Problem 12 - Calculus Concepts

Question

Fuel Efficiency The function \(g\) gives the fuel efficiency, in miles per gallon, of a car traveling \(v\) miles per hour. Write a sentence of interpretation for each of the following statements. a. \(g(55)=32.5\) and \(g^{\prime}(55)=-0.25\) b. \(g^{\prime}(45)=0.15\) and \(g^{\prime}(51)=0\) c. Sketch a possible graph of \(g\).

Step-by-Step Solution

Answer

The car's efficiency at 55 mph is 32.5 mpg, decreasing by 0.25 mpg/mph. At 45 mph, efficiency increases by 0.15 mpg/mph; at 51 mph, efficiency change stops. Graph peaks or levels at 51 mph.

Step 1: Interpret g(55) = 32.5

The value \(g(55) = 32.5\) indicates that when the car is traveling at a speed of 55 miles per hour, its fuel efficiency is 32.5 miles per gallon. This tells us how many miles the car can travel on one gallon of fuel at this specific speed.

Step 2: Interpret g'(55) = -0.25

The derivative \(g'(55) = -0.25\) represents the rate of change of fuel efficiency with respect to speed at \(v = 55\) miles per hour. The negative sign indicates that, at 55 mph, increasing the speed slightly will decrease fuel efficiency by 0.25 miles per gallon per additional mile per hour.

Step 3: Interpret g'(45) = 0.15

The value \(g'(45) = 0.15\) shows that at a speed of 45 miles per hour, the fuel efficiency is increasing at a rate of 0.15 miles per gallon per mile per hour. This suggests that driving faster from this point improves fuel efficiency slightly.

Step 4: Interpret g'(51) = 0

The value \(g'(51) = 0\) indicates that at 51 miles per hour, the fuel efficiency is neither increasing nor decreasing. This suggests a possible local maximum or plateau in the function \(g\), where changes in speed do not affect the fuel efficiency.

Step 5: Sketch a Graph of g

To sketch a possible graph of \(g\), plot speed \(v\) on the x-axis and fuel efficiency \(g(v)\) on the y-axis. At \(v = 45\), remember that \(g'(45) > 0\), so the graph should be increasing. At \(v = 51\), \(g'(51) = 0\), indicating a peak or flat point. At \(v = 55\), \(g'(55) < 0\), suggesting a decrease in fuel efficiency, so the graph should start sloping downwards. A reasonable graph may peak around 51 mph and slope slightly downward as speed increases to 55 mph.

Key Concepts

Fuel Efficiency

Fuel efficiency measures how effectively a vehicle converts fuel into distance traveled, typically expressed as miles per gallon (mpg). Understanding this concept is crucial for optimizing driving habits and reducing fuel costs.
In our exercise, the function \( g \) denotes the fuel efficiency of a car based on its speed \( v \). For example, \( g(55) = 32.5 \) means at 55 mph, the car can travel 32.5 miles on one gallon of fuel. This measurement helps drivers make informed decisions about their speed to maximize fuel savings.
Fuel efficiency is not constant; it varies with speed. Vehicles often reach optimal fuel efficiency at certain speeds, beyond which efficiency decreases. Therefore, knowing this optimal speed range is beneficial for both economical and environmental reasons.
Key takeaways include:

• Precision in speed can lead to significant savings in fuel costs.
• The chosen speed of travel has direct implications on fuel consumption and therefore overall efficiency.
• It's essential to understand how fuel efficiency varies with speed for optimal vehicle operation.

Rates of Change

Rates of change tell us how a quantity, like fuel efficiency, changes over time or with varying conditions, such as speed. In calculus, this is often represented by the derivative of a function.
The derivative \( g'(v) \) gives us insight into how sensitive fuel efficiency is at different speeds. For instance, \( g'(55) = -0.25 \) indicates that fuel efficiency decreases as speed increases past 55 mph. A negative rate suggests a reduction in fuel efficiency when speed increases, which is crucial information for drivers seeking to minimize fuel consumption.
Conversely, \( g'(45) = 0.15 \) suggests that at 45 mph, fuel efficiency is improving, while \( g'(51) = 0 \) shows a stable point where efficiency neither increases nor decreases.
Understanding the rates of change helps drivers:

• Identify optimal speeds for fuel efficiency.
• Predict how adjustments in speed affect fuel consumption.
• Make strategic decisions that affect driving habits and costs.

Graph Interpretation

Interpreting graphs is an essential skill in understanding the behavior of functions like \( g(v) \), which represents fuel efficiency over a range of speeds. Graphs provide a visual representation of how fuel efficiency changes, helping to pinpoint crucial points like maximum efficiency.
In our problem, we sketch a graph with speed on the x-axis and fuel efficiency on the y-axis. The graph increases at \( v = 45 \) due to \( g'(45) > 0 \), peaks or levels off at \( v = 51 \) where \( g'(51) = 0 \), and slopes downward at \( v = 55 \) where \( g'(55) < 0 \).
This graphical insight assists drivers in quickly grasping:

• Optimal speed ranges that enhance fuel efficiency.
• Points where speed adjustments do not impact efficiency.
• Speed intervals to avoid to prevent inefficiency.

By interpreting these graphs, drivers become better equipped to adjust their driving habits for improved fuel economy.

Derivatives in Real Life

Derivatives serve as powerful tools in real life by helping us understand how certain variables change with respect to others. They measure rates of change, providing valuable insights across various fields, from engineering to economics.
In the context of our exercise, derivatives like \( g'(v) \) reveal how fuel efficiency varies at different speeds. This application of derivatives aids in resolving real-world dilemmas such as optimizing vehicle performance.
Beyond automotive contexts, derivatives are used to:

• Model financial markets and predict economic trends.
• Analyze the growth rates of populations or investments.
• Locate maximum or minimum points for optimization problems.

In essence, derivatives bridge theoretical math with practical problems, making them indispensable for everyday decision-making.