Problem 58 - Calculus and its applications

Question

Find f such that: $$ f^{\prime}(x)=\frac{2}{\sqrt[3]{x}}, \quad f(1)=1 $$

Step-by-Step Solution

Answer

The function is \( f(x) = 3x^{2/3} - 2 \).

Step 1: Understand the Problem

We are given the derivative of a function, \( f'(x) = \frac{2}{\sqrt[3]{x}} \), and we need to find the original function, \( f(x) \). Additionally, we have the condition that \( f(1) = 1 \), which will help us find the constant of integration.

Step 2: Find the Antiderivative

To find \( f(x) \), we need to integrate \( f'(x) \). The expression \( \frac{2}{\sqrt[3]{x}} \) can be rewritten as \( 2x^{-1/3} \). The integral becomes: \[\int f'(x) \, dx = \int 2x^{-1/3} \, dx.\] Use the power rule for integrals: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), applying it here gives us:\[f(x) = 2 \cdot \frac{x^{2/3}}{2/3} + C = 3x^{2/3} + C.\]

Step 3: Use the Initial Condition

We know that \( f(1) = 1 \). Substitute \( x = 1 \) into \( f(x) = 3x^{2/3} + C \):\[1 = 3 \cdot 1^{2/3} + C.\]This simplifies to:\[1 = 3 + C.\]Solving this equation for \( C \) gives \( C = -2 \).

Step 4: Write the Final Function

Substitute \( C = -2 \) back into the expression for \( f(x) \): \[f(x) = 3x^{2/3} - 2.\] This is the function that satisfies both the derivative and the initial condition.

Key Concepts

Antiderivative

When tackling a problem involving derivatives like this, our goal is to find the original function from its derivative, known as finding the "antiderivative." The antiderivative is essentially the reverse of taking a derivative, meaning we are trying to find a function whose derivative gives us the function we started with. In integration, this is symbolized by the integral sign \( \int \), and our task is to

• Convert the given derivative expression into a form we can integrate easily.
• Remember that the process results in a function with a constant \( C \), known as the constant of integration.

For example, if you need to find the antiderivative of \( \frac{2}{\sqrt[3]{x}} \), rewriting it as \( 2x^{-1/3} \) makes it ready for integration. Each antiderivative we find is combined with a constant \( + C \) because integrating destroys specific information about particular solutions.

Power Rule

The power rule for finding antiderivatives is a fundamental tool in calculus. It helps us integrate functions of the form \( x^n \). The rule states that

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) when \( n eq -1 \).

This formula is incredibly useful for simplifying and integrating expressions with powers of \( x \). Let's see how we apply the power rule to integrate \( 2x^{-1/3} \):1. Identify the exponent: Here, \( n = -1/3 \).2. Use the power rule: \( \int 2x^{-1/3} \, dx = 2 \cdot \frac{x^{-1/3 + 1}}{-1/3 + 1} + C = 2 \cdot \frac{x^{2/3}}{2/3} + C \).3. Simplify to get \( 3x^{2/3} + C \).Applying the power rule transforms the task of integration into a straightforward arithmetic operation, allowing us to complete the integration process efficiently.

Initial Condition

An initial condition helps us find the constant of integration, \( C \), turning our general solution into a specific one. In this problem, the initial condition is given as \( f(1) = 1 \), which provides a specific value of \( x \) that makes it possible to compute \( C \).Here's how to use an initial condition:

• Substitute the values \( x = 1 \) and \( f(x) = 1 \) into the general antiderivative \( f(x) = 3x^{2/3} + C \).
• Solve the equation to isolate and find \( C \):

Substitute the condition:

• \( 1 = 3 \cdot 1^{2/3} + C \).

Simplify and solve:

• \( 1 = 3 + C \).
• \( C = -2 \).

Using the initial condition narrows down the infinite possibilities of \( C \) to a unique solution, making our final function \( f(x) = 3x^{2/3} - 2 \), which satisfies both the derivative and the initial condition.