Problem 28 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0\). Is there only one possible solution? $$ f(x)=-7 x $$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = -\frac{7}{2}x^2 \) and it is unique.

Step 1: Integrate the function

To find an antiderivative of the function, we need to integrate the given derivative. The function is given as \( f(x) = -7x \). The integral of \( -7x \) with respect to \( x \) is \( \int -7x \, dx = -\frac{7}{2}x^2 + C \), where \( C \) is a constant of integration.

Step 2: Apply the initial condition

We know from the problem that \( F(0) = 0 \). Plug \( x = 0 \) into the antiderivative expression \( F(x) = -\frac{7}{2}x^2 + C \), which becomes \( F(0) = -\frac{7}{2} \cdot 0^2 + C = C \). Since \( F(0) = 0 \), this implies that \( C = 0 \).

Step 3: Conclusion

With \( C = 0 \), the particular solution that satisfies the given conditions is \( F(x) = -\frac{7}{2}x^2 \).

Step 4: Determine uniqueness

Because the condition \( F(0) = 0 \) leads to a specific value of \( C \), this antiderivative is unique and no other solution is possible under the given condition.

Key Concepts

Integration

Integration is the mathematical process of finding an antiderivative or the area under the curve of a function. This is the reverse operation of differentiation.
When we integrate a function like \( f(x) = -7x \), we determine its antiderivative, which is a function whose derivative gives back the original function \( f(x) \). The integration process is expressed as \( \int f(x)\, dx \).
For \( f(x) = -7x \), the integral becomes:

• \( \int -7x\, dx = -\frac{7}{2}x^2 + C \)

This expression includes the term \( C \), known as the constant of integration. Integration can be thought of as 'adding up' tiny pieces to find the area under a curve, and in this context, it's used to find a function whose derivative matches the given function.

Initial Condition

An initial condition is a piece of information given alongside a differential or integral problem. It helps to determine a unique solution from many possible ones.
In the problem above, the initial condition is \( F(0) = 0 \). This condition requires that the particular antiderivative we find must satisfy the equation when \( x = 0 \).
When we consider the antiderivative \( F(x) = -\frac{7}{2}x^2 + C \), substituting \( x = 0 \) gives us:

• \( F(0) = -\frac{7}{2} \cdot 0^2 + C = C \)
• Since \( F(0) = 0 \), we solve for \( C \) and find \( C = 0 \)

Using initial conditions is essential because an indefinite integral includes an arbitrary constant \( C \), which can lead to many possible solutions.

Constant of Integration

In the process of integration, the constant of integration, denoted as \( C \), appears because the derivative of a constant is zero.
This means when we find an antiderivative, there are infinitely many functions differing by a constant value, all having the same derivative.
For example, integrating \( f(x) = -7x \) results in the antiderivative \( F(x) = -\frac{7}{2}x^2 + C \). The \( C \) represents any constant value that can be added to \( -\frac{7}{2}x^2 \) without affecting its derivative.
The initial condition \( F(0) = 0 \) was used to determine the exact value of \( C \) needed here, making \( C = 0 \). Once found, \( C \) fixes the antiderivative to one unique solution specific to the given condition.
The constant of integration explains why indefinite integrals potentially represent a family of functions, not just one.