Problem 90 - Applied Calculus

Question

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=f(x)\) and \(F(0)=5\) $$f(x)=6 e^{3 x}$$

Step-by-Step Solution

Answer

\( F(x) = 2e^{3x} + 3 \).

Step 1: Understanding the Given Function

We start with the given function, which is the derivative, \( f(x) = 6 e^{3x} \). We aim to find the original function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Antiderivative Rule for Exponential Functions

To find the antiderivative of \( 6e^{3x} \), we use the rule for integrating exponential functions: \( \int ae^{bx} \, dx = \frac{a}{b}e^{bx} + C \). Here, \( a = 6 \) and \( b = 3 \).

Step 3: Apply the Antiderivative Rule

Apply the rule: \( F(x) = \int 6e^{3x} \, dx = \frac{6}{3}e^{3x} + C \). Simplify to get: \( F(x) = 2e^{3x} + C \).

Step 4: Solve for the Constant C

We know that \( F(0) = 5 \). Substitute \( x = 0 \) into \( F(x) = 2e^{3x} + C \): \( 5 = 2e^{0} + C \). Simplifying, \( 5 = 2(1) + C \) gives \( C = 3 \).

Step 5: Write Down the Final Antiderivative

Now we have all the components. The antiderivative is \( F(x) = 2e^{3x} + 3 \).

Key Concepts

Exponential Functions

Exponential functions are fundamental in calculus because they describe many natural phenomena. These functions are of the form \(f(x) = ae^{bx}\), where \(a\) and \(b\) are constants and \(e\) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions grow rapidly and are crucial for modeling growth and decay in fields like biology, finance, and physics.

• In our exercise, the function \(f(x) = 6e^{3x}\) is an exponential function where \(a = 6\) and \(b = 3\).
• These functions have the property that their derivative is also an exponential function, which makes them particularly nice to work with in calculus.

Understanding these properties helps in calculating the antiderivatives and predicting the behavior of the function.

Constant of Integration

The constant of integration, represented by \(C\), is an important part of finding an antiderivative. When you compute an antiderivative, you are essentially reversing the differentiation process. However, since derivative rules remove constants (because the derivative of a constant is zero), we add a constant \(C\) when finding the antiderivative.

• In mathematical terms, if \(F'(x) = f(x)\), then \(F(x) = \int f(x) \, dx = F(x) + C\).
• In our exercise, after integrating, we found that the antiderivative was \(F(x) = 2e^{3x} + C\).

To solve for \(C\), we use initial conditions, like \(F(0) = 5\) in this case, to find that \(C = 3\). This ensures our antiderivative fits any specific conditions given in a problem.

Calculus Problem Solving

Solving calculus problems involves using rules and logic to find unknowns. To solve an antiderivative problem, you need to:

• Identify the function to be integrated. For exponential functions, use the rule \(\int ae^{bx} \, dx = \frac{a}{b}e^{bx} + C\).
• Apply the integration rule to find a general antiderivative.
• Use given conditions to solve for the constant of integration \(C\).

In our example, we:

• Identified \(f(x) = 6e^{3x}\), where we used \(a = 6\) and \(b = 3\).
• Applied the antiderivative rule to get \(F(x) = 2e^{3x} + C\).
• Solved \(F(0) = 5\) to find \(C\).

Following these steps ensures that students can successfully tackle calculus exercises around antiderivatives.