Problem 31 - Calculus Concepts

Question

Consider a function \(f\) whose rate of change with respect to \(x\) is constant. a. Write a differential equation describing the rate of change of this function. b. Write a general solution for the differential equation. c. Verify that the general solution for part \(b\) is indeed a solution by substituting it into the differential equation and obtaining an identity.

Step-by-Step Solution

Answer

a. \(\frac{df}{dx} = k\); b. \(f(x) = kx + C\); c. Substitution confirms the solution.

Step 1: Define the Differential Equation

Since the rate of change of the function \(f\) with respect to \(x\) is constant, we can say that the derivative \(\frac{df}{dx}\) is equal to some constant \(k\). Thus, the differential equation describing the rate of change of this function is \(\frac{df}{dx} = k\).

Step 2: Find the General Solution

To find the general solution of the differential equation \(\frac{df}{dx} = k\), we integrate both sides with respect to \(x\). The integration yields \(f(x) = kx + C\), where \(C\) is the constant of integration. This represents the general solution of the differential equation.

Step 3: Verify the Solution

To verify that \(f(x) = kx + C\) is a solution to the differential equation, substitute \(f(x)\) into the left-hand side of the equation. Compute \(\frac{df}{dx}\) of \(f(x) = kx + C\), which gives \(\frac{d}{dx}(kx + C) = k\). This equals the right-hand side of the original differential equation (\(\frac{df}{dx} = k\)). Since they are equal, it confirms that \(f(x) = kx + C\) is indeed a solution.

Key Concepts

Rate of Change

The rate of change of a function describes how quickly the function's output, or value, is changing at any given point. In mathematical terms, the rate of change is often represented by the derivative, denoted as \( \frac{df}{dx} \), where \( f \) is the function and \( x \) is an independent variable.
In this problem, we are given a function whose rate of change with respect to \( x \) is constant. This uniformity implies that no matter what value \( x \) takes, the rate at which \( f(x) \) changes remains the same, represented by a constant \( k \).
This concept is important because it simplifies the analysis of the function. Instead of dealing with a complex relation, we only have to account for a straight line's slope. In real-world terms, a constant rate of change means that for every unit increase in \( x \), \( f(x) \) increases by \( k \) units.

Constant Rate Differential Equation

A constant rate differential equation is a type of differential equation where the rate of change, or derivative, is set to a constant value. Such equations typically have the form: \( \frac{df}{dx} = k \), where \( k \) is a constant.
This form of differential equation is straightforward because it expresses the idea that the change in \( f \) as \( x \) changes is always \( k \). It doesn't depend on \( x \,\) or \( f(x) \); it's purely constant.
These equations are often encountered in various contexts like motion with a constant speed or growth/decay processes where the rate is unchanging. In our exercise, the differential equation \( \frac{df}{dx} = k \) models such a scenario, making it simple to handle with a predictable behavior.

Integration Techniques

Finding the solution to a differential equation like \( \frac{df}{dx} = k \) involves integration. Integration is a mathematical way of finding the original function given its derivative.
When we integrate both sides of the expression with respect to \( x \), we are essentially "undoing" the differentiation. The integration of the left side gives us \( f(x) \), and that of the right side \( \int{k \, dx} = kx + C \), where \( C \) is the constant of integration.
The constant \( C \) accounts for any initial value that the function \( f(x) \) might have, corresponding to any vertical shift from the graph of \( f(x) = kx \). This integration process is one of the key techniques to solve differential equations and derive a general solution.
Remembering that every integration comes with a constant, \( C \), is crucial, because it represents the family of all solutions the differential equation might have under different initial conditions.