Problem 1 - Applied Calculus

Question

Decide if the function is an antiderivative of \(f(x)=2 e^{2 x}\) $$F(x)=e^{2 x}+5$$

Step-by-Step Solution

Answer

Yes, \( F(x)=e^{2x}+5 \) is an antiderivative of \( f(x)=2e^{2x} \).

Step 1: Identify the Original Function

The original function we are given is \( f(x) = 2e^{2x} \). Our task is to determine if \( F(x) = e^{2x} + 5 \) is an antiderivative of \( f(x) \).

Step 2: Differentiate the Given Function

Let's find the derivative of \( F(x) \). We need to differentiate each term of \( F(x) = e^{2x} + 5 \).- The derivative of \( e^{2x} \) with respect to \( x \) is obtained using the chain rule: \( \frac{d}{dx}[e^{2x}] = 2e^{2x} \).- The derivative of a constant, like 5, is 0.

Step 3: Write the Derivative of F(x)

The derivative of \( F(x) \), combining the derivatives from Step 2, is:\[ F'(x) = 2e^{2x} + 0 = 2e^{2x} \]

Step 4: Compare with the Original Function

The derivative \( F'(x) \) matches exactly with \( f(x) = 2e^{2x} \). This confirms \( F(x) \) is an antiderivative of \( f(x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to find the derivative of composite functions. It simplifies the process of differentiating functions that are nested within each other. To apply the chain rule, we consider the outer and inner functions separately.

For the function \( e^{2x} \):

• The outer function is \( e^u \), where \( u = 2x \).
• The inner function is \( u = 2x \).

Applying the chain rule involves differentiating the outer function in terms of the inner function and then multiplying by the derivative of the inner function.

Here are the steps:

• Differentiate the outer function: \( \frac{d}{du}[e^u] = e^u \).
• Differentiate the inner function: \( \frac{d}{dx}[2x] = 2 \).
• Multiply the results: \( e^u \times 2 = 2e^{2x} \).

This result demonstrates the use of the chain rule to calculate the derivative of \( e^{2x} \), showing why \( \frac{d}{dx}[e^{2x}] = 2e^{2x} \).

Derivative

Derivatives represent the rate of change of a function with respect to a variable. They form the basis for much of calculus, making it possible to learn about the behavior of functions.

The process of differentiation can be applied to any function to find its derivative, which describes how the function's output changes as the input changes.

For example, when looking at the function \( F(x) = e^{2x} + 5 \), we differentiate each component:

• The term \( e^{2x} \) is differentiated using the chain rule, resulting in \( 2e^{2x} \).
• The constant term, \( 5 \), has a derivative of 0 since constants do not change.

After differentiating, you combine the results to obtain \( F'(x) = 2e^{2x} \), indicating that \( F(x) \) changes in the same manner as the original function \( f(x) = 2e^{2x} \).

Understanding how to find derivatives allows you to assess whether an antiderivative, like \( F(x) \), correctly represents a given derivative such as \( f(x) \).

Exponential Function

Exponential functions are a class of functions where a constant base is raised to a variable exponent. They are widely used due to their growth properties and occur naturally in many areas like population growth and finance.

The basic form of an exponential function is \( f(x) = a^x \). When the base is Euler's number \( e \), the function written is \( e^x \).

This exercise involves the exponential function \( e^{2x} \). Here, the rate of growth depends on the exponent, and this growth is described by the derivative calculated using the chain rule.

Some key properties of exponential functions include:

• They always have a positive growth rate, except when multiplied by zero.
• Their derivatives are proportional to their value, a feature unique to exponential functions.

In contrast to polynomial functions, exponential functions exhibit incredibly fast growth, making them powerful in mathematical modeling and real-life scenarios.