Problem 6 - Applied Calculus

Question

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=e^{2 x}+\int_{0}^{1} e^{2 t} d t$$

Step-by-Step Solution

Answer

Yes, the function \( F(x) \) is an antiderivative of \( f(x) = 2e^{2x} \).

Step 1: Evaluate the Constant Term

Calculate the constant term by evaluating the integral \( \int_{0}^{1} e^{2t} \, dt \). To solve this integral, find the antiderivative of \( e^{2t} \), which is \( \frac{1}{2}e^{2t} \). Now, apply the limits of integration from 0 to 1:\[ \left[ \frac{1}{2}e^{2t} \right]_{0}^{1} = \frac{1}{2}e^{2 \times 1} - \frac{1}{2}e^{2 \times 0} = \frac{1}{2}e^{2} - \frac{1}{2}e^{0} = \frac{1}{2}e^{2} - \frac{1}{2} \]

Step 2: Simplify F(x)

Now that we have evaluated the constant part, substitute it into the function \( F(x) \):\[ F(x) = e^{2x} + \frac{1}{2}e^{2} - \frac{1}{2} \]

Step 3: Take the Derivative of F(x)

Find \( F'(x) \) by differentiating \( F(x) \):\[ F(x) = e^{2x} + \frac{1}{2}e^{2} - \frac{1}{2} \]The derivative of \( e^{2x} \) with respect to \( x \) is \( 2e^{2x} \).The derivative of a constant number is 0, so the term \( \frac{1}{2}e^{2} - \frac{1}{2} \) contributes nothing to the derivative.Thus, \( F'(x) = 2e^{2x} \).

Step 4: Verify the Antiderivative Condition

Compare \( F'(x) \) with \( f(x) \).Given \( f(x) = 2e^{2x} \), and we found \( F'(x) = 2e^{2x} \).Since \( F'(x) = f(x) \), \( F(x) \) is indeed an antiderivative of \( f(x) \).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. This exercise involves understanding whether a function is an antiderivative, which is a core concept in calculus. An antiderivative of a function is another function whose derivative equals the original function. Essentially, it is the reverse process of differentiation.

To verify whether a given function is an antiderivative, we differentiate the function and check if it results in the original function. This requires proficiency in both differentiation and integration, two fundamental operations in calculus.

• Differentiation focuses on finding the rate of change of a function.
• Integration allows us to find the accumulation of values or areas under curves.

In this exercise, identifying the antiderivative involves both calculating a definite integral and differentiating a function correctly, linking together these two crucial concepts.

Exponential Functions

Exponential functions are a type of mathematical function where the variable appears in the exponent. They have the form \(a^x\), where \(a\) is a constant. In calculus, one particular exponential function of interest is the natural exponential function, \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828.

Exponential functions are unique because their rates of change are proportional to their values. This characteristic makes them widely used in modeling growth processes, such as population growth and radioactive decay.

• The derivative of \(e^x\) is \(e^x\), maintaining its form.
• The antiderivative of \(e^x\) is also \(e^x + C\), where \(C\) is the constant of integration.

In this exercise, the function \(f(x) = 2e^{2x}\) is an exponential function, and recognizing this helps in easily computing derivatives and antiderivatives, which are pivotal in the solution process.

Differentiation

Differentiation is a fundamental operation in calculus, used to find the rate at which a quantity changes. It provides us with the derivative of a function, offering insights into the function's behavior. Differentiation involves applying rules such as the power rule, product rule, or chain rule to obtain the derivative.

In this particular exercise, differentiation is used to verify whether \(F(x)\) is an antiderivative of \(f(x) = 2e^{2x}\). The steps show that differentiating \(F(x)\) yields \(F'(x) = 2e^{2x}\). Finds support in the differentiation of the exponential function:

• The derivative of \(e^{kx}\) with respect to \(x\) is \(ke^{kx}\).
• Constant terms become zero upon differentiation.

These rules applied here confirm that \(F(x)\) is indeed an antiderivative of \(f(x)\), ensuring that differentiation has been performed correctly and aligning it with the broader calculus concepts at play.