Question

General relative grow th rates Define the relative growth rate of the function \(f\) over the time interval \(T\) to be the relative change in f over an interval of length \(T\) : $$ R_{T}=\frac{f(t+T)-f(t)}{f(t)} $$ Show that for the exponential function \(y(t)=y_{0} e^{t},\) the relative growth rate \(R_{T}\), for fixed \(T\), is constant for all \(t\).

Short answer

Answer: The relative growth rate of the exponential function \(y(t) = y_0e^t\) is \(R_{T} = e^T - 1\), and it is constant for all \(t\).

Step-by-step solution

Substitute the exponential function into the formula for \(R_{T}\)

We are given the exponential function \(y(t) = y_0e^t\) and the formula for the relative growth rate \(R_{T}\): $$ R_{T} = \frac{f(t+T)-f(t)}{f(t)} $$ Substitute \(y(t)\) into the formula for \(R_{T}\): $$ R_{T} = \frac{y_0e^{t+T} - y_0e^t}{y_0e^t} $$

Simplify the expression

Now, we will simplify the expression for \(R_{T}\). First, we can factor out \(y_0e^t\) from the numerator: $$ R_{T} = \frac{y_0e^t(e^T - 1)}{y_0e^t} $$ Now, we can cancel the common factor \(y_0e^t\) in both the numerator and the denominator: $$ R_{T} = e^T - 1 $$

Show that \(R_{T}\) is constant for all \(t\)

From Step 2, we have found the expression for \(R_{T}\): $$ R_{T} = e^T - 1 $$ Notice that this expression does not contain the variable \(t\). Therefore, \(R_{T}\) is constant and does not depend on \(t\). Consequently, we have shown that the relative growth rate \(R_{T}\) of the exponential function \(y(t) = y_0e^t\) is constant for all \(t\).

Key concepts

Relative Growth Rate

The concept of a relative growth rate is central when examining changes over time, particularly within exponential functions. The relative growth rate, symbolized as \(R_T\), helps us understand how much a function grows relative to its size over a given time interval \(T\).

• This rate is calculated by analyzing the relative change in the function over that interval.
• The formula \(R_T = \frac{f(t+T) - f(t)}{f(t)}\) quantifies this relationship, emphasizing proportional change rather than absolute change.

A useful application of the relative growth rate is to measure how exponential functions evolve. In many real-life scenarios involving growth, such as populations or investments, constants like \(T\) are fixed time durations, and the relative growth rate offers insight into how quickly scenarios expand within those constraints. Consequently, understanding how to calculate and interpret \(R_T\) is vital for analyzing any function's behavior over time.

Constant Function

Although the term "constant function" typically refers to a function that always returns the same value, in the context of growth rates, it has a different implication. Here, a function being 'constant' means its rate of change is stable across any interval of time \(t\).

• In the formula derived from exponential functions, \(R_T = e^T - 1\), notice how \(t\) does not appear.
• Thus, the rate of change is unaffected by where we start measuring (any \(t\)), but is instead tied to \(T\), showing that the function's relative growth rate is consistently the same.

This constancy means that, for the exponential function, the proportionate increase remains the same over equal intervals, leading to predictable and uniform growth patterns. It's a key feature that defines exponential growth as fundamentally different from linear growth, where the amount added changes with the interval.

Exponential Growth

Exponential growth describes a situation where the rate of growth is proportional to the current size of the quantity. It's a common phenomenon in various situations, including population growth, compound interest in finance, and certain viral processes.

• The mathematical model, \(y(t) = y_0 e^t\), hinges on the natural exponential function \(e\), which provides a base for continuous growth.
• This equation shows how the function grows multiplicatively, with each step involving a multiplication by \(e\).

The beauty of exponential growth is its simplicity in mathematical modeling, coupled with its profound implications. With a constant \(R_T\), exponential growth can rapidly transform small values into large ones, given enough time. Its exponential nature means doubling periods shorten as the quantity grows, explaining why such growth can "explode" in a practical sense.