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July 19th, 2017, 07:04 AM   #1
EDL
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Question on exponential functions

I am taking an online college algebra class and we just started exponential and logarithmic functions.

Our instructor is always urging us to think about the math and not just plug numbers into a formula and spit out answers, so I am and I have questions!

If I were to plot a basic exponential that illustrates the growth rate of computing power over time, let's say $\displaystyle y=3^x$, then the x axis is the time line (and asymptote) and the y axis is the power of computers.

Since x is an asymptote, then no matter which year I pick, y will always have a positive value.

How can this be? If look at a point in time 300 years ago, I get a positive value, even when computers didn't even exist.

There is obviously a flaw in my understanding, or there is something else to this that I don't know. I'm guessing both. What am I missing?
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July 19th, 2017, 07:42 AM   #2
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In terms of time, -300 (which I'm assuming is what you used) doesn't make sense - it's
below 0 so there is really no such time. For the function you gave you might want to
restrict the domain of $x$ to $x\in[0,\infty)$.

A part of mathematical modeling is knowing when the model applies and when it doesn't.
Then we apply our model where it makes sense. Applying the domain of $x$ that I gave
above suggests that computing power grows without bound, but does that really apply?
Is there an upper bound and if there is, what is it?
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July 19th, 2017, 09:31 AM   #3
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Originally Posted by EDL View Post
I am taking an online college algebra class and we just started exponential and logarithmic functions.

Our instructor is always urging us to think about the math and not just plug numbers into a formula and spit out answers, so I am and I have questions!

If I were to plot a basic exponential that illustrates the growth rate of computing power over time, let's say $\displaystyle y=3^x$, then the x axis is the time line (and asymptote) and the y axis is the power of computers.

Since x is an asymptote, then no matter which year I pick, y will always have a positive value.

How can this be? If look at a point in time 300 years ago, I get a positive value, even when computers didn't even exist.

There is obviously a flaw in my understanding, or there is something else to this that I don't know. I'm guessing both. What am I missing?
Things is called extrapolation, and is very dangerous. Basically, you have a very accurate mathematical model which you have shown to hold. But any model has restrictions. As such, when you use the model you need to be aware of these restrictions and not simply try to assume the model holds everywhere.

In your case, the model is probably accurate over a 50 years period, but not before.

When arguing with creationists, they will very often resort to some kind of unwarranted extrapolation. For example, they will say that carbon dating gives inaccurate results, which is true, but they apply carbon data where it is not allowed.

Somewhat more fun I had with a creationist is that they said that SAT scores rised over time. In the past, the SAT scores were a lot less than now. So they fit it in a nice linear equation and found that 10000 years ago, the SAT score had to be negative, which is absurd. The error is that the nice linear equation holds for a certain time period. Outside that, the model may be violated and give nonsensical results, or it might give sensical results which are false.
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July 19th, 2017, 12:15 PM   #4
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Quote:
Originally Posted by EDL View Post
Our instructor is always urging us to think about the math and not just plug numbers into a formula and spit out answers, so I am and I have questions!
Off topic... But that is the best way to understand and get the most out of mathematics. Good on you!
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July 19th, 2017, 05:54 PM   #5
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Originally Posted by greg1313 View Post
In terms of time, -300 (which I'm assuming is what you used) doesn't make sense - it's
below 0 so there is really no such time. For the function you gave you might want to
restrict the domain of $x$ to $x\in[0,\infty)$.

A part of mathematical modeling is knowing when the model applies and when it doesn't.
Then we apply our model where it makes sense. Applying the domain of $x$ that I gave
above suggests that computing power grows without bound, but does that really apply?
Is there an upper bound and if there is, what is it?
That explanation actually makes it perfectly clear for me now, so thank you.

If our domain starts at 0, the y value is 1 (any number to the 0 exponent is always 1). This makes sense because 0 represents the moment in time the first computer was created and it has a value of 1 because there is nothing to compare it to and no more time has elapsed or increases in computing power derived at that point. In essence, (0,1) is the computer's birthday!

As for boundless growth, I don't have a clue. Is there a limit? Engineers seem to keep finding ways for increasing computing power, even after we've been hearing for quite a number of years now that we're approaching the limits of silicon.

If quantum computing ever gets figured out and producible, that'll certainly boost the curve a fair bit. I suspect at some point they'll do it. If there is one thing I have near absolute faith in, is man's ability to increase technology.
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July 19th, 2017, 09:30 PM   #6
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That explanation actually makes it perfectly clear for me now, so thank you.

If our domain starts at 0, the y value is 1 (any number to the 0 exponent is always 1). This makes sense because 0 represents the moment in time the first computer was created and it has a value of 1 because there is nothing to compare it to and no more time has elapsed or increases in computing power derived at that point. In essence, (0,1) is the computer's birthday!

As for boundless growth, I don't have a clue. Is there a limit? Engineers seem to keep finding ways for increasing computing power, even after we've been hearing for quite a number of years now that we're approaching the limits of silicon.

If quantum computing ever gets figured out and producible, that'll certainly boost the curve a fair bit. I suspect at some point they'll do it. If there is one thing I have near absolute faith in, is man's ability to increase technology.

Of course there is a limit. You can't have more computers than particles in the universe for example. Google how many particles there are in the (observable) universe, and find out in how many years your mathematical model reaches this!
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July 20th, 2017, 10:51 AM   #7
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Of course, but it's also not like we're just creating more and more computers without ever retiring (recycling) the parts and materials from old ones.
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July 20th, 2017, 10:59 AM   #8
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Things is called extrapolation, and is very dangerous. Basically, you have a very accurate mathematical model which you have shown to hold. But any model has restrictions. As such, when you use the model you need to be aware of these restrictions and not simply try to assume the model holds everywhere.

In your case, the model is probably accurate over a 50 years period, but not before.

When arguing with creationists, they will very often resort to some kind of unwarranted extrapolation. For example, they will say that carbon dating gives inaccurate results, which is true, but they apply carbon data where it is not allowed.

Somewhat more fun I had with a creationist is that they said that SAT scores rised over time. In the past, the SAT scores were a lot less than now. So they fit it in a nice linear equation and found that 10000 years ago, the SAT score had to be negative, which is absurd. The error is that the nice linear equation holds for a certain time period. Outside that, the model may be violated and give nonsensical results, or it might give sensical results which are false.
Heh, well, if we're going to discuss creationists, then we're kinda getting away from math in general, but I'll just say that I believe science and religion tie together just fine. There is a scientific explanation for everything.

My personal view is that the bible is to humans as the owners manual is to a new car. It explains how to operate the car and the various maintenance periods, but it does not in any way define or describe in great detail HOW the car was created or built, nor anything about the nuts and bolts in general; that would be the shop manual that would describe that.

So, I see science as just attempting to discover the nuts and bolts of the universe...our attempt to write the shop manual, as it were.

Some people also assume that science is "in the business" of disproving God, and I completely disagree. I don't believe that is their goal at all. Of course, there will be questions and as science advances attempts will be made to make those connections, but it's not what scientists set out to do.
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July 20th, 2017, 11:57 AM   #9
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My personal view . . .
That's fine; however, the "official" line of many faiths is that you can't have a personal view that conflicts with any of various key tenets of the faith. If they didn't have such a line, it would make it meaningless for you to say that you adhere to such a faith, as most people wouldn't know your personal view and hence wouldn't understand what your adherence meant. Also, of course, the personal views of most people change from time to time on at least some issues, so whatever you currently think is right you might consider to be wrong a bit later.
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July 20th, 2017, 12:02 PM   #10
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Heh, well, if we're going to discuss creationists, then we're kinda getting away from math in general, but I'll just say that I believe science and religion tie together just fine. There is a scientific explanation for everything.

My personal view is that the bible is to humans as the owners manual is to a new car. It explains how to operate the car and the various maintenance periods, but it does not in any way define or describe in great detail HOW the car was created or built, nor anything about the nuts and bolts in general; that would be the shop manual that would describe that.

So, I see science as just attempting to discover the nuts and bolts of the universe...our attempt to write the shop manual, as it were.

Some people also assume that science is "in the business" of disproving God, and I completely disagree. I don't believe that is their goal at all. Of course, there will be questions and as science advances attempts will be made to make those connections, but it's not what scientists set out to do.
The existence of God has nothing to do with creationism really. In the sense that one can believe in God and also accept the theory of evolution. The theory of evolution is an established scientific fact. So creationism is provably wrong, unless you're going to jump through an amazing amount of weird hoops. The existence of God is not at all affected by the theory of evolution. See for example the Catholic church which accepted the theory of evolution as truth many years ago.
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