Question

Multiple Choice In which of the following models is \(d y / d t\) directly proportional to \(y\) ? $$\begin{aligned} \text { I. } y &=e^{k t}+C \\ \text { II. } & y=C e^{k t} \\\ \text { III. } & y=28^{k t} \end{aligned}$$ (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III

Short answer

The correct answer is (B) II only.

Step-by-step solution

Differentiate Model I

Differentiate \(y = e^{kt} + C\) with respect to \(t\). Using the chain rule, the derivative of a constant \(C\) is zero, and the derivative of \(e^{kt}\) is \(ke^{kt}\). So, \(dy/dt = ke^{kt} + 0\). Here, \(dy/dt\) is not directly proportional to \(y\), because multiplying \(y\) by a constant \(k\) does not yield the derivative.

Differentiate Model II

Differentiate \(y = Ce^{kt}\) with respect to \(t\). Using the chain rule and the constant rule, the derivative is \(dy/dt = Cke^{kt}\). Here, \(dy/dt\) is directly proportional to \(y\), because multiplying \(y\) by a constant \(k\) yields the derivative.

Differentiate Model III

Differentiate \(y = 28^{kt}\) with respect to \(t\). The basic rule of differentiation for exponential functions results in \(dy/dt = k \cdot \ln(28) \cdot 28^{kt}\). Here, \(dy/dt\) is not directly proportional to \(y\), because multiplying \(y\) by a constant \(k\) does not yield the derivative.

Key concepts

Exponential Growth Models

Exponential growth models describe how quantities increase over time at a rate that is proportional to their current value. Such models take the form of the equation
\( y(t) = C e^{kt} \),
where \( y(t) \) represents the quantity at time \( t \), \( C \) is a constant representing the initial amount, \( e \) is the base of natural logarithms, and \( k \) is the growth rate constant. If \( k > 0 \), the model describes exponential growth, and if \( k < 0 \), it represents exponential decay.

In such models, if you calculate the derivative of \( y(t) \) with respect to time \( t \) and the result is proportional to the function \( y(t) \) itself, you are observing what's called 'proportional growth.' To determine whether there is proportional growth, we look for a situation where \( \frac{dy}{dt} = ky(t) \) for some constant \( k \). This means that the rate of change of the quantity is proportional to the quantity itself. In the context of our original exercise, only Model II exhibits this characteristic.

Chain Rule Differentiation

The chain rule is a fundamental method in differential calculus used to find derivatives of composite functions. When we have a function that is formed by the composition of two or more functions, the derivative of this composite function is found by multiplying the derivative of the outer function by the derivative of the inner function.

For instance, if we have a function \( y = h(g(t)) \) and we want to differentiate \( y \) with respect to \( t \) using the chain rule, we would do the following:
\( \frac{dy}{dt} = \frac{dh}{dg} \cdot \frac{dg}{dt} \).
This formula is crucial when dealing with exponential functions where the exponent itself is a function of \( t \) as seen in the textbook problem. It allows us to handle the differentiation of functions like \( e^{kt} \), which are common in many fields including physics, biology, and economics. The step-by-step solutions for Models I and II illustrate the use of the chain rule where the differentiation of the exponential element of the models was necessary.

Proportional Relationships in Calculus

Proportional relationships in calculus refer to the direct relationship between two quantities where one quantity is a constant multiple of the other. This is succinctly described by the equation \( y = kx \) for some constant \( k \) and variable \( x \). In the context of derivatives, this means the rate of change of one variable is directly proportional to the variable itself.

When it comes to exponential growth, a proportional relationship indicates a particularly simple and elegant behavior: the rate of change of the system at any time is proportional to the value of the system at that time. This relationship is a characteristic signature of exponential growth or decay processes, as observed in Model II of our original exercise. Here, because the derivative \( \frac{dy}{dt} \) is a constant multiple of \( y \) itself, we can say that \( y \) (the output of the function) changes at a rate proportional to its current value. Understanding this principle helps to comprehend a wide range of natural phenomena, including population dynamics and radioactive decay.