagine a quadratic graph on the acceleration graph.(but only the 1st quadrant of it) Now just like peddling down the accelerator in a car can be described by a line w/ some slope on the acceleration graph, how would you describe a real situation which matches w/ my description of the acceleration graph?

My answer is 'who cares'.

Actually my answer is: that still makes no more sense: what has an quadrant of a graph (whatever that might be) got to do with anything at all. What you've written does not make any sense at, just as an example of the English language. How on Earth does a graph of anything have a 'line on it'. Do you mean the tangent at some point? If so, say it. Perhaps I am just overly reacting to the fact that you won't bother to type the word 'with'. Further, what 'description' of what 'acceleration' graph?

Go ask an engineer or someone who actually deals with 'real life', if you want an answer about real life applications of maths.

 

[Robokapp]

yeah...when you drive your car, at very low speeds it accelerates very fast...but as you're accelerating on the highway, even if you gas it all the way, the speed increases...but much much slower. so...the acceleration must be not that big. so...if you graph the acceleration it'd be a parabola...something that raises fast first, and gradually loses the slope value...


a negative linear jerk, makes a quadratical increasing acceleration...imagine the graph of y=-x^2 from...-10 to -2. it's having positive slopes, but the slopes are smaller and smaller. those slopes would be the "jerk".

 

[Skhandelwal]

oh crap, if I knew how to draw here, it would much simpler to demonstrate it by a graph. Anyways, all I was saying is that if the function increases at an exponential rate on an acceleration vs time graph, then do you think you could come up w/ an example that could match the following description in real life?(as you might know, getting in contact w/ real life engineer is not easy)

ex. if on an acceleration vs time graph, I see the graph y=x. That could match the description in real life as in a person who is peddling down the accelerator to increase the speed of the car, but since his speed on peddling down is increasing, the graph has some slope(it didn't have to y=x, just a linear rate graph)

If his rate were constant, I would get a horiz. line on the graph on some y value which would be the constant rate of acceleration. Do you follow me?

 

[matt grime]

Anyways, all I was saying is that if the function

what function. Explain yourself as if we know nothing about what you're talking about.

increases at an exponential rate on an acceleration vs time graph,

so, we've got a function that is increasing on some graph. What function, what graph? What does it even mean for something to 'increase at an exponential rate on a graph'? Nothing as far as I know.

then do you think you could come up w/ an example that could match the following description in real life?(as you might know, getting in contact w/ real life engineer is not easy)

really? not thought about posting on the engineering forum here?

ex. if on an acceleration vs time graph, I see the graph y=x. That could match the description in real life as in a person who is peddling down the accelerator to increase the speed of the car, but since his speed on peddling down is increasing, the graph has some slope(it didn't have to y=x, just a linear rate graph)

If his rate were constant, I would get a horiz. line on the graph on some y value which would be the constant rate of acceleration. Do you follow me?

So if the graph were y=f(x), then someone accelerating at the rate f(x) might do it, d'ya reckon? (Where x could be, what, time, displacement, christ knows what.)

 

[Swapnil]

Sorry Skhandelwal, I kinda diluted the difference between a line with a constant slope and a curve on an acceleration-time graph.
Let me start over again. Here is the part where I talk about no acceleration and then constant acceleration.

Say you are in an intergallactic spaceship which is taveling at a constant velocity (i.e. a flat horizontal line [y=0] on your acceleration-time graph). How do you know this? Because you are experiencing no force.
Now say the ship starts accelerating at a constant rate (i.e. still a flat horizontal line [but y=constant] in the accleration-time graph). How do you detect this? Because you are feeling a constant force (maybe a throbbing pain in your chest).

Now say the spaceship is going through a change in acceleration but the rate of change is constant (i.e. a slant line with some non-zero slope in the acceleration-time graph) How do you know? Because each instant of time you are experiencing more and more force (i.e. the pain in your chest would throbb more and more every second!). But that pain would inscrease at a constant rate (for example, the pain gets doubled every second).

Now say that the spaceship is going through a change in acceleration but that rate of change increases every instant (i.e. an increasing curve in the acceleration-time graph). How do you know this? Because each instead of time you be experiencing more and more force (thus more pain). But that pain would not increase at a constant rate but the rate of increase of pain would be more and more every second (for example, the pain gets doubled the first second, tripled the second second, and quadrupled the third second, etc).

That's basically how you can visualize the 4 aspects of acceleration: 1) no acceleration; 2) constant acceleration; 3) linear acceleration; 4) quadratic acceleration.

In summary, since acceleration is directly proportional to force and force is directly proportional to pain. Just think of acceleration-time graph as a pain-time graph which tells you how much and at what rate is your pain increasing/decreasing.

 

[Data]

Anyways, all I was saying is that if the function increases at an exponential rate on an acceleration vs time graph, then do you think you could come up w/ an example that could match the following description in real life?

The example that I gave is an example of a situation in which you'd have an exponentially decaying acceleration vs. time graph, if you make the common approximation that air resistance is linear in speed.

You can't have an situation with an exponentially increasing acceleration-time graph in the long run, because then eventually you'd get faster than the speed of light!

 

[Skhandelwal]

Would all the acceleration graph have the same tangential slope at the point I hit the speed of light or that would depends on the rate of acceleration and initial velocity?

Sorry Data, I posted just after you did, so I failed to read it.

 

[Data]

Nothing massive can be accelerated to the speed of light. All massless particles that we know of today travel at the speed of light, and there is no reason to believe that there are massless particles that travel slower than the speed of light. So that question doesn't make much sense.

Anyways, this should really be one of the physics forums.

 

[LeonhardEuler]

Nvm that, btw, what does an increasing curve(like a half of the positive quadratic) in an acceleration graph indicate?(I know technically but hoping you guys could give me a real life example.

If I'm understanding you, you are asking for a real life example of an object whose acceleration increases at a non-constant rate. An example would be this:
You have a fixed charge. There is another charge some distance away from it which is free to move. Suppose they are of opposite sign so the attract. The force is given by Coulomb's law:
$$F=\frac{kq_1q_2}{x^2}=m\frac{d^2x}{dt^2}$$
with x being the distance between them. The free charge will begin to move closer to the fixed one. As it does, the force, and therefore the acceleration (F=ma) grows. The acceleration will not grow at a linear rate. In fact it will approach infinity as the free charge approaces the fixed one (if they are point charges, not a realistic assumption).

 

[Skhandelwal]

I got it, thanks a lot.