Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.12a

Chapter 3, Problem 3.6.12a

Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).

Verified step by step guidance

Identify the position function s = f(t), which represents the number of ground miles from Seattle at time t hours after take-off.

To find the average velocity over the interval [0, 1.5], use the formula for average velocity: \( \text{Average Velocity} = \frac{f(b) - f(a)}{b - a} \), where a and b are the endpoints of the time interval.

In this problem, a = 0 and b = 1.5, so substitute these values into the formula: \( \text{Average Velocity} = \frac{f(1.5) - f(0)}{1.5 - 0} \).

Determine the values of f(1.5) and f(0) from the position function or the given figure, which represent the distances at t = 1.5 hours and t = 0 hours, respectively.

Substitute the values of f(1.5) and f(0) into the average velocity formula to find the average velocity over the first 1.5 hours of the trip.

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Key Concepts

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

Position Function

The position function, denoted as s = f(t), describes the location of an object over time. In this context, it represents the distance of the airliner from Seattle as a function of time since take-off. Understanding this function is crucial for analyzing the motion of the airliner and calculating various rates of change, such as velocity.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. For the airliner, it can be calculated by finding the change in position over the specified time interval (0 ≤ t ≤ 1.5 hours). This concept is essential for understanding how fast the plane is moving on average during the first part of its journey.

Calculus of Motion

The calculus of motion involves using derivatives and integrals to analyze the movement of objects. In this scenario, it helps in determining the average velocity and understanding the relationship between position, velocity, and time. Familiarity with these principles allows for a deeper insight into the dynamics of the airliner's trip.

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