Calculate exponential growth when rate r is changing

Vodkacannon

The formula f(t) = I(1 + r)^t gives the value of something after it has grown for t years, at rate r, with an initial value I.

What if the growth rate r is continuously increasing?
What is the formula now?
Does it relate somehow to x^x?

haruspex

It depends on how it increases with t. Is it linear with t? Geometric? Logarithmic? ...

HallsofIvy

It is, of course, [itex]f(t)= I(1+ r(t))^t[/itex]. And, yes, that has similar properties to [itex]x^x[/itex], especially if r(t) is linear.

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Vodkacannon	Calculate exponential growth when rate r is changing Tweet The formula f(t) = I(1 + r)^t gives the value of something after it has grown for t years, at rate r, with an initial value I. What if the growth rate r is continuously increasing? What is the formula now? Does it relate somehow to x^x?

haruspex Recognitions: Homework Help Science Advisor	It depends on how it increases with t. Is it linear with t? Geometric? Logarithmic? ...

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	It is, of course, [itex]f(t)= I(1+ r(t))^t[/itex]. And, yes, that has similar properties to [itex]x^x[/itex], especially if r(t) is linear.