Problem 88 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\) $$f(x)=x^{2}+1$$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = \frac{x^3}{3} + x + 5 \).

Step 1: Integrate the Function

To find the antiderivative \( F(x) \) of \( f(x) = x^2 + 1 \), we need to integrate \( f(x) \). The integral of \( x^2 \) is \( \frac{x^3}{3} \) and the integral of \( 1 \) is \( x \). Therefore, \( F(x) = \frac{x^3}{3} + x + C \), where \( C \) is the constant of integration.

Step 2: Apply the Initial Condition

We know that \( F(0) = 5 \). We substitute \( x = 0 \) into the equation \( F(x) = \frac{x^3}{3} + x + C \) to find \( C \). This gives us \( F(0) = 0 + 0 + C = C \). Since \( F(0) = 5 \), \( C = 5 \).

Step 3: Write the Final Antiderivative

Now that we have determined \( C = 5 \), the antiderivative \( F(x) \) of \( f(x) = x^2 + 1 \) satisfying the condition \( F(0) = 5 \) is \( F(x) = \frac{x^3}{3} + x + 5 \).

Key Concepts

Integration

Integration is a fundamental concept in calculus that allows us to find the antiderivative, or the area under the curve, of a given function. When we integrate a function such as \( f(x) = x^2 + 1 \), we are essentially reversing the process of differentiation to recover the original function, often called \( F(x) \). This function is the antiderivative.
The process involves finding a function whose derivative is the given function. For \( x^2 \), the antiderivative is \( \frac{x^3}{3} \), and for a constant like \( 1 \), the antiderivative is \( x \). Therefore, the integral of \( x^2 + 1 \) results in:

• \( F(x) = \frac{x^3}{3} + x + C \)

The \( C \) here symbolizes the constant of integration, which we'll explore later. This integration helps solve many mathematical problems, including those involving motion, area, and volume.
Learning integration is crucial for understanding how systems change over time or across different dimensions.

Initial Condition

An initial condition is a specific value that allows us to determine the constant in an indefinite integral. In our exercise, we find that \( F(0) = 5 \) is provided as an initial condition. This means when \( x = 0 \), the antiderivative \( F(x) \) should be equal to \( 5 \).

This crucial piece of information helps us pinpoint the exact antiderivative among the infinite set of possible functions that differ by a constant. Without this, the antiderivative remains general.

• Substitute \( x = 0 \) into \( F(x) = \frac{x^3}{3} + x + C \).
• This gives us \( 0 + 0 + C = 5 \).
• Thus, \( C = 5 \).

Having an initial condition can be thought of like setting a starting point or a reference that aligns the function to a real-world scenario. This ensures that mathematical solutions are applicable to real-world situations, maintaining consistency with initial measurements or conditions.

Constant of Integration

The constant of integration, denoted typically as \( C \), plays a vital role when calculating indefinite integrals. It represents the unknown constant that arises because the derivative of a constant is zero, meaning multiple antiderivatives can exist for a single derivative function.

When we perform integration to find \( F(x) \), the \( C \) accounts for all possible vertical shifts of the function graph that result in the same derivative. Thus, when integrating \( x^2 + 1 \), writing \( \int (x^2+1) \, dx = \frac{x^3}{3} + x + C \) acknowledges this ambiguity.

• The constant of integration ensures the solution covers all potential functions.
• To find \( C \), specific conditions like initial conditions are needed.
• In this case, \( F(0) = 5 \) provides clarity and allows us to solve \( C = 5 \).

Once \( C \) is determined using initial conditions or boundary values, the antiderivative becomes uniquely defined. This makes it easier to apply mathematical solutions accurately to concrete problems.