Problem 28 - Applied Calculus

Question

Annual net sales, in billion of dollars, for the Hershey Company, the largest US producer of chocolate, is a function \(S=f(t)\) of time, \(t,\) in years since 2000 (a) Interpret the statements \(f(8)=5.1\) and \(f^{\prime}(8)=\) 0.22 in terms of Hershey sales. \(^{13}\) (b) Estimate \(f(12)\) and interpret it in terms of Hershey sales.

Step-by-Step Solution

Answer

\( f(8) \) shows sales in 2008; \( f^{\prime}(8) \) shows sales increased at $0.22 billion/year; estimate \( f(12) = 5.98 \) by 2012.

Step 1: Interpret \( f(8) = 5.1 \)

The function value \( f(8) = 5.1 \) indicates that 8 years since the year 2000, which is the year 2008, the net sales of the Hershey Company were 5.1 billion dollars. This provides a snapshot of the company’s sales in that specific year.

Step 2: Interpret \( f^{\prime}(8) = 0.22 \)

The derivative \( f^{\prime}(8) = 0.22 \) represents the rate of change of sales in 2008. Specifically, it indicates that the sales were increasing by 0.22 billion dollars per year around the year 2008. This suggests a growth trend in sales for that period.

Step 3: Estimate \( f(12) \)

To estimate \( f(12) \), we can use the information that sales were increasing at a rate of 0.22 billion dollars per year from 2008 onward. Since \( t = 12 \) corresponds to the year 2012 (4 years after 2008), we calculate:\[ f(12) \approx f(8) + (4) \times f^{\prime}(8) = 5.1 + 4 \times 0.22 = 5.98 \]Thus, the estimated net sales for Hershey in 2012 is approximately 5.98 billion dollars.

Step 4: Interpret \( f(12) \)

Based on our estimation, \( f(12) = 5.98 \) implies that by the year 2012, Hershey's net sales were approximately 5.98 billion dollars, assuming the growth rate remained constant from 2008 to 2012.

Key Concepts

Derivative

In calculus, the term "Derivative" plays a crucial role in understanding how a function changes. It's like a measurement of change at any given point. For instance, when you hear about the derivative of a function, it refers to the instantaneous rate at which one quantity changes with respect to another. Think of it as the speedometer in a car: it tells you how fast you're going at any specific moment.
In our example, we are determining the derivative of Hershey's sales function at a specific time, say in 2008. The notation used is typically a small prime symbol, like in the expression \( f'(8) = 0.22 \). This means the sales were increasing by \(0.22\) billion per year around the year 2008.

• Derivatives help us understand trends, such as if sales are increasing or decreasing at any given time.
• They're fundamental in fields ranging from economics to engineering, anywhere change is occurring.
• Understanding derivatives can help not only in prediction but also in optimizing processes.

At its core, a derivative is about change, and learning how this change manifests provides insights into analyzing numerous real-world situations effectively.

Function Notation

Function notation provides a clean and clear way to express relationships between variables. If you ever see a notation such as \( f(t) \), it's a simple yet powerful way to convey how one variable depends on another. Here \( f \) represents our function, and \( t \) indicates the variable we're observing. This notation is concise and gets to the heart of what calculus is all about: modeling real-world phenomena and their changes over time.
Using function notation, we can describe how Hershey's sales evolve as time progresses. For example, \( f(8) = 5.1 \) tells us that in the year 2008, Hershey's sales reached 5.1 billion dollars. This direct interpretation aids in easily understanding data at given points without excessive calculations.
Function notation is universal and becomes a powerful tool in:

• Mathematical modeling
• Predicting outcomes
• Communicating complex relationships simply

Embracing function notation is like gaining a new language in the realm of mathematics, one that simplifies and streamlines communication of mathematical concepts.

Interpreting Mathematical Models

Interpreting mathematical models can demystify complicated relationships and predict future trends. A mathematical model is essentially a representation of the real world, using functions, equations, and data points. It turns complex scenarios into understandable, abstract concepts.
In our Hershey sales example, the function \( S = f(t) \) serves as a model depicting how sales change over time. By analyzing statements like \( f(8) = 5.1 \) and \( f'(8) = 0.22 \), we can gauge the health of the business, recognize past patterns, and make educated guesses about the future. This small model aids business leaders in understanding and planning strategies.
Interpreting models involves:

• Translating mathematical expressions into real-world insights.
• Looking beyond the numbers to understand their implications.
• Using estimates, like \( f(12) \), to predict outcomes under current assumptions.

These interpretations are an invaluable part of decision-making processes across various industries, not just business. They support informed choices by providing a "big picture" view, helping to break down complex situations into understandable components.