a function that gradually increses, and then picks up exp. rate

[computerages]

Hello!

 

I was wondering if someone can give me a functions that increases gradually in the beginning, and then picks up an exponential rate after some time. It would be much similar to an exponential function such as x^2, but it would increase gradually for a certain time in the beginning.

 

Thanks!

[Aeternus]

[math]x^2[/math] is not an exponential function or even a function exhibiting exponential growth.

 

See http://en.wikipedia.org/wiki/Exponential_growth .

 

I imagine the answer to your question is really going to depend on what exactly you want and what you mean by "in the beginning" and gradual growth. Can you go into a bit more detail about what it is you want to derive?

[Bignose]

I guess you don't want:

 

[math]f(t) = \begin{cases} e t^2 \text{ if } t <= 1,\\ e^t \text{ if } t >1 \end{cases}[/math]

 

Like Aeternus said, you probably need to provide more info to get better answers

[computerages]

please see the attachment to understand what i mean.. as u can see the graph suddently begins to exp. grow but in the begining just increases gradually (not exactly linearly though)....

 

I hope that makes sense.....

[Cap'n Refsmmat]

So you mean something like

 

[math]y = - \frac{1}{x-10}[/math]

[Aeternus]

If you don't mind it being symmetric about the y axis and only want strong growth not assymptotic like Capn's, then you might also consider something similar to what Bignose suggested -

 

[math]

y = e^{(\frac{x}{a})^b} - 1

[/math]

 

Where you can change [math]a[/math] to change where it begins growing very quickly and change [math]b[/math] to change how quickly it grows at that point and how gradually it grows before that.

 

To generalise what Capn said a bit, if you want assymptotic behaviour, similar to what he said you can do

 

[math]

y = \frac{1}{b \cdot x-b \cdot a}

[/math]

 

Where [math]a[/math] and [math]b[/math] have similar effects as above.

[YT2095]

doesn`t n! do that also?

[ajb]

doesn`t n! do that also?

 

For natural numbers yes. But looking at the plot given, he is looking for a function on the real numbers.

 

But you could use the Gamma function on [math]\mathbb{R}^{+}[/math] Try, [math]y[x] = \Gamma[x+1][/math]

Edited May 29, 2008 by ajb
multiple post merged

[Riogho]

2^x

[ajb]

2^x

 

Or indeed, [math] y[x] = a^{x} +b[/math] for [math] a > 1[/math] and [math]b \in \mathbb{R}[/math]

 

The rate of growth is [math] y'[x] = a^{x}\log[a][/math].