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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 18d
Chapter 3, Problem 18d

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


d. When does the particle stand still for more than an instant?
graph

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1
Examine the graph of velocity v = f(t) over time t. The particle stands still when its velocity is zero.
Identify the sections of the graph where the velocity is zero. These are the points where the graph intersects the horizontal axis (v = 0).
Observe the graph and note that the velocity is zero between t = 4 seconds and t = 5 seconds.
Since the velocity is zero for a duration from t = 4 to t = 5 seconds, the particle stands still for more than an instant during this interval.
Conclude that the particle stands still for more than an instant between t = 4 seconds and t = 5 seconds.

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Key Concepts

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

Velocity and Motion

Velocity is the rate of change of position with respect to time, indicating how fast and in what direction an object is moving. In the context of the graph, the velocity function v = f(t) shows how the particle's speed varies over time. When the velocity is zero, the particle is momentarily at rest, which is crucial for determining when it stands still.
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Derivatives Applied To Velocity

Graph Interpretation

Interpreting graphs involves understanding the relationship between the axes and the data represented. In this case, the x-axis represents time (t in seconds), while the y-axis represents velocity (v). Analyzing the graph allows us to identify intervals where the velocity is zero, indicating when the particle is at rest, and to determine if it remains at rest for more than an instant.
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Graphing The Derivative

Critical Points and Intervals

Critical points in a function occur where the function's value is zero or undefined, which in this case relates to the velocity function. By identifying these points on the graph, we can determine intervals where the particle is stationary. If the velocity remains zero over an interval rather than just at isolated points, the particle stands still for more than an instant.
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Critical Points
Related Practice
Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


a. Find the average cost per machine of producing the first 100 washing machines.

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Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


a. When does the particle move forward? Move backward? Speed up? Slow down?

Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


c. When does the particle move at its greatest speed?

Textbook Question

In Exercises 19–22, find the values of the derivatives.


dr/dθ |θ₌₀ if r = 2/√(4 – θ)

Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


b. When is the particle’s acceleration positive? Negative? Zero?