Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Is there an example of two functions, \(u(x)\) and \(v(x),\) for which \(\left[\frac{u(x)}{v(x)}\right]^{\prime}=\frac{u^{\prime}(x)}{v^{\prime}(x)} ?\)

Step-by-Step Solution

Answer

Yes, example: \(u(x) = c, v(x) = cx+d\).

Step 1: Understand the Problem

We are asked to identify functions \( u(x) \) and \( v(x) \) such that the derivative of their quotient equals the quotient of their derivatives. This means \( \left[ \frac{u(x)}{v(x)} \right]^{\prime} = \frac{u^{\prime}(x)}{v^{\prime}(x)} \).

Step 2: Recall the Quotient Rule

The derivative of a quotient is given by the Quotient Rule: \( \left[ \frac{u(x)}{v(x)} \right]^{\prime} = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{[v(x)]^2} \). We need this expression to be equal to \( \frac{u^{\prime}(x)}{v^{\prime}(x)} \).

Step 3: Set the Two Expressions Equal

Set the quotient rule expression equal to the given condition: \( \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{[v(x)]^2} = \frac{u^{\prime}(x)}{v^{\prime}(x)} \).

Step 4: Cross-Multiply

Cross-multiply to get: \( u^{\prime}(x)v(x)v^{\prime}(x) = u(x)[v^{\prime}(x)]^2 + u^{prime}(x)[v(x)]^2 \).

Step 5: Analyze the Equation

Analyze the structure of the equation after cross-multiplying. This is generally a complex condition to satisfy for arbitrary functions.

Step 6: Evaluate Specific Cases

Consider simple functions. Let \( u(x) = c \) and \( v(x) = cx + d \), where \( c \) and \( d \) are constants. Then \( u^{\prime}(x) = 0 \) and \( v^{\prime}(x) = c \). Substitute into the equations: \( \left[ \frac{c}{cx+d} \right]^{\prime} = \frac{0}{c} = 0 \), and also \( \frac{0}{c} = 0 \). These match, so it works.

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental technique in calculus for finding the derivative of a quotient of two functions. It's particularly useful when you have a function that can be expressed as one function divided by another.
To apply the Quotient Rule, if you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their quotient \( \frac{u(x)}{v(x)} \) is given by the formula:
\[ \left[ \frac{u(x)}{v(x)} \right]^{\prime} = \frac{u^{\prime}(x) v(x) - u(x) v^{\prime}(x)}{[v(x)]^2} \]
In this formula:

• \( u^{\prime}(x) \) is the derivative of \( u(x) \).
• \( v^{\prime}(x) \) is the derivative of \( v(x) \).
• The denominator \( [v(x)]^2 \) ensures we aren't dividing by zero.

The Quotient Rule is a bit more complex than its counterpart, the Product Rule, but it's essential when dealing with division in derivatives.

Derivative

Derivatives are a cornerstone of calculus, reflecting how a function changes as its input changes. Put simply, the derivative of a function at any point gives the slope of the tangent line to the curve at that point.
To find a derivative:

• Determine the rate of change, or slope, of the function.
• Use derivative rules, like the power rule, product rule, or quotient rule to calculate derivatives analytically.

Derivatives not only tell you about rates of change but also about the function's behavior, such as finding maxima, minima, and points of inflection.
Understanding derivatives enables deeper insights into the nature of functions and their graph properties.

Function Analysis

Function analysis involves examining the properties and behavior of functions. This often includes studying the derivatives of the function for better understanding.
Key elements of function analysis include:

• Continuity: Evaluate where the function is smooth or where it has jumps or breaks.
• Critical Points: Identify where the derivative is zero or undefined, which can indicate peaks, troughs, or flat points on the graph.
• Intervals of Increase/Decrease: Use the derivative to see where the function is going up or down.
• Concavity: Use the second derivative to determine the curvature.

Function analysis helps not just in understanding individual functions, but in comparing them and their interaction. This analysis is essential for higher-level calculus and real-world application of functions.

Mathematical Problem Solving

Mathematical problem-solving is a process that involves several steps to find solutions to complex problems. It often requires a blend of creativity and technical skill.
Key strategies include:

• Understanding the Problem: Fully grasping the question and what's being asked.
• Breaking down the Problem: Decompose the problem into smaller, more manageable parts.
• Exploring Possible Solutions: Consider different methods, such as substituting simpler functions or using known rules like the Quotient Rule.
• Verification: Check that the solutions satisfy all conditions and re-evaluate if necessary.

In the exercise at hand, mathematical problem solving was used to determine specific functions \( u(x) \) and \( v(x) \) for which \( \left[\frac{u(x)}{v(x)}\right]^{\prime}=\frac{u^{\prime}(x)}{v^{\prime}(x)} \) holds true. This entailed both an understanding of the calculus principles involved and creative thinking to identify cases that meet the given criteria.