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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 1b
Capitolo 2, Problema 1b

Which of the following functions are continuous for all values in their domain? Justify your answers.


b. n(t)=number of quarters needed to park legally in a metered parking space for t minutes

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Step 1: Understand the function n(t). The function n(t) represents the number of quarters needed to park legally in a metered parking space for t minutes. This is a real-world scenario where the function outputs discrete values (whole numbers) based on the input time t.
Step 2: Consider the nature of the function. Since n(t) outputs the number of quarters, it is a step function. A step function is a piecewise constant function that jumps from one value to another without taking on any intermediate values.
Step 3: Analyze the continuity of step functions. A function is continuous at a point if there is no interruption or jump at that point. For step functions, there are jumps at certain points (e.g., when the number of quarters needed changes), which means they are not continuous at those points.
Step 4: Determine the continuity of n(t). Since n(t) is a step function, it is not continuous at the points where the number of quarters needed changes. These are the points where the function jumps from one integer value to another.
Step 5: Conclude about the continuity of n(t). Since n(t) is not continuous at the points where it jumps, it is not continuous for all values in its domain. It is only continuous within the intervals between these jumps.

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Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over its entire domain, it must be continuous at every point in that domain. This means there are no breaks, jumps, or asymptotes in the function's graph.
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Intro to Continuity

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. Understanding how these functions behave at the boundaries of their pieces is crucial for determining continuity. If a piecewise function has different rules for different intervals, one must check the limits and values at the transition points to ensure continuity.
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Piecewise Functions

Real-World Applications of Functions

In real-world scenarios, such as the number of quarters needed for parking, functions can represent discrete quantities. Understanding how these functions behave in practical contexts helps in assessing their continuity. For example, if the number of quarters changes abruptly based on time, it may indicate discontinuity, as the function cannot take on every value smoothly.
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Real World Application
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