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 3.6.12d

Chapter 3, Problem 3.6.12d

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>
d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.

Verified step by step guidance

To determine the velocity of the airliner at noon (t = 6), we need to find the derivative of the position function s = f(t) with respect to time t. The derivative, f'(t), represents the velocity of the airliner at any given time t.

Evaluate the derivative f'(t) at t = 6 to find the velocity at noon. This involves substituting t = 6 into the derivative function f'(t).

The velocity is negative because the airliner is on its return trip to Seattle. A negative velocity indicates that the direction of travel is towards the starting point, which in this case is Seattle.

Conceptually, velocity is a vector quantity, meaning it has both magnitude and direction. A negative value signifies movement in the opposite direction to the positive axis, which is typically defined as the direction away from the starting point.

Ensure that the units of the velocity are consistent with the units used in the position function. If the position function is in miles, the velocity will be in miles per hour.

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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, as it provides the necessary data to determine both velocity and acceleration.

Velocity

Velocity is defined as the rate of change of position with respect to time, mathematically expressed as v(t) = f'(t). It indicates both the speed and direction of an object's movement. In this scenario, calculating the velocity at a specific time, such as noon (t = 6), helps to understand how fast the airliner is traveling and in which direction relative to Seattle.

Negative Velocity

A negative velocity indicates that the object is moving in the opposite direction to the defined positive direction, which in this case is towards Seattle. When the airliner is returning to Seattle after reaching Minneapolis, its velocity becomes negative. This concept is essential for interpreting the motion of the airliner and understanding the implications of its position function at different times.

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Related Practice

Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

d. Graph P' and use the graph to estimate the year in which the population is growing fastest.

Textbook Question

Tangents and inverses Suppose L(x)=ax+b (with a≠0) is the equation of the line tangent to the graph of a one-to-one function f at (x0,y0). Also, suppose M(x)=cx+d is the equation of the line tangent to the graph of f^−1 at (y0,x0).

c. Prove that L^−1(x)=M(x).

Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(x²−1)sin^−1 x on [−1,1]

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

d. f'(1)

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Textbook Question

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

d. At what time is the magnitude of the flow rate a minimum? A maximum?

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The lines tangent to the graph of y=sin x on the interval [−π/2,π/2] have a maximum slope of 1.