Problem 87 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\) $$f(x)=6 x-5$$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = 3x^2 - 5x + 5 \).

Step 1: Recognize the Problem

We need to find the function \( F(x) \) such that its derivative is \( f(x) = 6x - 5 \), and \( F(0) = 5 \). This is an antidifferentiation problem with an initial condition.

Step 2: Find the General Antiderivative

To find the antiderivative of \( f(x) = 6x - 5 \), integrate term by term:\[ F(x) = \int (6x - 5) \, dx = \int 6x \, dx - \int 5 \, dx. \]Start by integrating \( 6x \):\[ \int 6x \, dx = 6 \cdot \frac{x^2}{2} = 3x^2. \]Now integrate \( -5 \):\[ \int -5 \, dx = -5x. \]Combine the results:\[ F(x) = 3x^2 - 5x + C, \]where \( C \) is the constant of integration.

Step 3: Use the Initial Condition

Apply the initial condition \( F(0) = 5 \) to find \( C \):Substitute \( x = 0 \) into the equation:\[ F(0) = 3(0)^2 - 5(0) + C = C. \]Since \( F(0) = 5 \), we set \( C = 5 \).

Step 4: Write the Particular Antiderivative

Substitute \( C = 5 \) back into the general antiderivative to get the specific antiderivative that satisfies the initial condition:\[ F(x) = 3x^2 - 5x + 5. \]

Key Concepts

Understanding Initial Value Problem

An initial value problem in calculus involves finding a function from its derivative, given a specific value that the function must satisfy. Here, the problem requires determining a function, in this case, an antiderivative, where the slope of the tangent at any point is defined by another function (like a blueprint). By also specifying a value of the function at a particular point, called the initial condition (such as \(F(0) = 5\)), we zero in on a particular solution among infinitely many possibilities.

• This condition acts as an anchor, determining the constant of integration, which appears after performing an indefinite integration.
• It provides the unique answer needed for specific applications in physics, engineering, and other fields where knowing initial conditions is essential.

Understanding this concept is critical, as it transitions the focus from general math to its application in real-world problems.

The Process of Integration

Integration is the reverse process of differentiation. It involves finding a function whose derivative is the given function. In calculus, it helps determine the accumulation of quantities.

• The indefinite integral \( F(x) = \int f(x) \,dx \) finds a general form of the antiderivative, which includes an arbitrary constant \( C \).
• This constant arises because the derivative of any constant is zero, meaning it can be any real number unless further information defines it.

In the example provided, integration is carried out term by term on \(6x - 5\). First, each component (like \(6x\) and \(-5\)) is integrated separately, then combined:
Integrate \(6x\):\[ \int 6x \, dx = 3x^2 \]
Integrate \(-5\):\[ \int -5 \, dx = -5x \]
Thus, the general antiderivative is \(F(x) = 3x^2 - 5x + C\).

Applying Calculus Problem Solving

In calculus problem solving, understanding how to apply differentiation and integration is key. When given an initial value problem, applying these concepts strategically is the way forward.

• You begin by recognizing what is known and what needs to be found—this directs your approach to solve the problem methodically.
• The next logical step is computation, like finding the general antiderivative by integrating the given function. The calculations should always be double-checked for accuracy.
• Finally, using the initial condition, you calculate any remaining unknowns, typically solving for the constant of integration \( C \) by substituting the initial condition into the derived general solution.

For the given problem, utilizing the initial condition \( F(0) = 5 \) was crucial in determining \( C \). The calculated antiderivative, \( F(x) = 3x^2 - 5x + 5 \), then satisfies both the differential equation and the initial condition, showcasing how calculus tools combine for problem-solving in real scenarios.