Problem 45 - Calculus Single Variable

Question

In each of Exercises \(41-48\), use the given information to find \(F(c)\). $$ F^{\prime}(x)=e^{x}, \quad F(2)=2+e^{2}, \quad c=3 $$

Step-by-Step Solution

Answer

\( F(3) = e^3 + 2 \)

Step 1: Understanding the Problem

We need to find the value of the function \( F(c) \) where \( c=3 \). We are given the derivative of the function \( F'(x) = e^x \) and the value \( F(2) = 2 + e^2 \).

Step 2: Integrating the Derivative

Since \( F'(x) = e^x \), integrating \( F'(x) \) will yield \( F(x) \). The integral of \( e^x \) is \( e^x + C \), where \( C \) is the constant of integration. So, \( F(x) = e^x + C \).

Step 3: Using the Initial Condition

We use the condition \( F(2) = 2 + e^2 \) to find the constant \( C \). Substitute \( x = 2 \) into the equation, \( F(2) = e^2 + C = 2 + e^2 \).

Step 4: Finding the Constant \( C \)

Solving the equation \( e^2 + C = 2 + e^2 \), we subtract \( e^2 \) from both sides to get \( C = 2 \).

Step 5: Substituting Back to Find \( F(x) \)

Now that we know \( C = 2 \), substitute back to get the function: \( F(x) = e^x + 2 \).

Step 6: Evaluating \( F(c) \)

Substitute \( c = 3 \) into the function to find \( F(3) \). Calculating, we have \( F(3) = e^3 + 2 \).

Key Concepts

Integration: Finding the Original Function

Integration is a fundamental concept in calculus used for finding the original function from its derivative. The process essentially reverses differentiation. When you know the derivative of a function, like in our exercise where the derivative is given as \( F'(x) = e^x \), integration helps us determine the function itself. When you integrate the exponential function \( e^x \), the result is \( e^x + C \). The \( C \) represents the constant of integration which accounts for any constant value that could have been differentiated to zero in the original function. This constant is crucial as it ensures the calculated function fits all points, not just the form of the equation. In our example, integrating \( F'(x) = e^x \) gives us \( F(x) = e^x + C \). Keep in mind that integration is not just about reversing derivatives; it's a tool to find accumulation and areas under curves, making it an essential skill in mathematics.

Exponential Function: An Important Math Element

Exponential functions, like \( e^x \), play a vital role in calculus. They are functions of the form \( a^x \) where \( a \) is a positive constant. The base \( e \) is a special number approximately equal to 2.71828, and it has unique properties that make it frequently appear in calculus. One of the most important properties of exponential functions is the fact that the derivative of \( e^x \) is \( e^x \). This reinforces their importance since it provides a seamless connection between differentiation and the function itself. This feature also makes exponential functions naturally arise in growth and decay processes, modeling various real-world scenarios such as population dynamics, radioactive decay, and financial computations. Understanding how to work with exponential functions opens doors to analyzing natural processes and mathematical projections.

Initial Conditions: Pinning Down the Constant

Initial conditions are crucial for determining the specific form of a function after integration. After integrating a derivative, you end up with a general function that includes an unknown constant of integration \( C \). To find \( C \), initial conditions which tell you the value of a function at a certain point, are used. These conditions are like clues that help us zero in on the exact form of the function. In our problem, the initial condition \( F(2) = 2 + e^2 \) is used to solve for \( C \). By plugging \( x = 2 \) into the integrated function \( F(x) = e^x + C \) and setting it equal to \( 2 + e^2 \), we solved for \( C \). Initial conditions not only ensure that the calculated function is accurate at a specific point but also make the function complete and usable in further calculations. They provide the essential anchor that ties theoretical calculus to practical application.