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Math Help - exponential growth

  1. #1
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    exponential growth

    I do not understand how to do either of these problems but i need to know how to do them in order to do my homework...could anyone please take me through these, i dont really understand. Thank you very much
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    As far as I understand it,

    P(t) = 100e^{0.2197t}
    is the same as
    P(t) = 100 * 10 ^ {0.2197 * t} [100 times 10, to the power of {0.2197 times t(time)} ]

    a) asks you to use a graphing utility (such as a graphical calculator, or a computer application) to plot this graph -- good news that you do not need to do it manually on graph paper!!
    though, not so good news if you do not have / do not know how to use one

    b) then asks you to pick off points 0, 5, and 10 - if the value of time is in seconds, then it would be asking you for the values at 0seconds, 5seconds and 10seconds

    and finally, c) asks you to show algebraic proof for your answers in b) .. in other words, use the formula with the values of t being 0, 5, and 10


    68 is a very similar question, however, it specifies much more detail




    btw, I'm not a perfect mathematician, so I stand corrected if I have made any mistakes here; this is just my interpretation of the questions.
    Last edited by kwah; August 19th 2007 at 03:34 PM.
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    thank you that makes a lot more sense the way you put that, i think i got it now! i will post again if i have any trouble
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    youre welcome - and I just noticed that t is time in HOURS.

    I recommend that you do some research into nth terms - even if it is just for fun
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    when i try to put these problems into the calculator, it does not graph them? maybe i am doing something wrong...i go to y= and put in the equation
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    Quote Originally Posted by kwah View Post
    As far as I understand it,

    P(t) = 100e0.2197t
    is the same as
    P(t) = 100 * 10 ^ (0.2197 * t) [100 times 10, to the power of (0.2197 times t(time) ]
    What??

    This is what you said:
    P(t) = 100 \cdot e^{0.2197t} = 100 \cdot 10^{0.2197t}

    This is not correct!

    First of all, you have the equation for P(t), why change it to base 10? None of the numbers work out to be nice, so you have no advantage in changing the base: your calculator is doing the heavy work anyway.

    Second,
    P(t) = 100 \cdot e^{0.2197t} = 100 \cdot 10^{0.2197t/ln(10)}
    if you really do want to change the base.

    -Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by aikenfan View Post
    when i try to put these problems into the calculator, it does not graph them? maybe i am doing something wrong...i go to y= and put in the equation
    There might be two problems. The first is simple and, depending on your type of calculator, may be irrelevant: make sure you are using the variable "x" and not "t."

    The more likely problem is one of scale. Either go to your "Zoom" feature (F2 on my calculator) and select "Zoom Fit" or go to your "Window" screen and reset the max y value as 900 or 1000.

    -Dan
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    Quote Originally Posted by topsquark View Post
    What??

    This is what you said:
    P(t) = 100 \cdot e^{0.2197t} = 100 \cdot 10^{0.2197t}

    This is not correct!
    P(t) = 100 \cdot e^{0.2197t}

    To me, that means 100 [decimal point] e ^ 0.2197t
    That is NOT what I said.

    I was using * to represent the multiplication symbol, since x looks like the algebraic x.
    Am I mistaken in doing this? ie, is this where we differ?


    Quote Originally Posted by topsquark View Post
    First of all, you have the equation for P(t), why change it to base 10? None of the numbers work out to be nice, so you have no advantage in changing the base: your calculator is doing the heavy work anyway.
    I accept that there is no need to change it, however, I was trying to explain the meaning of e; though, it appears that I've made a mistake - is this not correct? (where X represents the multiplication symbol)

    1000 = 1.0 X 10^{3} = 1e3


    I know I'm still learning, but I've been using the e symbol in standard form calculations to represent x10^ without any problems, so I assume this is relevant? My sincerest apologies if it turns out otherwise?


    I am relatively new to this forum so am unaware of the [ma th]...[/ma th] codes, but I have since edited my previous post to correctly display the powers
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by kwah View Post
    P(t) = 100 \cdot e^{0.2197t}

    To me, that means 100 [decimal point] e ^ 0.2197t
    That is NOT what I said.

    I was using * to represent the multiplication symbol, since x looks like the algebraic x.
    Am I mistaken in doing this? ie, is this where we differ?



    I accept that there is no need to change it, however, I was trying to explain the meaning of e; though, it appears that I've made a mistake - is this not correct? (where X represents the multiplication symbol)

    1000 = 1.0 X 10^{3} = 1e3


    I know I'm still learning, but I've been using the e symbol in standard form calculations to represent x10^ without any problems, so I assume this is relevant? My sincerest apologies if it turns out otherwise?


    I am relatively new to this forum so am unaware of the [ma th]...[/ma th] codes, but I have since edited my previous post to correctly display the powers
    Ahhhhhhhhh! I see the problem.

    When looking at exponential functions e = 2.718281828459..., so the function is:
    P(t) = 100 * (2.718281828459...)^{0.2197t}

    Also, I was using \cdot as a multiplication symbol to separate the "100" and the base I was using:
    100 \cdot e^{\text{whatever}}

    This is distinguished from a decimal point:
    100.00 \cdot e^{\text{whatever}}
    by its position. Though I grant there was no decimal point to compare it to, so I can see why the error might be made.

    Edit: Upon some reflection, I should mention this. As I understand it, some Europeans (I think) use 100 \cdot 23 to represent 100.23. All I can say is that my way is better because I said so!

    Edit II: Somewhere on your calculator should be an "e^" or an " e^x" button. That's how you input this into the calculator.

    -Dan
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    Apologies, my mistake.

    Is \cdot used to represent multiplication? If so, why?
    Like I said, I'm new here and I'm still learning - but I've never seen this used before?

    I just noticed in the topic review section, you have edited your post. Yes, I'm European (UK) and have been taught that all dots are decimal points, and I have never come across anything different.
    Also, all commas ( , ) are used to mark thousands - eg, 1,000,000.00


    Quote Originally Posted by topsquark View Post
    When looking at exponential functions e = 2.718281828459..., so the function is:
    P(t) = 100 * (2.718281828459...)^{0.2197t}

    Edit II: Somewhere on your calculator should be an "e^" or an " e^x" button. That's how you input this into the calculator.
    This is confusing. Before your second edit, I took e to represent 2.718281828459... similarly to \pi representing 3.142..
    However, your Edit II has confused me somewhat. lol
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  11. #11
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    Quote Originally Posted by kwah View Post
    Apologies, my mistake.

    Is \cdot used to represent multiplication? If so, why?
    Like I said, I'm new here and I'm still learning - but I've never seen this used before?

    I just noticed in the topic review section, you have edited your post. Yes, I'm European (UK) and have been taught that all dots are decimal points, and I have never come across anything different.
    Also, all commas ( , ) are used to mark thousands - eg, 1,000,000.00




    This is confusing. Before your second edit, I took e to represent 2.718281828459... similarly to \pi representing 3.142..
    However, your Edit II has confused me somewhat. lol
    To get a value for e^2 punch " e^x 2" enter into your calculator. (You should get an answer of 7.38906)

    The \cdot is a LaTeX version of the multiplication operator. I use it when I feel confusion might arise if I just "mash" the two factors together. (For example 3 times 6 can be written as (3)(6), but it is shorter to type 3 \cdot 6. Many on the forum here use this convention.)

    -Dan
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    Upon closer inspection, I see that we have been talking about different things.

    It seems e (as in e^{x}) and E (as in EXP) are two different things.

    Yes, you are right about e^1 having a value of 2.718281828...

    BUT, the Capital E on my calculator is marked as EXP on the keypad - I assume this to mean exponential, and has a value of x10^ as I initially thought.
    This is causing further confusion in my mind because I accepted the difference between e^{x} and E; however, it appears that E stands for EXP (exponential?) which this topic is about .. ?


    LaTeX version of the multiplication operator?? Could you please provide a reference for these LaTeX variations? I feel awkward asking you to explain all these topic-irrelevant questions.


    I don't know whether to apologise and cut this discussion short because I've hijacked this thread with my own confusion, or whether to continue questioning?
    Sorry again.
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  13. #13
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    Quote Originally Posted by kwah View Post
    Upon closer inspection, I see that we have been talking about different things.

    It seems e (as in e^{x}) and E (as in EXP) are two different things.

    Yes, you are right about e^1 having a value of 2.718281828...

    BUT, the Capital E on my calculator is marked as EXP on the keypad - I assume this to mean exponential, and has a value of x10^ as I initially thought.
    This is causing further confusion in my mind because I accepted the difference between e^{x} and E; however, it appears that E stands for EXP (exponential?) which this topic is about .. ?


    LaTeX version of the multiplication operator?? Could you please provide a reference for these LaTeX variations? I feel awkward asking you to explain all these topic-irrelevant questions.


    I don't know whether to apologise and cut this discussion short because I've hijacked this thread with my own confusion, or whether to continue questioning?
    Sorry again.
    Hello,

    1. have a look here: http://www.mathhelpforum.com/math-he...-tutorial.html

    2. You probably use a casio calculator(?). then you find the e^x-function at \boxed{\text{SHIFT}}\boxed{\text{ln}}

    3. In quite a lot (continental) european countries the comma is used as the separator between integer and fraction. The point is used to separate multiples of 10^3.
    For example the number 120,345,789.12345 is written in Germany (or Austria, Switzerland,...) as 120.345.789,12345
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by kwah View Post
    Upon closer inspection, I see that we have been talking about different things.

    It seems e (as in e^{x}) and E (as in EXP) are two different things.

    Yes, you are right about e^1 having a value of 2.718281828...

    BUT, the Capital E on my calculator is marked as EXP on the keypad - I assume this to mean exponential, and has a value of x10^ as I initially thought.
    This is causing further confusion in my mind because I accepted the difference between e^{x} and E; however, it appears that E stands for EXP (exponential?) which this topic is about .. ?

    I don't know whether to apologise and cut this discussion short because I've hijacked this thread with my own confusion, or whether to continue questioning?
    Sorry again.
    First of all, always feel free to ask questions!

    Yes, the base e that I have been talking about is the same as EXP on your calculator. I had momentarily forgotten that abbreviation.

    However, beware of making the mistake that the way your calculator writes things out is the same as the way a Mathematician would write them out. Only if the argument is very ugly have I seen something like
    P(t) = 100 * EXP( \text{whatever} )
    rarely in a case where the argument is as simple as this one.

    Another example of this is that I have never seen anyone write out 1000 as 1e3, except on homework solutions. A Mathematician would either write this simply as 1000 or would write something like 1 \times 10^3, using so-called "scientific notation."

    -Dan
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    Quote Originally Posted by topsquark View Post
    First of all, always feel free to ask questions!

    Yes, the base e that I have been talking about is the same as EXP on your calculator. I had momentarily forgotten that abbreviation.

    However, beware of making the mistake that the way your calculator writes things out is the same as the way a Mathematician would write them out. Only if the argument is very ugly have I seen something like
    P(t) = 100 * EXP( \text{whatever} )
    rarely in a case where the argument is as simple as this one.

    Another example of this is that I have never seen anyone write out 1000 as 1e3, except on homework solutions. A Mathematician would either write this simply as 1000 or would write something like 1 \times 10^3, using so-called "scientific notation."

    -Dan
    Right ... let me get this straight - when I use e^{x} on my casio calculator, it shows as:
    1e3 = 20.08553692
    However, if I use the EXP button, it shows on my display as:
    1E3 = 1,000

    Do you see why I keep asking about the different meanings?
    On the calculator display:
    - a lower-case e is equal to 0.20855... ,
    - but the UPPER-case E is a short form of x10^

    However, you said "I have never seen anyone write out 1000 as 1e3, except on homework solutions", suggesting that the lower-case example as you put it, is equal to my calculator's UPPER-case e .. ?


    And yes, I agree that 1000 would never really be written as 1e3 (or 1E3, whichever is correct), it is the most simple example I could think of -- I could see no need complicate matters further ..
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