Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.11a

Chapter 3, Problem 3.6.11a

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>
a. Determine the average velocity of the car during the first 45 minutes of the trip.

Verified step by step guidance

First, understand that the average velocity is defined as the change in position divided by the change in time. Mathematically, this is expressed as: \( \text{Average Velocity} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} \), where \( t_1 \) and \( t_2 \) are the initial and final times, respectively.

Convert the time interval from minutes to hours since the position function \( s = f(t) \) is given in terms of hours. The first 45 minutes is equivalent to 0.75 hours.

Identify the initial time \( t_1 \) and the final time \( t_2 \). Here, \( t_1 = 0 \) hours (9:00 A.M.) and \( t_2 = 0.75 \) hours (9:45 A.M.).

Evaluate the position function \( f(t) \) at \( t_1 \) and \( t_2 \) to find \( f(t_1) \) and \( f(t_2) \). These values represent the car's position at the start and end of the 45-minute interval.

Substitute \( f(t_1) \), \( f(t_2) \), \( t_1 \), and \( t_2 \) into the average velocity formula to calculate the average velocity over the first 45 minutes.

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Key Concepts

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

Position Function

A position function, denoted as s = f(t), describes the location of an object over time. In this context, it indicates the distance traveled by the patrol car in miles, t hours after it leaves the station. Understanding this function is crucial for analyzing the car's movement and calculating its velocity.

Average Velocity

Average velocity is defined as the change in position divided by the change in time. It can be calculated using the formula: Average Velocity = (s(t2) - s(t1)) / (t2 - t1). In this scenario, it involves finding the car's position at two different times within the first 45 minutes and applying this formula to determine how fast the car is traveling on average during that period.

Time Conversion

In this problem, time is measured in hours, but the trip duration is given in minutes (45 minutes). To perform calculations accurately, it is essential to convert minutes into hours. This conversion is done by dividing the number of minutes by 60, which allows for consistent units when calculating average velocity.

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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).

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The volume V of a sphere of radius r changes over time t.

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{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by

f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>

Let f (x) = √x.

a. Find the exact value of f' (4).

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