Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 49b

Chapter 4, Problem 49b

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.
b. At what speed does the gas mileage function have its maximum?

Verified step by step guidance

To find the speed at which the gas mileage function has its maximum, we need to understand the relationship between speed and gas mileage. Typically, gas mileage (miles per gallon) is a function of speed, and it can be represented as a function f(v), where v is the speed of the vehicle.

The maximum gas mileage occurs at the speed where the derivative of the gas mileage function with respect to speed is zero. This is because the derivative represents the rate of change, and a zero derivative indicates a local maximum or minimum.

First, express the gas mileage function f(v) in terms of speed v. This function is often derived from empirical data or a model that relates speed to fuel efficiency.

Next, find the derivative of the gas mileage function, f'(v), with respect to speed v. This involves applying differentiation rules to the function f(v).

Finally, solve the equation f'(v) = 0 to find the critical points. Evaluate these points to determine which one corresponds to the maximum gas mileage. This involves checking the second derivative or analyzing the behavior of the function around the critical points.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gas Mileage Function

The gas mileage function describes how the fuel efficiency of a vehicle varies with speed. Typically, this function increases to a certain point (optimal speed) and then decreases as speed continues to rise. Understanding this function is crucial for determining the speed at which gas mileage is maximized, which directly impacts travel costs.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to apply techniques such as taking the derivative of the gas mileage function and setting it to zero to find critical points. This process helps identify the speed that minimizes fuel costs by maximizing gas mileage.

Derivatives

Derivatives represent the rate of change of a function with respect to a variable. In this scenario, the derivative of the gas mileage function will indicate how mileage changes as speed varies. By analyzing the derivative, we can determine where the function reaches its maximum value, which is essential for solving the problem.

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Derivatives

Related Practice

Textbook Question

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs $p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns $w/hour.

e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

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Textbook Question

Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.

Textbook Question

Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is \$30 minus \$0.25 for every ticket sold. If gas and other miscellaneous costs are \$200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x²/(x² - 1) on [-4,4]

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Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x√(4 - x²) on [-2,2]

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.b. At what speed does the gas mileage function have its maximum?

Verified video answer for a similar problem:

Key Concepts

Gas Mileage Function

Optimization

Derivatives

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.
b. At what speed does the gas mileage function have its maximum?