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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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 18c
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?
Verified step by step guidance1
Examine the graph of velocity v = f(t) over time t. The graph shows how the velocity of the particle changes with time.
Identify the points on the graph where the velocity reaches its maximum and minimum values. The greatest speed corresponds to the maximum absolute value of velocity, regardless of direction.
Observe that the graph has peaks and valleys. The highest peak or lowest valley indicates the greatest speed.
Determine the time at which these maximum or minimum values occur by looking at the horizontal axis (time) corresponding to these points.
Conclude that the particle moves at its greatest speed at the time where the graph reaches the highest or lowest point, which is the maximum absolute value of velocity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity
Velocity is a vector quantity that refers to the rate of change of an object's position with respect to time. In the context of the graph, the velocity function v = f(t) indicates how fast and in which direction the particle is moving at any given time t. Positive values indicate motion in one direction, while negative values indicate motion in the opposite direction.
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Derivatives Applied To Velocity
Speed
Speed is the magnitude of velocity and is a scalar quantity, meaning it only has a value without direction. The greatest speed of the particle corresponds to the maximum absolute value of the velocity function on the graph. This can be determined by identifying the highest peaks and lowest troughs of the graph, as these points represent the fastest movement of the particle.
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06:29Derivatives Applied To Velocity
Graph Interpretation
Interpreting graphs is crucial in calculus, especially when analyzing motion. The graph of velocity versus time allows us to visualize how the particle's speed changes over time. By examining the shape of the graph, we can identify intervals of increasing or decreasing speed, as well as moments when the particle is at rest (when velocity equals zero), which aids in determining when the particle moves at its greatest speed.
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Related Practice
Textbook Question
Understanding Motion from Graphs
Launching a Rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands.
The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following.
f. When was the rocket’s acceleration greatest?
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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?
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Textbook Question
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?
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Textbook Question
In Exercises 19–22, find the values of the derivatives.
dr/dθ |θ₌₀ if r = 2/√(4 – θ)
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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?
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