Problem 46 - Applied Calculus

Question

Show that the relative rate of change of a quotient \(f / g\) is the difference between the relative rates of change of \(f\) and \(g\).

Step-by-Step Solution

Answer

The relative rate of change of \( f/g \) is the difference \( \frac{f'}{f} - \frac{g'}{g} \).

Step 1: Define Relative Rates of Change

The relative rate of change of a function is defined as \( \frac{f'(x)}{f(x)} \) for a function \( f(x) \). This represents how fast the function changes relative to its value at a point.

Step 2: Write Down the Quotient Rule

Given two differentiable functions \( f(x) \) and \( g(x) \), the derivative of their quotient \( \frac{f(x)}{g(x)} \) is given by the quotient rule: \[ \left( \frac{f}{g} \right)' = \frac{f' g - f g'}{g^2}. \]

Step 3: Compute the Derivative of the Quotient

Using the quotient rule, find the derivative \( \left( \frac{f}{g} \right)' = \frac{f' g - f g'}{g^2} \). This gives us the rate of change of the quotient \( f/g \).

Step 4: Express Relative Rate of Change of Quotient

The relative rate of change of \( \frac{f}{g} \) is \( \frac{1}{f/g} \cdot \left( \frac{f}{g} \right)' = \frac{g}{f} \cdot \frac{f' g - f g'}{g^2} = \frac{f' g - f g'}{f g}. \)

Step 5: Simplify the Expression

Simplify \( \frac{f' g - f g'}{f g} \) to get \( \frac{f'}{f} - \frac{g'}{g} \). This shows the relative rate of change of \( f/g \) is the difference between the relative rates of change of \( f \) and \( g \).

Key Concepts

Quotient Rule

The Quotient Rule is a formula used in calculus to find the derivative of the quotient of two differentiable functions. When dealing with functions that are divided by each other, like \( \frac{f(x)}{g(x)} \), the quotient rule provides a way to compute their derivatives easily. The rule is based on the idea that if both functions, \( f \) and \( g \), are differentiable, then their rate of change can be expressed as: \[\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}.\]Here's how it works:

• \( f' \) is the derivative of \( f \), representing the rate at which \( f(x) \) changes.
• \( g' \) is the derivative of \( g \), representing the rate at which \( g(x) \) changes.
• The numerator, \( f'g - fg' \), accounts for changes in the numerator and denominator separately.
• The denominator, \( g^2 \), is the original \( g(x) \) squared, to adjust the expression for any impact changes in \( g \) might have on the rate.

This rule is quite handy, especially when you need to find how fast a ratio of two functions is changing.

Calculus

Calculus is a branch of mathematics that studies continuous change. It is foundational for understanding how things evolve over time. This field can be divided into two main branches: differential calculus, which focuses on rates of change and slopes, and integral calculus, which deals with accumulation and areas. Differential calculus is particularly relevant here as it includes methods for calculating derivatives. Derivatives tell us the rate at which a quantity is changing at any given point, and this is crucial for problems like our exercise where we are interested in the relative rate of change of a quotient. Learning calculus helps in:

• Understanding the fundamental principles of change and motion.
• Solving practical problems in physics and engineering by modeling rates of change.
• Providing tools for optimization and analysis of real-life scenarios.

Overall, calculus provides the mathematical groundwork needed for advanced study in many fields.

Differentiable Functions

A function is said to be differentiable if it has a derivative at every point in its domain. This means that the function's graph is smooth and without any sharp turns or discontinuities. Differentiability is a key requirement when using tools like the quotient rule in calculus.When a function is differentiable:

• It has a well-defined slope or tangent at every point.
• It can be locally approximated by a linear function, which is its tangent line.
• Calculations involving its rate of change become more predictable and reliable.

In our exercise, both \( f \) and \( g \) are differentiable, which allows us to apply the quotient rule confidently. Without differentiability, we wouldn't be able to determine the relative rates of change reliably or work out other calculus-related computations effectively.