Problem 44 - Calculus for Biology and Medicine

Question

We are not always given the function of interest in explicit form.y\( is related to \)x\( by an implicit equation. Determine using implicit differentiation and the first derivative test whether \)y\( is an increasing or a decreasing function of \)x\( \)x y=e^{-y}, \quad x>0$

Step-by-Step Solution

Answer

\( y \) is decreasing when \( y > 0 \) and increasing when \( y < 0 \) for \( x > 0 \).

Step 1: Differentiate Implicitly

Start with the implicit function given: \( xy = e^{-y} \). To find how \( y \) changes with respect to \( x \), differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so you'll use the product rule for the left side and chain rule for the right side.On the left, differentiate \( xy \):\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]On the right, differentiate \( e^{-y} \):\[ \frac{d}{dx}(e^{-y}) = -e^{-y} \frac{dy}{dx} \]

Step 2: Solve for \( \frac{dy}{dx} \)

Set the differentiated left side equal to the differentiated right side:\[ x \frac{dy}{dx} + y = -e^{-y} \frac{dy}{dx} \]Rearrange terms to isolate \( \frac{dy}{dx} \):\[ x \frac{dy}{dx} + e^{-y} \frac{dy}{dx} = -y \]Factor out \( \frac{dy}{dx} \):\[ \frac{dy}{dx}(x + e^{-y}) = -y \]Solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{-y}{x + e^{-y}} \]

Step 3: Analyze the Sign of \( \frac{dy}{dx} \)

To determine if \( y \) is increasing or decreasing with respect to \( x \), check the sign of \( \frac{dy}{dx} \).Since \( x > 0 \), both \( x \) and \( e^{-y} \) are positive. Therefore, \( x + e^{-y} > 0 \) always. The numerator is \( -y \). Since we are not given a specific range for \( y \), analyze possibilities:- If \( y > 0 \), then \(-y < 0\), leading \( \frac{dy}{dx} \) to be negative; thus, \( y \) is decreasing.- If \( y < 0 \), then \(-y > 0\), leading \( \frac{dy}{dx} \) to be positive; thus, \( y \) is increasing.

Key Concepts

Calculus

Calculus is one of the essential branches of mathematics that helps us understand change through derivatives and integration. The focus of this exercise is implicit differentiation, a method used when you have equations involving variables implicitly related, not solved for one specific variable. Let's break down how calculus helps us understand implicit equations.

Implicit Differentiation
When the function between two variables is not expressed explicitly, we use implicit differentiation. In our problem, the function is given by the equation \( xy = e^{-y} \). Here, \( y \) is not easily isolated on one side, making it suitable for implicit differentiation:

• Differentiate both sides with respect to \( x \).
• Apply the product rule to products of functions and chain rule when differentiating compositions of functions.
• The goal is to express \( \frac{dy}{dx} \), which represents the rate of change of \( y \) with respect to \( x \).

In our example, implicit differentiation allows us to derive \( \frac{dy}{dx} = \frac{-y}{x + e^{-y}} \), giving a way to analyze how \( y \) changes as \( x \) varies.

First Derivative Test

The first derivative test is a powerful tool in calculus to determine where a function increases or decreases, providing insights into its behavior. After differentiating and finding the first derivative of a function, the challenge is to analyze the sign.

Applying the First Derivative Test
Using the first derivative test on \( \frac{dy}{dx} = \frac{-y}{x + e^{-y}} \) helps determine if \( y \) is increasing or decreasing with respect to \( x \) based on the sign of \( \frac{dy}{dx} \):

• If \( \frac{dy}{dx} > 0 \), the function \( y \) is increasing. This occurs when \(-y > 0\), meaning \( y < 0 \).
• If \( \frac{dy}{dx} < 0 \), the function \( y \) is decreasing. This happens when \(-y < 0\), therefore \( y > 0 \).

Since \( x > 0 \) and the sum \( x + e^{-y} \) is positive, the test rests solely on the sign of \(-y\). The first derivative test thus tells us how \( y \) behaves based on its value, providing an avenue to understand functions in calculus better.

Function Analysis

Function analysis involves studying a function's properties to understand its graphical behavior thoroughly. Through calculus and tests like the first derivative, we explore characteristics like monotonicity, critical points, and intervals of increase or decrease.

Understanding Through Implicit Equations
When given an implicit equation like \( xy = e^{-y}\), applying function analysis means diving deep into how changes in \( x\) affect \( y \). Implicit differentiation provides the derivative, and the derivative's sign tells us about changes:

• Behavior for \( y > 0 \): We saw that \( \frac{dy}{dx} < 0 \), indicating \( y \) decreases as \( x \) increases. This section is a decreasing interval.
• Behavior for \( y < 0 \): Here, \( \frac{dy}{dx} > 0 \), so \( y \) increases with \( x\), marking an increasing interval.

Function analysis using implicit differentiation and the first derivative test in calculus provides a comprehensive picture of a function's behavior without needing a graph or explicit equation, empowering deeper understanding of mathematical relationships.