Problem 46 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0 .\) Is there only one possible solution? $$f(x)=\frac{1}{4} x$$

Step-by-Step Solution

Answer

\( F(x) = \frac{1}{8}x^2 \) is the unique antiderivative satisfying the given conditions.

Step 1: Identify the Function

We are given the function \( f(x) = \frac{1}{4}x \) and need to find its antiderivative \( F(x) \). The function is a linear function and the task is to integrate it.

Step 2: Apply the Basic Integration Rule

Recall that the antiderivative of \( ax^n \) is \( \frac{a}{n+1}x^{n+1} + C \). Here, \( a = \frac{1}{4} \) and \( n = 1 \). So, the antiderivative is given by \( F(x) = \frac{1}{4}\frac{x^2}{2} + C = \frac{1}{8}x^2 + C \).

Step 3: Use Initial Condition

We have the initial condition \( F(0) = 0 \). Substitute \( x = 0 \) in \( F(x) = \frac{1}{8}x^2 + C \) to find \( C \). This gives \( 0 = \frac{1}{8}(0)^2 + C \), which simplifies to \( C = 0 \).

Step 4: Write the Particular Antiderivative

With \( C = 0 \), the antiderivative function is \( F(x) = \frac{1}{8}x^2 \).

Step 5: Consider the Uniqueness of the Solution

Since \( C \) is determined uniquely by the condition \( F(0)=0 \), there is only one solution that satisfies both the derivative condition and the initial condition.

Key Concepts

Initial Condition

An initial condition in mathematics refers to specific values provided for a function or its derivative. These values help determine the particular solution to a differential equation or antiderivative problem. In the exercise `Find an antiderivative`, the initial condition given is \( F(0) = 0 \).

Initial conditions are crucial because they allow us to find the constant of integration, which is represented as \( C \). The constant \( C \) emerges as a result of integrating a function, since antiderivatives family solutions differ by a constant. However, with an initial condition, we can find the exact value of \( C \).

Here's how we applied this in our exercise:

• First, we found the general antiderivative of \( f(x) = \frac{1}{4}x \) which includes the constant \( C \).
• Then, we used the initial condition \( F(0) = 0 \) to solve for \( C \).
• Substituting \( x = 0 \) into \( F(x) = \frac{1}{8}x^2 + C \), we found that \( C = 0 \).

This process allows us to narrow down the infinite possible antiderivative functions to a single, particular function that precisely satisfies the given initial condition.

Basic Integration Rule

To find an antiderivative, we often rely on the basic integration rule, which is fundamental to calculus. The rule states that the antiderivative of a function \( ax^n \) is \( \frac{a}{n+1}x^{n+1} + C \). This formula is derived from reversing the process of differentiation.

In our specific exercise, we approached the function \( f(x) = \frac{1}{4}x \) with simplicity:

• We first identified this as a linear function, making it easier to integrate.
• We applied the basic rule, where \( a = \frac{1}{4} \) and \( n = 1 \), resulting in the general form \( F(x) = \frac{1}{8}x^2 + C \).

This application of the basic integration rule is a straightforward method, and crucial for students to practice since it applies to many types of functions they will encounter in calculus. It is through mastering this rule that we can confidently navigate through finding antiderivatives.

Unique Solution

A unique solution in the context of an antiderivative problem refers to one specific function that fulfills both the integration requirement and any given conditions—like the initial condition. In the original exercise, we were tasked with finding the antiderivative of \( f(x) = \frac{1}{4}x \) that also satisfies \( F(0) = 0 \).

Here's why our solution is unique:

• The integration step provides us with a general solution, \( F(x) = \frac{1}{8}x^2 + C \), which represents all possible antiderivatives.
• By using the initial condition, we solved for the constant \( C = 0 \).
• Incorporating \( C = 0 \) reduces the general solution to the particular one: \( F(x) = \frac{1}{8}x^2 \).

Since there are no other constants that satisfy the condition \( F(0) = 0 \), this solution is indeed unique. This characteristic emphasizes how specific conditions guide us to a particular function rather than a family of functions, ensuring that we have precisely the right fit for the problem at hand.