Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 33c

Capitolo 1, Problema 33c

State whether each function is increasing, decreasing, or neither.

c. Height above Earth’s sea level as a function of atmospheric pressure (assumed nonzero)

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Understand the relationship between height above sea level and atmospheric pressure: As you ascend in altitude, atmospheric pressure generally decreases. This is because there is less air above you exerting pressure.

Consider the function: Height above sea level as a function of atmospheric pressure. We need to determine if this function is increasing, decreasing, or neither.

Analyze the behavior of the function: As atmospheric pressure decreases, height above sea level increases. This suggests a negative correlation between the two variables.

Determine the nature of the function: Since height increases as atmospheric pressure decreases, the function is decreasing with respect to atmospheric pressure.

Conclude the analysis: The function is decreasing because an increase in height corresponds to a decrease in atmospheric pressure.

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Monotonic Functions

A function is considered monotonic if it is either entirely non-increasing or non-decreasing over its domain. This concept is crucial for determining whether a function is increasing, decreasing, or neither. An increasing function has a positive slope, while a decreasing function has a negative slope. Understanding monotonicity helps in analyzing the behavior of functions in relation to their inputs.

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Properties of Functions

Derivative and Its Sign

The derivative of a function provides information about its rate of change. If the derivative is positive over an interval, the function is increasing; if negative, it is decreasing. For the given function, analyzing the derivative with respect to atmospheric pressure will reveal how height changes as pressure varies. This relationship is fundamental in calculus for understanding function behavior.

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Derivatives

Inverse Relationships

In some contexts, functions can exhibit inverse relationships, where an increase in one variable leads to a decrease in another. In this case, as atmospheric pressure decreases, height above sea level increases, indicating a negative correlation. Recognizing such relationships is essential for interpreting the behavior of functions in real-world scenarios, particularly in physics and environmental science.

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Inverse Cosine

Pratica correlata

Domanda del libro di testo

Find the largest interval on which the given function is increasing.

d. R(x) = √ 2x - 1

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Domanda del libro di testo

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?

s = −tan πt

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Domanda del libro di testo

State whether each function is increasing, decreasing, or neither.

d. Kinetic energy as a function of a particle’s velocity

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Domanda del libro di testo

[Technology Exercise]

a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

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