Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(C(w)\) be the amount (in dollars) it costs you to mail your grandmother a firstclass package weighing \(w\) ounces. Suppose you just mailed her a birthday present that weighed \(A\) ounces. Describe in words the practical meaning of each of the following expressions. (a) \(C(A)\) (b) \(C(2 A)\) (c) \(2 C(A)\) (d) \(C(A+1)\) (e) \(C(A)+1\)

Step-by-Step Solution

Answer

The expressions \(C(A)\), \(C(2A)\), \(2C(A)\), \(C(A+1)\), and \(C(A)+1\) represent the cost of mailing the present, double the present's weight, two presents of original weight, a present plus one extra ounce, and extra one dollar to the present's cost, respectively.

Step 1: Interpreting Expression a - \(C(A)\)

To determine the cost of mailing a package that weighs \(A\) ounces, the cost function \(C(w)\) should be applied. Here \(w\) is replaced by \(A\). Hence, \(C(A)\) refers to the cost to mail the present package to your grandmother.

Step 2: Interpreting Expression b - \(C(2A)\)

The weight of the package is doubled in this case, i.e., \(2A\) ounces. Hence, \(C(2A)\) accounts for the cost of mailing a package that is twice the weight of the birthday present.

Step 3: Interpreting Expression c - \(2C(A)\)

This signifies the double amount of the cost that was required to send the birthday present. So, \(2C(A)\) is the cost of mailing two packages, each weighing \(A\) ounces.

Step 4: Interpreting Expression d - \(C(A+1)\)

This illustrates the cost of sending a package that is one ounce heavier than the original birthday present. Hence, \(C(A+1)\) is the cost of mailing a package weighing \(A+1\) ounces.

Step 5: Interpreting Expression e - \(C(A)+1\)

This indicates that one dollar is added to the original cost of mailing the birthday present. Hence, \(C(A)+1\) represents the cost of mailing the birthday present plus an extra dollar.

Key Concepts

Cost Function Interpretation

Understanding the cost function in calculus is essential when dealing with practical problems involving expenses. For instance, let's consider the expression C(A), which in a practical sense means the cost of mailing a package that weighs A ounces. It exemplifies how we use a function to associate a particular weight with its corresponding mailing cost.

When dealing with C(2A), the interpretation shifts to calculating the cost of an item that's twice as heavy. Similarly, 2C(A) implies the total cost for two individual items, each of the same weight as the initial package - a concept known colloquially as 'bulk pricing' or 'combined shipping'. For C(A+1), we're exploring the cost of sending a slightly heavier package, demonstrating how even small changes in the variable can affect the overall cost. The functional notation C(A)+1 reflects an additional charge perhaps due to handling or insurance. This way, functional notation lets us play with variables and constants to understand different scenarios.

Variable Manipulation in Functions

Manipulating variables within functions allows us to explore how changes in one quantity affect another. This is a cornerstone of algebra and is vividly seen in our cost function example. Take the expression C(A+1) from our mailing scenario. Here's where variable manipulation comes into play; adding one to the weight variable A provides insight into how the cost is affected by an incremental weight increase.

By applying such alterations to the input A, we can predict changes in cost due to different package weights. This is a powerful technique, especially when creating models or making decisions based on varying conditions. When we double the original weight to 2A, we manipulate the variable to analyze a hypothetical situation. This operation is crucial in many fields like economics, engineering, and science where functions model real-world situations, and changing variables help in making predictions and informed decisions.

Practical Application of Functions

The cost function is just one of many practical applications of functions in real life. They serve as models to connect different quantities, providing a way to predict and analyze outcomes. Functions can represent anything from the trajectory of a ball in physics to the population growth in biology, or the interest accumulation in finance.

Our previous example prominently demonstrated that functions are not just abstract mathematical constructs but have a tangible impact on day-to-day activities. For businesses, functions can predict profits based on sales, for meteorologists, they can predict weather changes, and in health care, they can calculate medication doses based on a patient's weight. The list is endless. By understanding functional notations and how to manipulate variables within them, we unlock the ability to analyze and influence the world around us in profound ways.