Problem 49 - Calculus and its applications

Question

Find f such that: $$ f^{\prime}(x)=x^{2}-4, \quad f(0)=7 $$

Step-by-Step Solution

Answer

The function is \( f(x) = \frac{x^3}{3} - 4x + 7 \).

Step 1: Identify the Integral

The problem gives us the derivative of the function, which is \( f'(x) = x^2 - 4 \). To find \( f(x) \), we need to integrate \( f'(x) \). Thus, \[ f(x) = \int (x^2 - 4) \, dx \]

Step 2: Integrate the Function

Integrate the expression \( x^2 - 4 \) term by term. The integral of \( x^2 \) is \( \frac{x^3}{3} \), and the integral of \(-4\) is \(-4x\). Including the constant of integration \( C \), we have: \[ f(x) = \frac{x^3}{3} - 4x + C \]

Step 3: Use the Initial Condition

We use the initial condition \( f(0) = 7 \) to find the constant \( C \). Substituting \( x = 0 \) into the integrated function, we get: \[ f(0) = \frac{0^3}{3} - 4(0) + C = 7 \]This simplifies to \( C = 7 \).

Step 4: Write the Final Function

Substitute \( C = 7 \) back into the function from Step 2. Thus, the function \( f(x) \) is:\[ f(x) = \frac{x^3}{3} - 4x + 7 \]

Key Concepts

Integration Techniques

Integration techniques are methods used to determine the antiderivative of a function. In this exercise, we start with the derivative, which is given as \( f'(x) = x^2 - 4 \). To find the original function \( f(x) \), we need to perform integration.

• Term-by-Term Integration: This technique involves integrating each term of the function separately. Here, we have two terms: \( x^2 \) and \( -4 \).
• Power Rule: The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Applying this to \( x^2 \), we get the integral \( \frac{x^3}{3} \).
• Constant Rule: The integral of a constant \( a \) is \( ax \). So, for \( -4 \), the integral is \( -4x \).
• Constant of Integration: When performing indefinite integration, always add a constant \( C \) to your result. This accounts for any horizontal shifts in the original function.

Ultimately, our integral becomes \( f(x) = \frac{x^3}{3} - 4x + C \). This expression embodies our integrated terms and the constant of integration.

Initial Value Problem

An initial value problem involves finding a function given its derivative and an initial condition. This exercise included that initial condition as \( f(0) = 7 \).

• Determine the Constant: To find \( C \), we substitute the initial values into the integrated function.
• Calculating with the Initial Condition: We use \( f(0) = 7 \) and substitute \( x = 0 \) into our integrated function. This gives us \( 0 + 0 + C = 7 \), leading to \( C = 7 \).

This allows us to clarify the specific antiderivative that satisfies not just the derivative equation but also the initial value condition. Solutions to initial value problems result in a precise function rather than a family of solutions.

Antiderivative

The antiderivative, often symbolized as the indefinite integral, represents the original function whose derivative is known. Here, we were tasked with finding the antiderivative of \( f'(x) = x^2 - 4 \).

• Reverse Operation: Whereas differentiation involves finding the rate of change, integration seeks to 'reverse' this process by finding the original function.
• Building the Antiderivative: Through integration, each component of the expression is transformed back to its antecedent form. From \( x^2 \), we derive \( \frac{x^3}{3} \); from \(-4\), we ever so systematically deduce \(-4x\).
• Incorporating Constants: The constant \( C \) is intrinsic to the general form of an antiderivative and influenced by boundary or initial conditions provided. In this context, \( f(0) = 7 \) specified \( C \) concretely as 7.

Ultimately, the antiderivative \( f(x) = \frac{x^3}{3} - 4x + 7 \) provides the complete original function, illustrating the elegance of moving from rates of change back to foundational expressions.