Problem 18 - Applied Calculus

Question

Let \(C(n)\) be a city's cost, in millions of dollars, for plowing the roads, when \(n\) inches of snow have fallen. Let \(c(n)=C^{\prime}(n) .\) Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given $$ \begin{array}{lll} e^{\prime}(n) & <0, & \int_{0}^{15} c(n) d n=7.5, & c(15)=0.7 \\ c(24) & =0.4, & C(15)=8, & C(24)=13 \end{array} $$ $$\int_{15}^{24} c(n) d n$$

Step-by-Step Solution

Answer

The increase in cost from 15 to 24 inches of snow is 5 million dollars.

Step 1: Understand the Relationship

First, recognize that the function \( C(n) \) represents the total cost of plowing, while \( c(n) = C'(n) \) represents the marginal cost, or the rate of change of the cost with respect to snow depth.

Step 2: Use Given Integrals

We are given \( \int_{0}^{15} c(n) \, dn = 7.5 \). This means that the total increase in cost from 0 to 15 inches of snow is 7.5 million dollars. We need to find \( \int_{15}^{24} c(n) \, dn \) to determine the total cost increase from 15 to 24 inches.

Step 3: Apply Fundamental Theorem of Calculus

The fundamental theorem of calculus relates the definite integral of a function to its antiderivative. Here, it states \( \int_{15}^{24} c(n) \, dn = C(24) - C(15) \).

Step 4: Substitute Given Values

Substitute the known costs into the expression: \( C(24) - C(15) = 13 - 8 \). This evaluates to \( 5 \).

Step 5: Interpret the Result

The integral \( \int_{15}^{24} c(n) \, dn = 5 \) million dollars. This means that the total increase in the cost of plowing when the snowfall increases from 15 inches to 24 inches is 5 million dollars.

Key Concepts

Integral Calculus

Integral calculus plays a crucial role in calculus by helping us understand changes over a continuum. It involves two main operations: computing definite and indefinite integrals. In this context, an **integral** is used to calculate the accumulation of quantities, such as cost, over an interval.

For example, given the function \( c(n) = C'(n) \), where \( C(n) \) is the total cost of plowing snow, the **definite integral** \( \int_{a}^{b} c(n) \ dn \) can be interpreted as the total increase in cost from \( a \) to \( b \) inches of snow. In our problem, we used the integral \( \int_{15}^{24} c(n) \ dn \) to find out how much the cost increased as snowfall went from 15 inches to 24 inches.

When you deal with integrals in calculus, there are a few crucial points to remember:

• **Function**: The function under the integral, here \( c(n) \), represents the rate of change, similar to a derivative.
• **Limits of Integration**: They tell you the interval over which you're summing continuous changes, such as snow depth or time.
• **Area Under the Curve**: In graphical terms, the integral represents the area under the curve of the given function over the specified interval.

Understanding these aspects solidifies the concept of integration as a tool for summing changes across a particular range, as we did to compute snow plowing costs.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) bridges the concepts of differentiation and integration. It states that if you have a continuous function and you integrate its derivative over an interval, you get the net change in the function over that interval.

There are two main parts to this theorem:

• **First Part**: It describes that the indefinite integral of a function is related to its antiderivative. Essentially, it tells us that differentiation and integration are inverse processes.
• **Second Part**: This is the part mostly used in problems like ours. It states that the integral of a function’s derivative over an interval can be calculated as the difference between the function's values at the boundaries of the interval, \( b-a \).

In our exercise, you find that \( \int_{15}^{24} c(n) \ dn = C(24) - C(15) \). This expression directly applies the second part of FTC.

The significance of this theorem in real-world problems is profound because it provides a method to calculate total quantities based on rates of change over time or different measures, like snow depth, allowing us to solve for total changes efficiently.

Marginal Cost

Marginal cost is a central concept in economics and calculus, particularly when analyzing costs in terms of changes in quantity. It represents how much the cost will change as you increase the quantity by one additional unit.

In this exercise, the marginal cost \( c(n) = C'(n) \) is the rate at which the cost changes with respect to the snow depth \( n \). It effectively tells us how costly it is to remove each additional inch of snow.

Understanding marginal cost involves:

• **Incremental Costs**: It shows the extra cost associated with a small increase in activity or production.
• **Efficiency**: Helps determine the optimal amount of snow that should be cleared to maintain cost efficiency.
• **Decision-Making**: Analyzing marginal costs aids in decisions regarding resource allocation for snow plowing or any other service.

In the context provided, \( c(n) \) helps to identify points where the costs may start to outweigh the benefits of clearing additional snow. Thus, marginal cost is not only critical in understanding and calculating costs but also essential for strategic financial planning.