Problem 25 - Calculus Single Variable

Question

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=6 e^{2 x}, F(0)=-1, c=1 / 2 $$

Step-by-Step Solution

Answer

\( F\left(\frac{1}{2}\right) = 3e - 4 \).

Step 1: Find the antiderivative of F'(x)

To find the function \( F(x) \), we must integrate its derivative \( F'(x) = 6e^{2x} \). The integral of \( e^{2x} \) is \( \frac{1}{2}e^{2x} \), so: \[ \int 6e^{2x} \, dx = 6 \cdot \frac{1}{2}e^{2x} + C = 3e^{2x} + C. \] Therefore, \( F(x) = 3e^{2x} + C \).

Step 2: Use the initial condition to find C

We know \( F(0) = -1 \). Substitute \( x = 0 \) into \( F(x) = 3e^{2x} + C \): \[ F(0) = 3e^{0} + C = 3 + C. \] Set this equal to \( -1 \) to find \( C \): \[ 3 + C = -1 \rightarrow C = -4. \] So, \( F(x) = 3e^{2x} - 4 \).

Step 3: Compute F(c) using c = 1/2

Now we need to compute \( F(c) \) where \( c = \frac{1}{2} \) using the function \( F(x) = 3e^{2x} - 4 \). Substitute \( x = \frac{1}{2} \) into the expression for \( F(x) \): \[ F\left(\frac{1}{2}\right) = 3e^{1} - 4 = 3e - 4. \] Therefore, \( F\left(\frac{1}{2}\right) = 3e - 4 \).

Key Concepts

Understanding Antiderivatives

An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. When we differentiate a function, we find its derivative. But when we integrate a derivative, we find the original function from which it was derived (up to an added constant). This constant is necessary because derivatives of constant terms are zero, so when going back to the original function, we must account for any constant that was lost in differentiation.
To find the antiderivative of a function like \( F'(x) = 6e^{2x} \), we apply integration techniques. The antiderivative of an exponential function is itself, multiplied by the reciprocal of the coefficient of \( x \) in the exponent. That’s why integrating \( e^{2x} \) gives us \( \frac{1}{2}e^{2x} \). Hence, multiplying by the 6 in front, we get:

• \( \int 6e^{2x} \, dx = 3e^{2x} + C \)

This results in the general form of \( F(x) = 3e^{2x} + C \), where \( C \) is the constant of integration.

What is a Derivative?

A derivative represents the rate of change or slope of a function at a given point. It describes how a function’s value changes as its input changes. Mathematically, a derivative of a function \( f(x) \) is represented as \( f'(x) \).
In the exercise, we started with \( F'(x) = 6e^{2x} \), indicating how \( F(x) \) changes with respect to \( x \). To imagine why this is crucial, think of a car's speed (derivative) changing as it travels (original function). Calculating derivatives allows us to predict behavior and make informed decisions based on changes.

The Role of Initial Conditions

Initial conditions help us determine the specific solution to a differential equation. While an antiderivative gives us a family of functions due to the constant \( C \), an initial condition pinpoints the exact function.
In this exercise, the initial condition given is \( F(0) = -1 \). This means that when \( x = 0 \), the value of the function \( F(x) \) is \(-1\). We use this to solve for \( C \) in the antiderivative:

• \( 3 + C = -1 \)
• \( C = -4 \)

So, with the initial condition applied, \( F(x) \) becomes \( F(x) = 3e^{2x} - 4 \). This way, the initial condition has helped us locate the precise version of \( F(x) \) needed.

Exploring Exponential Functions

Exponential functions are one of the most important types of functions in mathematics. They are characterized by a constant base raised to a variable exponent. A common form is \( f(x) = a \, e^{bx} \).
These functions grow at rates proportional to their current values, making them ubiquitous in modeling natural phenomena, such as population growth or radioactive decay. In our exercise, the function \( F'(x) = 6e^{2x} \) is an exponential function.

• The base \( e \) is the Euler's number, approximately 2.718, a fundamental constant used naturally in growth and decay processes.
• Multiplying by a constant, like 6, doesn't change the nature of the function but affects its growth rate.
• The constant in the exponent, here 2, determines how steeply the function grows.

Understanding these properties allows us to work effectively with exponential functions, whether integrating, differentiating, or solving application problems.