Problem 17 - Calculus for Biology and Medicine

Question

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=e^{-x}, x \in \mathbf{R} $$

Step-by-Step Solution

Answer

The function is always decreasing and concave up on \( x \in \mathbf{R} \).

Step 1: Calculate the First Derivative

The function given is \( y = e^{-x}\). To find the first derivative, apply the rule for exponential functions: \( \frac{d}{dx} e^{u} = e^{u} \cdot u' \). Here, \( u = -x \), so \( u' = -1 \). Thus, \( y' = \frac{d}{dx}e^{-x} = e^{-x} \cdot (-1) = -e^{-x} \).

Step 2: Use the First Derivative Test for Increasing/Decreasing

The first derivative \( y' = -e^{-x} \) describes the slope of the function. Since \( e^{-x} \) is always positive and has a multiplicative factor of \(-1\), \( y'\) is always negative for all \( x \in \mathbf{R} \). Thus, the function \( y = e^{-x} \) is always decreasing.

Step 3: Calculate the Second Derivative

To find concavity, calculate the second derivative. The first derivative is \( y' = -e^{-x} \). Differentiate again: \( y'' = \frac{d}{dx}(-e^{-x}) = -\frac{d}{dx}(e^{-x}) = -(-e^{-x}) = e^{-x} \).

Step 4: Use the Second Derivative Test for Concavity

The second derivative \( y'' = e^{-x} \) is always positive for all \( x \in \mathbf{R} \). A positive second derivative implies the function is concave up. Therefore, \( y = e^{-x} \) is concave up everywhere on its domain.

Key Concepts

First Derivative Test

The first derivative test is a useful tool in calculus that helps determine where a function increases or decreases. To use this test, we first find the derivative of the function, which tells us about the slope of the function at any point. Let's break it down step by step.

• Find the Derivative: For the function \( y = e^{-x} \), the first derivative is \( y' = -e^{-x} \). This was found using the rule for derivatives of exponential functions.
• Analyze the Derivative: Notice that \( -e^{-x} \) is always negative because \( e^{-x} \) is positive for all \( x \), and the negative sign changes the entire expression to a negative value.
• Conclusion: Since the derivative \( y' \) is negative everywhere, the function \( y = e^{-x} \) is always decreasing over its entire domain \( x \in \mathbf{R} \).

Second Derivative Test

The second derivative test helps us determine whether a function is concave up or concave down at particular intervals. Let's see how this works for the function \( y = e^{-x} \) through these steps:

• Calculate the Second Derivative: We differentiate the first derivative, \( y' = -e^{-x} \), which gives us \( y'' = e^{-x} \).
• Examine the Second Derivative: The expression \( e^{-x} \) is always positive no matter the value of \( x \). This is because exponential functions never dip below zero.
• Draw Conclusions: Since \( y'' > 0 \) for all \( x \), the function \( y = e^{-x} \) is concave up everywhere within the domain \( x \in \mathbf{R} \).

Concavity

Concavity is an important concept in calculus that tells us the direction in which a function curves. Here's how concavity influences our understanding of \( y = e^{-x} \):

• Understanding Concavity: A function is said to be concave up if it curves upwards like a smiley face. It is concave down if it curves downwards like a frown.
• Connection with the Second Derivative: Concavity is determined using the second derivative. If \( y'' > 0 \), the function is concave up, and if \( y'' < 0 \), it is concave down.
• For Our Function: Since for \( y = e^{-x} \) the second derivative \( y'' = e^{-x} \) is always positive, the curve is always concave up throughout the domain \( x \in \mathbf{R} \).