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Chapter 10: Problem 42

Describe a real-life scenario in your room that can be modeled by a discontinuous function.

Short Answer

Expert verified
A real-life scenario in a room that can be modeled by a discontinuous function is the position of a window throughout the day. We define \(x\) as the time during the day in hours and \(f(x)\) as the position of the window with respect to time. Given: - Window is closed from 0 to 4 hours (midnight to 4 AM) - Window is slightly open from 4 to 7 hours (4 AM to 7 AM) - Window is fully open from 7 to 20 hours (7 AM to 8 PM) - Window is closed again from 20 to 24 hours (8 PM to midnight) The discontinuous function to represent this scenario is: \[ f(x) = \begin{cases} 0, & \mbox{if}\ 0\leq x<4 \\ 0.5, & \mbox{if}\ 4\leq x<7 \\ 1, & \mbox{if}\ 7\leq x<20 \\ 0, & \mbox{if}\ 20\leq x\leq24 \end{cases} \] This function exhibits discontinuity at \(x=4\), \(x=7\), and \(x=20\) when the window position changes suddenly.

Step by step solution

01

Identify a real-life scenario in your room

Observe the situations around your room, such as thermostat settings, light switches, or other day-to-day activities. For this example, consider the position of a window in the room being either open, closed, or slightly open.
02

Define the variables involved

In our scenario, we have two variables. Let 'x' be the time during the day in hours and 'f(x)' be the position of the window with respect to time.
03

Explain the discontinuity

A discontinuous function occurs when there is a break in the graph, meaning the relationship between the variables is not smooth or uninterrupted. In our window example, discontinuity occurs when the window is suddenly opened or closed, causing a jump in the graph of the function.
04

Model the scenario using a discontinuous function

We can model the window position as a piecewise function based on different time frames throughout the day. Let's say that: - From 0 to 4 hours (midnight to 4 AM), the window is closed. - From 4 to 7 hours (4 AM to 7 AM), the window is slightly open. - From 7 to 20 hours (7 AM to 8 PM), the window is fully open. - From 20 to 24 hours (8 PM to midnight), the window is closed. We can define the discontinuous function f(x) as follows: f(x) = \[ \begin{cases} 0, & \mbox{if}\ 0\leq x<4 \\ 0.5, & \mbox{if}\ 4\leq x<7 \\ 1, & \mbox{if}\ 7\leq x<20 \\ 0, & \mbox{if}\ 20\leq x\leq24 \end{cases} \]
05

Describe the function's behavior

This piecewise-defined function f(x) represents the position of the window during different time frames throughout the day, with discontinuity happening at 4, 7, and 20 hours. At these points, the position of the window suddenly changes, illustrating how the function models a real-life scenario within the context of the example.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Piecewise Function
A piecewise function is a mathematical expression that behaves differently based on the input value, dividing the function into distinct "pieces". This is especially useful when modeling situations where the output suddenly changes due to alterations in the input conditions. For example, when you look at the behavior of the window in the room over a 24-hour period, the window position could be modeled using different functions for each time segment of the day.

In our provided scenario, the function was defined with distinct behaviors at certain times:
  • From 0 to 4 hours: The window remains closed, resulting in an output of 0.
  • From 4 to 7 hours: The window is slightly open, modeled with an output of 0.5.
  • From 7 to 20 hours: The window is fully open, thus the function outputs 1.
  • From 20 to 24 hours: The window closes again, reverting to an output of 0.
This structure allows us to effectively describe the window's position as a piecewise function, exhibiting different states at various times. Such models are vital in capturing real-world phenomena that don't follow a single linear path.
Real-life Scenario
Real-life scenarios often involve situations where changes are not gradual but happen suddenly and distinctly, which can be aptly modeled using discontinuous functions. Take our window example; it reflects a situation many of us face daily. In your room, different activities can affect how objects behave or are positioned at different times.

Other similar scenarios could include:
  • The temperature setting on a room thermostat that switches between preset temperatures at specific intervals.
  • A digital clock that shows sudden jumps from one time data point to the next.
  • The light status in a room where it toggles between on, dimmed, and off at various scheduled times.
These real-life situations illustrate how discontinuities appear naturally and are best captured through piecewise representations. Understanding these scenarios not only enhances our comprehension of real-world dynamics but also aids in predicting behaviors and optimizing outcomes within our environments.
Modeling Discontinuity
Modeling discontinuity involves representing abrupt changes in a system through mathematical functions. In the piecewise function for the window, discontinuities occur at the points where transitions happen from one time segment to another. This type of modeling is essential in accurately reflecting real-life situations where shifts are not smooth or gradual.

Key steps in modeling discontinuity include:
  • Identifying the points where behavior changes, which are the discontinuity points. In our scenario, these are at 4, 7, and 20 hours.
  • Defining different constants or functions for each interval between the discontinuity points.
  • Ensuring the function accurately models the behavior over each segment, thus capturing the intended real-life scenario.
This modeling provides a clear and simplified way to represent complex changes within a system. Allowing for better analysis, planning, and control in situations ranging from simple domestic settings to broader applications in engineering and the sciences.

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Most popular questions from this chapter

In Exercises 39-44, find the equation of the tangent to the graph at the indicated point. HINT [Compute the derivative algebraically; then see Example \(2(\mathrm{~b})\) in Section \(10.5 .]\) $$ f(x)=x^{2}-3 ; a=2 $$

(Compare Exercise 29 of Section 10.4.) The percentage of mortgages issued in the United States that are subprime (normally classified as risky) can be approximated by $$ A(t)=\frac{15}{1+8.6(1.8)^{-t}} \quad(0 \leq t \leq 9) $$ where \(t\) is the number of years since the start of 2000 . a. Estimate \(A(6)\) and \(A^{\prime}(6)\). (Round answers to two significant digits.) What do the answers tell you about subprime mortgages? b. \(\mathrm{T}\) Graph the extrapolated function and its derivative for \(0 \leq t \leq 16\) and use your graphs to describe how the derivative behaves as \(t\) becomes large. (Express this behavior in terms of limits if you have studied the sections on limits.) What does this tell you about subprime mortgages? HINT [See Example 5.]

Compute \(f^{\prime}(a)\) algebraically for the given value of a. HINT [See Example 1.] $$ f(x)=3 x^{2}+x ; a=1 $$

Compute \(f^{\prime}(a)\) algebraically for the given value of a. HINT [See Example 1.] $$ f(x)=-2 x+4 ; a=-1 $$

Use the balanced difference quotient formula, $$ f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a-h)}{2 h} $$ to compute \(f^{\prime}(3)\) when \(f(x)=x^{2}\). What do you find?
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