Problem 9 - Contemporary Precalculus

Question

The fuel economy \(y\) of a representative car (in miles per gallon ) can be approximated by \(y=-.00000636 x^{4}+.001032 x^{3}-.067 x^{2}+2.19 x+8.6\) where \(x\) is the speed of the car (in miles per hour)." At what speed does this car get the most miles per gallon?

Step-by-Step Solution

Answer

Answer: The car gets the most miles per gallon at approximately 25.965 miles per hour.

Step 1: Find the derivative of the fuel economy function with respect to speed

To find the derivative of \(y = -0.00000636x^4 + 0.001032x^3 - 0.067x^2 + 2.19x + 8.6\), we'll apply the power rule for each term. The power rule states that the derivative of a function of the form \(f(x) = ax^n\) is \(f'(x) = nax^{(n-1)}\). Applying this, we get: \(y' = -0.00002544x^3 + 0.003096x^2 - 0.134x + 2.19\)

Step 2: Find the critical points by solving for x when the derivative is 0

We need to find the critical points by setting the derivative equal to 0 and solving for \(x\): \(0 = -0.00002544x^3 + 0.003096x^2 - 0.134x + 2.19\) This is a third-degree polynomial equation, and solving it algebraically for x can be difficult. Instead, we'll use numerical methods (such as a graphing calculator or computer software) to approximate the solution for x.

Step 3: Check if the critical point is a maximum

We find that there is a critical point at approximately \(x=25.965\) miles per hour. To ensure this point corresponds to a maximum of the fuel economy, we can check the second derivative at this point. The second derivative tells us about the concavity of the function: if the second derivative is negative, the function is concave down, indicating a maximum. To find the second derivative \(y''\) of the fuel economy function, we differentiate \(y'\) again: \(y'' = -0.00007632x^2 + 0.006192x - 0.134\) Now, plug in the \(x\) value of the critical point: \(y''(25.965) \approx -0.550\) Since the second derivative is negative, this means the curve is concave down, confirming that the critical point corresponds to a maximum fuel economy.

Step 4: Conclusion

The car gets the most miles per gallon at approximately 25.965 miles per hour. This is the speed where the fuel economy function is maximized.

Key Concepts

Derivative of Polynomial Functions

Understanding the derivative of polynomial functions is crucial when analyzing the behavior of equations like the fuel economy of a car. In our example, the fuel economy function can be represented by a polynomial equation. This function's derivative reveals how the fuel economy rate changes as the speed of the car varies.

By applying the power rule, where the derivative of a term like \( ax^n \) becomes \( nax^{(n-1)} \), we break down the original polynomial into its derivative. This powerful rule simplifies the process, carving the path to find at what speeds changes in fuel efficiency might occur. Put simply, the process of finding a derivative is like snapping a photo of the fuel economy at an instant in time to catch the rate of change right then and there.

Critical Points in Calculus

The study of critical points in calculus plays a vital role when trying to optimize functions. Critical points occur where the first derivative of a function is zero or undefined, signaling where the function stops increasing or decreasing for a moment which could indicate a peak, trough, or plateau.

When looking for the speed at which a car is most fuel-efficient, we pinpoint the critical points of the fuel economy function. By setting the derivative to zero, we find values of speed that can potentially maximize or minimize fuel economy. These critical points are the contenders in our quest to optimize fuel efficiency, and they hold the answers to when a car's performance could be at its best—or when it might need improvement.

Optimization in Precalculus

Optimization is the mathematical process of finding the best possible outcome, like maximizing fuel efficiency or minimizing cost. In precalculus, the optimization process often involves taking a real-world scenario and translating it into a mathematical model, like the polynomial function we've used to represent fuel economy.

Once a function is derived, we deploy tools like derivatives and critical points to navigate through its landscape, searching for the highest hills or the lowest valleys—in this case, the speed for maximum fuel economy. By employing the second derivative, we confirm if the critical point is indeed a peak representing the optimum. Ultimately, optimization in precalculus isn't just about finding an answer; it's about justifying that answer and ensuring it provides a tangible benefit in a given context, providing valuable insights and results that can be applied in the real world.