Problem 4 - Applied Calculus

Question

Write the Leibniz notation for the derivative of the given function and include units. An employee's pay, \(P\), in dollars, for a week is a function of the number of hours worked, \(H\)

Step-by-Step Solution

Answer

The derivative \( \frac{dP}{dH} \) represents the rate of pay change in dollars per hour.

Step 1: Identify the Variables

In this problem, we have the function \( P \) representing the pay in dollars, which depends on \( H \), the number of hours worked. Thus, \( P \) is a function of \( H \).

Step 2: Express the Function Relation

The pay function can be generally described as \( P = f(H) \), where \( P \) is in dollars and \( H \) is in hours.

Step 3: Write the Leibniz Notation for Derivative

The Leibniz notation for the derivative of pay \( P \) with respect to hours worked \( H \) is expressed as \( \frac{dP}{dH} \). This represents the rate of change of pay with respect to the number of hours worked.

Step 4: Interpret the Units

Since \( P \) is in dollars and \( H \) is in hours, the derivative \( \frac{dP}{dH} \) has units of dollars per hour. This indicates how many dollars the pay increases (or decreases) for each additional hour worked.

Key Concepts

Understanding Derivatives

The derivative is a fundamental concept in calculus. It tells us how a function changes as its input changes. In simpler terms, it's like asking, "If I change this one thing just a little, how will that affect the outcome?"
For our specific example, the function in question is an employee's pay based on the hours they work. The derivative, written as \( \frac{dP}{dH} \), measures the change in pay with respect to changes in hours worked.
This mathematical tool helps us understand and predict how small changes in input (hours) affect the output (pay). Thus, the derivative can offer insights into the rate at which pay increases as more hours are worked.

Exploring the Pay Function

In this context, the pay function, represented as \( P = f(H) \), is a mathematical representation of how an employee's pay is connected to the hours they work. Here, \( P \) denotes the pay in dollars, while \( H \) represents the hours worked.
The function itself is a rule that calculates the pay for a given number of hours. Think of it like a machine: you input the hours worked, and it provides the corresponding output, which is the total pay for those hours. Understanding this function is vital, as it forms the basis for calculating not just pay, but also other possible outcomes when changes occur.

Rate of Change Explained

The rate of change is a term often used interchangeably with "derivative". It tells us how fast or slow one quantity changes concerning another. In our case, it's about understanding how pay changes as hours increase.
When we use the Leibniz notation \( \frac{dP}{dH} \), we are particularly interested in how the pay rate per hour changes. If this rate is constant, pay increases by the same amount for every additional hour worked. However, if the rate varies, the pay might increase more (or less) for different intervals of hours. This is crucial for businesses and employees alike, as it directly affects earnings and budgeting.

The Role of Units of Measurement

Units of measurement give meaning to our mathematical expressions, especially derivatives. In this scenario, we are dealing with dollars and hours.
The derivative \( \frac{dP}{dH} \), therefore, has units of "dollars per hour." This tells us the pay adjustment for each additional hour worked. It's essentially the pay rate, or how much extra an employee earns for working one more hour.

• "Units" turn abstract mathematical expressions into practical, understandable terms.
• They ensure that calculations reflect real-world scenarios accurately.

Knowing the units involved helps in making informed decisions based on the interpreted data.