Problem 27 - Calculus

Question

Sketch reasonable graphs for the following. Pay particular attention to the concavity of the graphs. (a) The total revenue generated by a car rental business, plotted against the amount spent on advertising. (b) The temperature of a cup of hot coffee standing in a room, plotted as a function of time.

Step-by-Step Solution

Answer

(a) Concave down, flat slope due to diminishing returns. (b) Initially steep, concave up, approaching room temperature.

Step 1: Analyze Total Revenue vs. Advertising Spend

Consider how the total revenue of a car rental business might initially increase with more advertising expenditure, reaching a point of diminishing returns. In the beginning, revenue increases significantly with initial advertising due to increased visibility and customer reach. As advertising continues to increase, the growth in revenue starts to slow down, indicating diminishing returns.

Step 2: Sketch the Graph for Revenue vs. Advertising

Draw the graph with advertising spend on the x-axis and total revenue on the y-axis. Start with the graph rising steeply, depicting a rapid increase in revenue. Gradually, the slope should flatten, indicating diminishing returns, forming a concave down curve.

Step 3: Analyze Temperature of Coffee vs. Time

Understand the cooling process of hot coffee: initially, the coffee temperature drops quickly, and over time, it cools more slowly as it approaches room temperature. This process can be described using Newton's Law of Cooling, where the rate of temperature change is proportional to the difference between the coffee's temperature and the room temperature.

Step 4: Sketch the Graph for Coffee Temperature vs. Time

Plot time on the x-axis and coffee temperature on the y-axis. Start the graph at the initial high temperature of the coffee, decreasing rapidly initially. As time passes, the rate of cooling reduces and the curve becomes concave up, asymptotically approaching room temperature.

Key Concepts

Concavity

Concavity in calculus helps us understand how a graph bends, providing insights into the rate of change's increase or decrease. When we draw graphs, especially in the context of business or temperature changes, knowing whether a curve is concave up or down is crucial. A concave down graph looks like an upside-down bowl. This appears in real-life scenarios, like when plotting revenue against advertising spend. Initially, revenue increases steeply, meaning high returns from advertising. However, as more money is poured into advertising, the returns stabilize and eventually slow down.

Recognizing a concave down curve can be vital in business contexts since it shows when additional investments begin to yield diminishing returns. This helps businesses avoid overinvestment at points where expenses aren't yielding proportionate gains.

- **Concave Down:** The slope of the graph decreases as you move along the x-axis.
- **Concave Up:** Think of a right-side-up bowl; here, slopes get steeper. However, this is less relevant in the diminishing returns context but crucial in understanding the curve’s behavior in other scenarios.

Diminishing Returns

The principle of diminishing returns is an essential concept in economics and business, explaining how increasing one input (like advertising) while holding others constant leads to reduced incremental gains. Imagine a car rental business spending more on advertising. Initially, it attracts many new customers because the visibility is fresh and more people learn about the service.

Yet, as advertising continues to increase beyond a point, the influx of new customers grows at a slower pace. The graph depicting this scenario would show a steep rise initially, flattening out as expenditures climb. This flattening indicates diminishing returns, as each new dollar spent brings fewer sales.

- **Key Points of Diminishing Returns in Graphs:** - Rapid initial increase in output (revenue or sales). - Gradual flattening as additional investments yield lesser increments. - Useful for businesses to know when additional spending is no longer efficient.

Newton's Law of Cooling

Newton's Law of Cooling provides an excellent framework for predicting how the temperature of an object changes over time in relation to its surroundings. If you've ever left a hot coffee to sit in a room, you've seen this phenomenon in action. Initially, the coffee cools rapidly due to the stark temperature differential between the coffee and the ambient air. As time progresses, this difference lessens, and thus the cooling rate diminishes.

In mathematical terms, the rate of temperature change \( \frac{dT}{dt} \) is proportional to the difference between the object's temperature \( T \) and the ambient temperature \( T_a \).

- **Graphing Newton's Law of Cooling:** - Starts with a steep descent when temperature is far from the surroundings. - Curve eventually flattens as it approaches room temperature, illustrating a concave up behavior. - Provides a basis for understanding real-world cooling and heating processes.