Equivalent growth functions The same exponential growth...

Key Concepts

Exponential Growth Models

Exponential growth models describe processes in which the rate of change of a quantity is proportional to its current value, leading to a pattern of rapid increase. These models can be expressed in different forms that are mathematically equivalent, allowing conversion between representations such as continuous growth (using the natural exponential function), discrete growth (using a fixed growth factor), and formulations involving doubling time.

Continuous and Discrete Growth Parameters

In the continuous growth model, the function is expressed as y(t) = y? e^(kt), where k is the constant continuous growth rate. In the discrete growth model, the function takes the form y(t) = y? (1 + r)^t, where r is the periodic growth rate. The relationship between these parameters is established through logarithmic transformation, specifically k = ln(1 + r), which allows conversion from one parameterization to the other.

Doubling Time

Doubling time, denoted as T?, is the time required for a quantity undergoing exponential growth to double in value. It provides an intuitive measure of growth speed and is mathematically related to the continuous growth rate by the formula T? = ln(2) / k. This relationship can also be used to connect the doubling time with the discrete growth rate through the transformations established between k and r.

Logarithmic Transformations

Logarithmic transformations are essential in converting between different exponential growth formulations. By taking the natural logarithm of both sides of an exponential equation, one can derive relationships such as k = ln(1 + r) and T? = ln(2) / k, which facilitate the conversion between continuous growth rates, discrete growth rates, and doubling time.

Transcript

00:01 Looking at three different ways to write our exponential growth function using three different constants, we're going to try to find each of these constants relationships to each other.

00:11 So we're going to find k in terms of r, r in terms of doubling time, and doubling time in terms of k.

00:21 This is going to be similar to a system of equations.

00:23 We're going to take two at a time and then simplify so we can get one thing on the left of the equation and everything else in terms of the other on the right side.

00:31 So we'll start with k of r.

00:34 So k in terms of r, we want k alone on the left side.

00:38 The two equations we'll pull for this are y of t is equal to y not e to the kt because we have k, and y of t is equal to y not times 1 plus r to the t because we have r in here.

00:56 We'll set these equal to each other because they are.

01:00 We'll get y0 e to the k t is equal to y not times 1 plus r to the t.

01:07 Our initial values will cancel.

01:10 And then we'll get e to the k t is equal to 1 plus r to the t.

01:16 Now we have an e with an exponent and our variable in the exponent here.

01:22 We'll take the natural log of both sides.

01:25 Natural log of e is 1.

01:27 We get k t is equal to t comes out front t ln 1 plus r now our t's here cancel we have them on either side of the equation so now we have k alone which is what we wanted in terms of r ln 1 plus r so that's our function of k terms of r which you could also write as this k of r now we'll look at r in terms of doubling time, which i will also call t2.

02:05 So our two equations here, we have y of t is equal to y not times 1 plus r to the t, and y of t is equal to y of t is equal to y not times 2 to the t over t 2.

02:21 We'll set them equal to each other so we can find r and t2.

02:26 We want r alone on the left side of the equation.

02:30 So we have y not times 1 plus r to the t equal to why not times 2 to the t over t 2.

02:41 We'll cancel our y -nots.

02:43 We'll get 1 plus r to the t equal to 2 to the t over t 2.

02:56 And we're going to rewrite this right side as 2 to the 1 over t 2.

03:02 T2 all to the t.

03:05 And now we have an exponent of t on both sides.

03:09 If we took the natural log and then put it to e again, they would cancel.

03:13 So this is a log rule that we also should just know.

03:16 We can cancel those two t's in the exponent...

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