Question

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 \(y(t)=y_{0} e^{k t},\) the relative growth rate \(R_{T}\) is constant for any \(T ;\) that is, choose any \(T\) and show that \(R_{T}\) is constant for all \(t\)

Short answer

Question: Show that the relative growth rate of the exponential function \(y(t) = y_0e^{kt}\) is constant for any value of \(t\). Answer: The relative growth rate, \(R_T\), of the exponential function is found to be \(R_T = e^{kT} - 1\), which is constant with respect to \(t\) as it depends only on \(T\) and \(k\).

Step-by-step solution

Calculate \(R_T\) for the exponential function \(y(t)\)

We are given the relative growth rate formula: $$R_{T}=\frac{f(t+T)-f(t)}{f(t)}$$ For the exponential function \(y(t) = y_0e^{kt}\), we can write the relative growth rate as: $$R_{T}=\frac{y_0e^{k(t+T)}-y_0e^{kt}}{y_0e^{kt}}$$

Simplify the expression to remove dependence on y_0

The \(y_0\) term appears in all terms of the fraction, so we can simplify the expression by dividing both the numerator and denominator by \(y_0\): $$R_{T}=\frac{e^{k(t+T)}-e^{kt}}{e^{kt}}$$

Simplify the exponential terms in the expression

Now, we notice that the term \(e^{kt}\) appears in both terms of the numerator and the denominator. We can simplify the expression by taking advantage of the properties of exponentials: $$ R_{T}=\frac{e^{kt}e^{kT}-e^{kt}}{e^{kt}} $$

Factor out the common term \(e^{kt}\) and simplify

Factor the common term \(e^{kt}\) out of the numerator, and then divide it out between the numerator and denominator to eliminate it: $$ R_{T}=\frac{e^{kt}(e^{kT}-1)}{e^{kt}} = e^{kT}-1 $$

Show that the relative growth rate is constant

Now that we have arrived at the expression for \(R_T\): $$ R_{T} = e^{kT}-1 $$ We can see that it is constant with respect to \(t\), as it only depends on \(T\) and \(k\). Thus, the relative growth rate \(R_T\) is constant for any \(T\) in the exponential function \(y(t) = y_0e^{kt}\).

Key concepts

Relative Growth Rate

The relative growth rate is an important concept when analyzing how a function grows over time. It measures how the function's value changes relative to its initial value over a specific time period. For any function \( f(t) \), the formula to find the relative growth rate \( R_T \) over a time interval \( T \) is given by: \[R_{T} = \frac{f(t+T) - f(t)}{f(t)}\]This formula tells us the proportionate change in \( f \) from \( t \) to \( t+T \). If \( R_T \) is constant, it indicates steady exponential growth without fluctuation over time.

• The numerator \( f(t+T) - f(t) \) represents the actual change in the function.
• The denominator \( f(t) \) standardizes this change, making it relative to the initial value.

For exponential functions, we often find that \( R_T \) remains consistent, illustrating predictable growth patterns.

Exponential Function

An exponential function is a powerful tool for modeling growth processes. It can be expressed in the form \( y(t) = y_0 e^{kt} \), where:

• \( y_0 \) is the initial value of the function.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.
• \( k \) is the growth rate constant, indicating how quickly the function grows.

Exponential functions are prevalent in real-world scenarios like population growth, financial investments, and radioactive decay.
As time \( t \) increases, the value of \( y(t) \) shifts significantly based on \( k \'s \) positivity or negativity. If \( k > 0 \), the function demonstrates positive growth, while \( k < 0 \) signifies decay.
This function is key to understanding consistent growth rates because its structure provides a constant relative growth rate over any period \( T \).

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. In proving that the relative growth rate \( R_T \) of our exponential function is constant, we relied on systematic simplifications and proper use of algebraic rules. Here’s how it was shown:

• Start with the formula for \( R_T \) using \( y(t) = y_0 e^{kt} \).
• Substitute \( y(t+T) = y_0 e^{k(t+T)} \) and simplify.


• The key step involves factoring out common terms and canceling them out, utilizing properties of exponents for simplification.

By step-by-step manipulation, the final result \( R_T = e^{kT} - 1 \) was derived, independent of \( t \), confirming constancy. Such processes are foundational in mathematics, ensuring that results are indisputable and clear.

Properties of Exponents

The properties of exponents are essential rules that dictate how expressions involving powers are simplified and manipulated. Exponential functions heavily rely on these properties due to their mathematical structure. Here are a few fundamental properties used in the solution:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{mn} \)

In the exercise, the property \( e^{kt} \times e^{kT} = e^{k(t+T)} \) was vital for simplifying the expression.
Understanding these properties allows us to break down complex expressions into manageable parts, making them easier to work with. Such simplifications are crucial in demonstrating proofs, like showing the constancy of the relative growth rate in exponential functions.