Err... what does it mean to increase (when something increases) at linear rate?

Err... what does it mean to increase (when something increases) at linear rate?
If you graph the dependent variable agianst the independent variable it will be a straight line.
What about an exponential rate?
Linear rate: the quantity changes by some fixed absolute amount per unit time. The graph looks like a straight line going up and to the right (if the amount is positive) or down and to the right (if the amount is negative):
(amount at next time step) = (amount at previous time step) + (some constant)
Exponential rate: the quantity changes by some fixed factor per unit time. The graph looks like a curve which increases ever more rapidly (if the factor is greater than 1) or decays to zero (if the factor is less than one):
(amount at next time step) = (amount at previous time step) * (some constant)
Last edited by btr; 05282014 at 07:16 PM.
Ok. Thank you. Now: Is it possible/correct to say "as x increases at a linear rate, y increases, in tandem, at an exponential rate"?
EDIT: Wait a second.... If I say y=x^2 doesn't that mean x is increasing linearly while y is increasing exponentially?
Um, if x is increasing at a linear rate, with what is it increasing with respect to? You are introducing a third variable here. With respect to your EDIT why would x be increasing at all?
I'm sorry, I'm a real mess with this stuff. But hm, can't x just increase linearly by itself? I don't understand what you mean by "with what is it increasing with respect to?"
If you say it increases linearly you are saying that it increases in proportion to something else, it might be time, it might be distance etc. for example if my hair grows linearly with respect to time, it grows at the same rate no whatever which month it is. If my hair grows linearly with respect to temperature, it grows faster when I am on holiday. It would make no sense to say my hair just grows linearly (in a straight line) without it being with respect to another variable.
Oh, I get it! Linear is a graph of a y=mx+b tipe equation! I'll get a straight line because y will increase at the same rate as x increases, right?
If y = mx, then y increases linearly with x. I wouldn't say "at the same rate", though, because if you increase x by 1 unit then y increases by m units.
In your earlier example, y = x^{2}, y is neither linearly nor exponentially increasing with x. It is quadratically increasing. For exponential growth you need something like y = a^{x}, where a is some constant.
Thanks! I think I'm beginning to really get it. When I say something increases linearly I'm implying a relation with some other value.
Then it makes no sense to say that as something increases lineraly, something else increases, in tandem, exponentially does it? Because that would imply 4 variables and I give only 2, right? I have to say what that something is increasing linearly to, and the same thing for the other something.
If you want to talk about two things (say, x and y) increasing together in some sense, you probably have in mind that both quantities depend on some third variable (call it t, say; perhaps t represents the amount of time elapsed or something).
For example, perhaps x = 3t and y = 2^{t}, in which case it makes sense to say that x increases linearly with t, while y increases exponentially with t.
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