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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 5a
Chapter 3, Problem 5a

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>


a. [0,2][0, 2]

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1
Identify the formula for average velocity over an interval [a, b], which is given by the change in position divided by the change in time: \( v_{avg} = \frac{s(b) - s(a)}{b - a} \).
Determine the values of \( s(a) \) and \( s(b) \) from the table for the interval [0, 2]. These represent the position of the object at times t = 0 and t = 2, respectively.
Substitute the values of \( s(0) \) and \( s(2) \) into the average velocity formula: \( v_{avg} = \frac{s(2) - s(0)}{2 - 0} \).
Simplify the expression by calculating the difference \( s(2) - s(0) \) to find the change in position.
Divide the change in position by the change in time (which is 2 seconds) to find the average velocity over the interval [0, 2].

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Key Concepts

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

Average Velocity

Average velocity is defined as the change in position divided by the time interval over which that change occurs. Mathematically, it is expressed as (s(t2) - s(t1)) / (t2 - t1), where s(t) represents the position function. This concept is crucial for understanding how an object's position changes over time and is fundamental in analyzing motion.
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Average Value of a Function

Position Function

The position function s(t) describes the location of an object at any given time t. It provides a mathematical representation of the object's trajectory along a line. Understanding this function is essential for calculating average velocity, as it allows us to determine the specific positions at the start and end of the time interval.
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Relations and Functions

Time Interval

A time interval is the duration over which motion is observed, typically denoted as [t1, t2]. In this context, it is the period during which the average velocity is calculated. Recognizing the significance of the time interval is important for accurately determining changes in position and understanding the object's motion dynamics.
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Interval of Convergence
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. 

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Textbook Question

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>


c. [0,1][0, 1]

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Textbook Question

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.

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e. h′(5)h^{\(\prime\)}\(\left\)(5\(\right\))

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Textbook Question

Use differentiation to verify each equation.

d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

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Textbook Question

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.

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Textbook Question

Let f(x) = sin x. What is the value of f′(π)?

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