Problem 47 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = \frac{2}{3}x^{3/2} \), and it is unique given the condition \( F(0) = 0 \).

Step 1: Identify the function to integrate

The derivative function given is \( f(x) = \sqrt{x} \). We need to find an antiderivative \( F(x) \) such that \( F'(x) = f(x) \) and \( F(0) = 0 \). An antiderivative of a function is found by integrating it.

Step 2: Rewrite the function for integration

Rewrite \( \sqrt{x} \) in a form that is easier to integrate. \( \sqrt{x} \) is equivalent to \( x^{1/2} \). This will make integration straightforward using the power rule for integration.

Step 3: Apply the power rule for integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Applying this to \( f(x) = x^{1/2} \), we get:\[F(x) = \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{(1/2) + 1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C\]

Step 4: Apply the initial condition

We are given that \( F(0) = 0 \). Substitute \( x = 0 \) into \( F(x) = \frac{2}{3}x^{3/2} + C \):\[0 = \frac{2}{3}(0)^{3/2} + C = 0 + C\]Thus, \( C = 0 \).

Step 5: Write the final antiderivative

Substitute \( C = 0 \) back into the expression for \( F(x) \):\[F(x) = \frac{2}{3}x^{3/2} + 0 = \frac{2}{3}x^{3/2}\]This is the antiderivative that satisfies \( F'(x) = f(x) \) and \( F(0) = 0 \).

Key Concepts

Integration

Integration is the process of finding an antiderivative, essentially reversing differentiation. When we integrate a function, we look for another function whose derivative is the original function. In this exercise, we start with a function \( f(x) = \sqrt{x} \) and our goal is to find a function \( F(x) \) such that its derivative gives us \( \sqrt{x} \). This process involves applying various rules and techniques of integration to achieve the desired antiderivative.Integrating is somewhat akin to piecing together a puzzle where each piece aligns perfectly as a derivative. For this exercise, identifying that \( f(x) = \sqrt{x} \) can be written as \( x^{1/2} \) makes it more straightforward to apply the power rule of integration. We often rewrite functions to make the integration process simpler. This entails expressing functions in a format that aligns with the available integration techniques, such as recognizing powers of \( x \).

Power Rule

The Power Rule for Integration is a fundamental method used in calculus to find antiderivatives. It states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This rule is quite powerful because it allows us to compute antiderivatives of functions with polynomial-like terms quickly.To apply this rule successfully, we first rewrite the given function in a form that exposes the power of \( x \). For \( f(x) = \sqrt{x} \), rewriting it as \( x^{1/2} \) was crucial. Applying the power rule involves increasing the exponent by one and dividing by the new exponent. In this case, it transformed \( x^{1/2} \) to \( \frac{x^{3/2}}{3/2} \).After completing the integration, we add a constant \( C \), which represents the integration constant. This constant is essential because it accounts for all possible vertical shifts of the antiderivative graph. Without this constant, we wouldn’t be able to verify the complete family of solutions for a given function.

Initial Condition

An initial condition is a requirement that specifies particular value(s) the antiderivative must have, given specific inputs. In this exercise, the initial condition \( F(0) = 0 \) provides the clue to determine the constant \( C \) in the equation obtained from integration.To find \( C \), we substitute the given condition into the antiderivative. If \( F(x) = \frac{2}{3}x^{3/2} + C \), substitute \( x = 0 \) and set \( F(0) = 0 \). This results in:

• \( 0 = \frac{2}{3}(0)^{3/2} + C \)
• Simplifying, we find \( C = 0 \)

With \( C \) known, substituting back gives the unique antiderivative \( F(x) = \frac{2}{3}x^{3/2} \). The initial condition is critical because it narrows down the infinite possible antiderivatives to a single, specific solution that meets the original problem requirement.