Problem 5 - Applied Calculus

Question

The cost, \(C=f(w)\), in dollars of buying a chemical is a function of the weight bought, \(w\), in pounds. (a) In the statement \(f(12)=5\), what are the units of the 12 ? What are the units of the 5 ? Explain what this is saying about the cost of buying the chemical. (b) Do you expect the derivative \(f^{t}\) to be positive or negative? Why? (c) In the statement \(f^{\prime}(12)=0.4\), what are the units of the 12 ? What are the units of the \(0.4\) ? Explain what this is saying about the cost of buying the chemical.

Step-by-Step Solution

Answer

(a) 12 is in pounds; 5 is in dollars. (b) Positive; cost likely increases with weight. (c) 12 is in pounds; 0.4 is in dollars per pound, indicating cost increase rate.

Step 1: Understanding Units in a Function

In the function notation \(f(w)\), \(w\) represents the input, which is the weight in pounds. Therefore, in \(f(12) = 5\), the number 12 represents pounds. The value of the function, 5, represents the cost in dollars. Therefore, this statement means that buying 12 pounds of the chemical costs \$5.

Step 2: Predicting the Sign of the Derivative

The derivative of the function \(f(w)\), denoted as \(f'(w)\), represents the rate of change of cost with respect to weight. Generally, we expect that buying more of a chemical leads to a higher total cost, implying that \(f'(w)\) is positive. Thus, the cost increases as the weight (amount) increases.

Step 3: Analyzing the Derivative Statement

In \(f'(12) = 0.4\), the 12 represents the weight in pounds, similar to the first statement. The derivative value 0.4 indicates the rate of change of cost per pound, with units of dollars per pound. This means that at 12 pounds, the cost is increasing by \$0.40 per additional pound.

Key Concepts

Function Notation

In calculus, function notation is a way to symbolize how one quantity depends on another. When you see a function such as \( f(w) \), it shows that the output or result "\( f(w)\)" is controlled by the input "\(w\)." Here, \( w \) is the weight of a chemical in pounds, while \( f(w) \) is the cost in dollars of that weight of the chemical.

To make things clearer, let's look at \( f(12) = 5 \). The number 12 signifies a specific weight — 12 pounds. The number 5 represents the cost at that weight — 5 dollars. Essentially, buying 12 pounds of the chemical costs 5 dollars.

Function notation is important as it concisely communicates a relationship between variables. It gives a clear way to describe how one quantity varies when another does.

Derivative Interpretation

Understanding derivatives is key in calculus. A derivative shows how one quantity changes as another changes. In our scenario, the derivative \( f'(w) \) represents the change in cost for each additional pound of chemical.

When considering the sign of a derivative like \( f'(w) \), we think about what it implies practically.

• If \( f'(w) \) is positive, the cost rises as you purchase more weight, an expected behavior for buying in larger quantities.
• If it's negative, it would mean costs decrease with more weight, which is not typical for buying chemicals or products.

In the problem, we expect \( f'(w) \) to be positive, reflecting that buying more weight leads to a higher total cost.

Units of Measurement

Understanding units in calculus problems is crucial for interpreting results correctly.

• In \( f(12) = 5 \), the number 12 is in pounds, and 5 is in dollars, showing weight and cost respectively.
• With \( f'(12) = 0.4 \), the derivative 0.4 has units of dollars per pound. Why? Because it measures how much the cost increases with each additional pound.

Let's break down \( f'(12) = 0.4 \): It tells us that at 12 pounds, each extra pound of chemical adds $0.40 to the total cost. Recognizing units helps in making sense of the rates and outcomes described by derivatives.