Problem 1 - Calculus Early Transcendentals

Question

In terms of relative growth rate, what is the defining property of exponential growth?

Step-by-Step Solution

Answer

Answer: The defining property of exponential growth in terms of relative growth rate is that the relative growth rate is constant and equal to the growth rate constant in the exponential growth formula, regardless of the population size or time.

Step 1: Defining Relative Growth Rate

Relative growth rate is the rate at which a population, quantity, or value increases or decreases with respect to its current size. Mathematically, it can be represented as the ratio of a function's derivative to the function itself. If we have a function describing population growth, P(t), then the relative growth rate (R) can be given as: R(t) = \frac{P'(t)}{P(t)}

Step 2: Understanding Exponential Growth

Exponential growth describes a process where the growth rate depends only on the current size of the population or quantity. In other words, the growth is proportional to the current value. A general formula for exponential growth can be given as: P(t) = P_0 * e^{rt} Where P(t) represents the population at time t, P_0 is the initial population, r is the growth rate constant, and e is the base of the natural logarithm (approximately 2.71828).

Step 3: Defining Property of Exponential Growth

Now, we need to find the relative growth rate of the exponential growth formula, so we can determine the defining property. We have P(t) = P_0 * e^{rt}, and to find P'(t), we'll differentiate with respect to time t: P'(t) = P_0 * r * e^{rt} Now we can find the relative growth rate R(t) by dividing P'(t) by P(t): R(t) = \frac{P'(t)}{P(t)} = \frac{P_0 * r * e^{rt}}{P_0 * e^{rt}} The P_0 and e^{rt} terms cancel out, leaving us with: R(t) = r

Step 4: Conclusion

The defining property of exponential growth in terms of relative growth rate is that the relative growth rate (R) is constant and equal to the growth rate constant (r) in the exponential growth formula, regardless of the population size or time.