Homework Help is Here – Start Your Trial Now!
Math
Calculus
As the speed of a train increases, the amount of power needed to maintain that speed increases, and the rate of power increase also increases. Let P = f(v) be the power, in megawatts, needed for the train to maintain a speed of v kilometers per hour. (a) Sketch a possible graph of f. (b) For each of the functions f, f', and f", decide whether the function is positive or negative, and explain your answers. (c) What requires a greater increase in power: increasing the train's speed from 100 to 150 kilometers per hour, or increasing the train's speed from 200 to 250 kilometers per hour? Explain, based on your answer to part (b).

As the speed of a train increases, the amount of power needed to maintain that speed increases, and the rate of power increase also increases. Let P = f(v) be the power, in megawatts, needed for the train to maintain a speed of v kilometers per hour. (a) Sketch a possible graph of f. (b) For each of the functions f, f', and f", decide whether the function is positive or negative, and explain your answers. (c) What requires a greater increase in power: increasing the train's speed from 100 to 150 kilometers per hour, or increasing the train's speed from 200 to 250 kilometers per hour? Explain, based on your answer to part (b).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
**Understanding Power and Speed Relationship in Trains** **Introduction:** As the speed of a train increases, more power is needed to maintain that speed. Furthermore, the rate of power increase also rises. Let \( P = f(v) \) represent the power, in megawatts, required for a train to maintain a speed of \( v \) kilometers per hour. **Tasks:** (a) **Sketch a Possible Graph of \( f \):** - Visualize a graph where the x-axis represents speed \( v \) (in kilometers per hour) and the y-axis represents power \( P \) (in megawatts). - The curve is expected to be upward-sloping, indicating that as speed increases, the power needed also increases. The curve might become steeper, depicting increasing rate of power consumption at higher speeds. (b) **Evaluate \( f, f', \) and \( f'' \):** - **\( f(v) \):** This function is positive since power required is a positive quantity as long as the train is moving. - **\( f'(v) \):** The first derivative indicates the rate of change of power with respect to speed. This should also be positive, reflecting the increase in power needed as speed increases. - **\( f''(v) \):** The second derivative reflects the rate of change of the rate of power increase. If \( f(v) \) becomes steeper, \( f''(v) \) is also positive, indicating that the rate of power increase is itself increasing. (c) **Analyze Power Increase Needs:** - Consider the power increase when speed shifts from 100 to 150 kilometers per hour compared to 200 to 250 kilometers per hour. - Since \( f'(v) \) is increasing, a greater increase in power is likely required for the higher speed range (200 to 250 km/h), because the curve becomes steeper as speed increases. This conceptual understanding helps analyze how increased speeds demand progressively more power, emphasizing efficiency considerations in train operations and design.
Transcribed Image Text:**Understanding Power and Speed Relationship in Trains** **Introduction:** As the speed of a train increases, more power is needed to maintain that speed. Furthermore, the rate of power increase also rises. Let \( P = f(v) \) represent the power, in megawatts, required for a train to maintain a speed of \( v \) kilometers per hour. **Tasks:** (a) **Sketch a Possible Graph of \( f \):** - Visualize a graph where the x-axis represents speed \( v \) (in kilometers per hour) and the y-axis represents power \( P \) (in megawatts). - The curve is expected to be upward-sloping, indicating that as speed increases, the power needed also increases. The curve might become steeper, depicting increasing rate of power consumption at higher speeds. (b) **Evaluate \( f, f', \) and \( f'' \):** - **\( f(v) \):** This function is positive since power required is a positive quantity as long as the train is moving. - **\( f'(v) \):** The first derivative indicates the rate of change of power with respect to speed. This should also be positive, reflecting the increase in power needed as speed increases. - **\( f''(v) \):** The second derivative reflects the rate of change of the rate of power increase. If \( f(v) \) becomes steeper, \( f''(v) \) is also positive, indicating that the rate of power increase is itself increasing. (c) **Analyze Power Increase Needs:** - Consider the power increase when speed shifts from 100 to 150 kilometers per hour compared to 200 to 250 kilometers per hour. - Since \( f'(v) \) is increasing, a greater increase in power is likely required for the higher speed range (200 to 250 km/h), because the curve becomes steeper as speed increases. This conceptual understanding helps analyze how increased speeds demand progressively more power, emphasizing efficiency considerations in train operations and design.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS