Textbook Question
At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.
a. Find an equation relating A to w.
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 5c
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.
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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 \( s(0) \) and \( s(1) \) from the table provided. These represent the position of the object at time \( t = 0 \) and \( t = 1 \) respectively.
Substitute the values of \( s(0) \) and \( s(1) \) into the average velocity formula: \( v_{avg} = \frac{s(1) - s(0)}{1 - 0} \).
Simplify the expression by calculating the difference \( s(1) - s(0) \) to find the change in position.
Divide the change in position by the change in time (which is 1 second in this case) to find the average velocity over the interval [0, 1].
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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 particularly useful in analyzing motion over specific intervals.
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Position Function
The position function, denoted as s(t), describes the location of an object at any given time t. It provides a mathematical representation of the object's movement along a line. Understanding the position function is essential for calculating average velocity, as it allows us to determine the positions at the start and end of the time interval in question.
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Time Interval
A time interval is the duration over which an event occurs, typically represented as [t1, t2]. In the context of average velocity, it is the period during which the object's position is measured. Recognizing the significance of the time interval is vital for accurately calculating average velocity, as it directly influences the values used in the formula.
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Related Practice
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>
a.
Textbook Question
Use differentiation to verify each equation.
d/dx(x / √1−x²) = 1 / (1−x²)^3/2.
Textbook Question
An equation of the line tangent to the graph of f at the point (2,7) is y = 4x−1. Find f(2) and f′(2).
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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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Let f(x) = sin x. What is the value of f′(π)?