Problem 41 - Calculus for Biology and Medicine

Question

We are not always given the function of interest in explicit form. In each of Problems \(41-44 y\) 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 \(x\) \(x^{2}+y^{2}=1, \quad 00\)

Step-by-Step Solution

Answer

The function \(y\) is decreasing for \(0 < x < 1\).

Step 1: Differentiate Implicitly

Given the equation \(x^2 + y^2 = 1\), we'll differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\). Differentiating gives \(2x + 2y \frac{dy}{dx} = 0\).

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

From \(2x + 2y \frac{dy}{dx} = 0\), isolate \(\frac{dy}{dx}\) by subtracting \(2x\) from both sides: \(2y \frac{dy}{dx} = -2x\). Then divide by \(2y\) to get \(\frac{dy}{dx} = -\frac{x}{y}\).

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

Given \(0 < x < 1\) and \(y > 0\), both \(-x\) and \(y\) are positive. Thus, \(\frac{dy}{dx} = -\frac{x}{y}\) is negative because it is the division of a negative numerator by a positive denominator.

Step 4: Conclusion on Function Behavior

Since \(\frac{dy}{dx}\) is negative for the given domain, it follows from the first derivative test that the function \(y\) is decreasing on the interval \(0 < x < 1\).

Key Concepts

First Derivative Test

The first derivative test is a common method used in calculus to determine the characteristics of a function, specifically whether the function increases or decreases over an interval. When we talk about increasing or decreasing functions, we refer to how the outputs of the function behave as the inputs move from left to right on a graph over a given interval.
To apply the first derivative test, we first find the first derivative of the function, which represents the rate of change of the function values. If a function's derivative is positive over an interval, this indicates that the function is increasing in that region. Conversely, if the derivative is negative, the function is decreasing. If the derivative equals zero, this could suggest a stationary point or a change in behavior but requires further analysis.
In this exercise, after performing implicit differentiation, we found that the derivative of the function, \(\frac{dy}{dx} = -\frac{x}{y}\), is negative when the conditions \(0 < x < 1\) and \(y > 0\) hold. Since the derivative is consistently negative across the interval, the first derivative test confirms that the function \(y\) is decreasing for \(0 < x < 1\). This insight helps us deeply understand how the function behaves just by looking at its derivative.

Implicit Equation

An implicit equation is one where the dependent variable is not isolated on one side of the equation. In simpler terms, the relationship between the variables is interconnected in such a way that the output variable, like \(y\), is not explicitly solved for in terms of the input variable, like \(x\). This contrasts with an explicit equation, where you'd have something like \(y = f(x)\).
In our exercise, the given equation \(x^2 + y^2 = 1\) is a classic example of an implicit equation. It defines a relationship between \(x\) and \(y\) without providing a direct formula for \(y\) in terms of \(x\). Solving for \(y\) directly in such cases can be complex or not possible, requiring techniques like implicit differentiation to explore the function's properties.
This type of equation is common in mathematical problems dealing with curves that don't neatly fit into the explicit format, like circles. Working through implicit equations teaches problem-solving techniques that go beyond simple algebra, thus broadening analytical skills.

Function Behavior Analysis

Analyzing function behavior helps us understand how a function behaves across its domain. This involves studying various characteristics of the function, such as increasing or decreasing behavior, rate of change, and extremum points.
To analyze the behavior of a function from an implicit equation like \(x^2 + y^2 = 1\), implicit differentiation proves vital. By deriving \(\frac{dy}{dx} = -\frac{x}{y}\), we gain insight into how \(y\) changes as \(x\) varies. Within the given interval \(0 < x < 1\) and under the condition \(y > 0\), the derivative is negative, indicating a decreasing trend.
Understanding function behavior isn't only about solving equations but about recognizing patterns of change and anticipating how variables interact. This skill is crucial for tackling more complex problems, as it allows us to predict and analyze how different forces and influences affect outcomes in real-world scenarios. Seeing these patterns can also simplify problem-solving and provide clearer strategies for approaching mathematical challenges.