Question

True or False? In Exercises \(67-70\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The differential equation modeling exponential growth is \(d y / d x=k y,\) where \(k\) is a constant.

Short answer

The statement is true

Step-by-step solution

Understanding the statement

In order to validate the statement, one needs to understand what exponential growth means and how it is represented mathematically. Exponential growth refers to an increase that becomes more rapid in proportion to the growing total number or size. In mathematics, it is generally represented by functions of the type \( y = a \cdot e^{kx} \), where \( a \) and \( k \) are constants, \( e \) is Euler's number, and \( x \) is the variable.

Identifying the differential equation

In differential equations, exponential growth is commonly modeled with the equation \( dy/dx = ky \), where \( k \) is a constant. This equation is derived from the exponential function used to represent exponential growth. If we derive \( y = a \cdot e^{kx} \) with respect to \( x \), we obtain \( dy/dx = a \cdot ke^{kx} = ky \), assuming that \( a = e^{0} = 1 \).

Verifying the statement

Upon comparing the differential equation provided in the statement with the one derived from the mathematical representation of exponential growth, we can conclude that they match. Thus, the statement is true.

Key concepts

Exponential Growth

Exponential growth is a fascinating and important concept in mathematics and sciences. It refers to situations where a quantity increases at a rate proportional to its current size. This means the larger the quantity, the faster it grows, which leads to an ever-increasing pace of growth that can quickly spiral out of control. A common example of exponential growth is seen in populations where resources are unlimited, like bacteria multiplying in ideal conditions. The mathematical representation typically used is the exponential function, defined as:

• \( y = a \cdot e^{kx} \)

Here, \( a \) is the initial quantity, \( e \) is Euler's number, and \( k \) is the growth constant. When \( k > 0 \), the function grows exponentially as \( x \) increases. Because of this rapid increase, exponential growth is an important concept in many fields, including biology, economics, and computer science.

Mathematical Modeling

Mathematical modeling involves using mathematical formulas and concepts to represent real-world phenomena. It helps simplify complex systems and predict behavior using mathematical structures. In the context of exponential growth, mathematical modeling uses differential equations to describe how a system evolves over time.

The differential equation \( \frac{dy}{dx} = ky \) is a common mathematical model for exponential growth. In this equation:

• \( dy/dx \) represents the rate of change of the quantity \( y \)
• \( k \) is a constant representing the growth rate

The solution of this differential equation gives us an exponential function, which models the growth behavior over time. This model shows how quantities can grow rapidly and is applicable in modeling population growth, radioactive decay, and interest compounding in finance.

Euler's Number

Euler's number, denoted as \( e \), is a fundamental constant approximately equal to 2.71828. It is a base of the natural logarithms and plays a crucial role in calculus and mathematical modeling.

One reason \( e \) is so important is its role in defining continuous growth. The function \( e^x \) has the unique property that its derivative is the same as the function itself, which is key for modeling processes that involve exponential growth. In the differential equation for exponential growth, the \( e^{kx} \) term arises naturally when integrating:

• Derivatives of exponentials with base \( e \) remain exponential.
• It simplifies calculations and is essential for modeling natural growth patterns.
• \( e \'s \) properties make it ubiquitous in growth models, statistics, and even in complex financial calculations.

Calculus

Calculus is the branch of mathematics that studies continuous change. It includes two main concepts: differentiation and integration. Both are crucial for understanding and modeling change in systems described by differential equations.

Differentiation, in calculus, involves finding the rate at which a quantity changes over time. It's used to derive the form \( \frac{dy}{dx} = ky \) from the exponential function \( y = a \cdot e^{kx} \). Integration is the reverse process, finding the original function from its derivative. These processes:

• Help solve differential equations.
• Identify growth patterns in data and predict future behavior.
• Are essential in fields ranging from physics to economics, where change over time must be modeled and understood.

Calculus provides the tools necessary for verifying statements about growth models, like the one we derived and confirmed to be true.