Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

\mathrm{\\{} E X P L ~ B e s i d e s ~ p r o v i d i n g ~ a n ~ e a s y ~ w a y ~ t o ~ d i f f e r e n t i a t e ~ p r o d u c t s , ~ l o g . ~ arithmic differentiation also provides a measure of the relative or fractional rate of change, defined as \(y^{\prime} / y .\) We explore this concept in Problems 39-42. Show that the relative rate of change of \(e^{k t}\) as a function of \(t\) is \(k\)

Step-by-Step Solution

Answer

The relative rate of change of \( e^{kt} \) is \( k \).

Step 1: Understand the function

We begin with the function given in the exercise, which is expressed as \( y = e^{kt} \). Here, \( e \) is the base of the natural logarithm, \( k \) is a constant, and \( t \) represents the variable.

Step 2: Differentiate the function

Next, we differentiate the function \( y = e^{kt} \) with respect to \( t \). Using the chain rule, the derivative \( y' \) of the function is \( y' = \frac{d}{dt}(e^{kt}) = k \times e^{kt} \).

Step 3: Relate it to the relative rate of change

The problem asks for the relative rate of change \( \frac{y'}{y} \). We substitute the expressions for \( y \) and \( y' \): \( \frac{y'}{y} = \frac{k \, e^{kt}}{e^{kt}} \). The \( e^{kt} \) terms cancel out, leaving \( \frac{y'}{y} = k \).

Step 4: Conclusion

The relative rate of change of the function \( e^{kt} \) with respect to \( t \) is \( k \). This means that for an exponential function of this form, the relative rate of change is constant and equal to the exponent's coefficient.

Key Concepts

Relative Rate of Change

One important concept introduced in the given problem is the relative rate of change. To put it simply, the relative rate of change is the rate at which a function changes concerning its current value. This is often expressed as \( \frac{y'}{y} \), where \( y' \) is the derivative of a function \( y \).Relative rate of change gives us a normalized measure of how a function's output varies compared to its original value. In practical terms, this means it provides insights into how quickly something is growing or decreasing relative to its size. For example, if we are considering an investment's growth or the rate at which a population increases, understanding the relative rate could offer significant insights.In the context of the exercise, the function \( y = e^{kt} \) has a relative rate of change of \( k \). This constant value \( k \) tells us that the function grows at a steady rate which is independent of time \( t \), making it a crucial tool for predicting outcomes over time in fields such as finance, biology, and physics.

Exponential Functions

Exponential functions are a family of functions that model growth or decay processes with constant proportional rates. They can be written in the form \( y = a \cdot e^{bt} \), where \( a \) is a constant, \( e \) is Euler's number (approximately 2.71828), and \( b \) determines the rate and direction of change.These functions represent phenomena where the rate of change of the function's value is directly proportional to the current function value. Common examples include population growth, radioactive decay, and continuously compounded interest.When dealing with exponential functions, a special property is the fact that the derivative of an exponential function is proportional to the function itself. In our context, with the function \( y = e^{kt} \), differentiating it yields \( y' = k \cdot e^{kt} \). This showcases why exponential functions are so integral to various scientific disciplines since they inherently model constant relative growth or decay.

Chain Rule

The concept of the Chain Rule is central when differentiating composite functions, ones that are made up of a function inside another function. It is best used when you have functions nested within each other, necessitating a method to derive them efficiently.The chain rule formula is given as \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \), where the variable \( u \) represents the inner function. Essentially, it states that one must multiply the derivative of the outer function by the derivative of the inner function. In simpler terms, it helps "unpack" functions that are inside other functions.In our exercise, the function \( y = e^{kt} \) benefits from the chain rule where \( e^{u} \) acts as the outer function and \( u = kt \) acts as the inner function. By applying the chain rule, the differentiation process becomes structured and more manageable, leading us to accurately determine that \( \, y' = k \cdot e^{kt} \). This makes the Chain Rule invaluable in tackling more complex functions, helping elucidate and solve intricate problems systematically.