Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=e^{-x}\)

Step-by-Step Solution

Answer

The function is decreasing for all real numbers.

Step 1: Understand the Monotonicity Theorem

The Monotonicity Theorem states that a function is increasing in an interval if its derivative is positive throughout that interval, and it is decreasing if its derivative is negative throughout that interval.

Step 2: Find the Derivative of the Function

To determine where the function is increasing or decreasing, we first find the derivative of the function. Given \[ f(x) = e^{-x} \]The derivative is \[ f'(x) = -e^{-x} \] by using the chain rule as the derivative of \( e^u \) is \( e^u \) multiplied by the derivative of \( u \), where \( u = -x \).

Step 3: Analyze the Derivative Sign

To know where \( f(x) \) is increasing or decreasing, examine the sign of \( f'(x) \). \[ f'(x) = -e^{-x} \] The term \( e^{-x} \) is always positive for any real number \( x \), but since there is a negative sign in front, \( f'(x) \) is always negative for all \( x \).

Step 4: Conclusion

Since \( f'(x) \) is negative for all real numbers \( x \), the function \( f(x) = e^{-x} \) is decreasing on the interval \((-, \infty, \infty)\).

Key Concepts

Derivative

A derivative is a fundamental tool in calculus used to determine how a function changes as its input changes. It can be seen as the rate of change or the slope of a function at any given point. Calculating the derivative involves differentiating the function.

• To find the derivative of a function like \( f(x) = e^{-x} \), we can use rules of differentiation, such as the chain rule.
• A derivative tells us whether a function is increasing or decreasing. If the derivative is positive, the function increases; if negative, the function decreases.

In our problem, the derivative was found to be \( f'(x) = -e^{-x} \). This indicates that \( f(x) = e^{-x} \) has a negative slope for all values of \( x \), meaning the function continuously decreases.

Increasing and Decreasing Intervals

Understanding where a function increases or decreases is key to analyzing its behavior over different intervals. We use the derivative to determine these regions.

• A function increases on an interval if the derivative is positive throughout that interval.
• Conversely, it decreases on an interval if the derivative is negative.

In the exercise, the derivative \( f'(x) = -e^{-x} \) is always negative since \( e^{-x} \) is inherently positive, and the entire expression is multiplied by -1. Thus, the original function \( f(x) = e^{-x} \) is decreasing across all real numbers.

Chain Rule

The chain rule is a technique in calculus used to differentiate composite functions. It helps us find the derivative of a function that is nested within another function.

• To use the chain rule, differentiate the outer function and multiply it by the derivative of the inner function.
• In this step, the chain rule allows us to take the derivative of \( e^{-x} \) efficiently.

By identifying \( u = -x \), the derivative of the inner function \( u \) is \(-1\). Therefore, the derivative of \( e^u \) (with respect to \( u \)) is \( e^u \). Applying the chain rule, the derivative becomes \( -e^{-x} \). This process highlights how the chain rule makes differentiating exponential functions with linear components straightforward.

Exponential Functions

Exponential functions are a significant category of functions where a constant base is raised to a variable exponent. These functions frequently appear in real-world applications, such as in population growth and compound interest.

• The exponential function \( e^x \) has special properties; its derivative is itself. This is unique and simplified using the chain rule.
• Understanding how to differentiate functions of the form \( e^u \) is crucial, especially when exploring and analyzing real-world phenomena.

In our exercise, we analyzed \( f(x) = e^{-x} \), a decreasing exponential function due to the negative sign in the exponent. This characteristic of exponential functions helps model decay processes or other situations where something decreases continuously over time.