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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...
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**Mathematics - Calculating Limits of Functions** --- ### Problem Statement **3.** The graphs of the piecewise linear functions \( f \) and \( g \) are shown. Let \( h \) be a function such that \( f(x) \leq h(x) \leq g(x) \) for all \( x \). What is \( \lim_{{x \to 2}} h(x) \)? **Options:** - A. \( 1 \) - B. \( 2 \) - C. \( 3 \) - D. The limit does not exist. --- ***Explanation of Graphs:*** *Note: In the original image, there is no actual graph provided, so this explanation assumes one has graphs from another source.* **Graphs Interpretation:** - The hypothetical graphs illustrate the piecewise linear functions \( f(x) \) and \( g(x) \). Assume that at \( x = 2 \), the graphs of \( f \) and \( g \) intersect or are very close. Analyze their behavior around \( x = 2 \). **Solution Approach:** 1. **Understanding the given condition:** - The function \( h(x) \) is always sandwiched between \( f(x) \) and \( g(x) \): \( f(x) \leq h(x) \leq g(x) \). 2. **Limit Definitions:** - \( \lim_{{x \to 2}} f(x) \) denotes the value that \( f \) approaches as \( x \) approaches 2. - \( \lim_{{x \to 2}} g(x) \) denotes the value that \( g \) approaches as \( x \) approaches 2. 3. **Using the Squeeze Theorem:** - Given \( f(x) \leq h(x) \leq g(x) \) for all \( x \), by the Squeeze Theorem, if \( \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} g(x) = L \), then \( \lim_{{x \to 2}} h(x) = L \). 4. **Calculating the Limits:** - Assume that the limits of \( f(x) \) and \( g(x) \) as \( x
Transcribed Image Text:**Mathematics - Calculating Limits of Functions** --- ### Problem Statement **3.** The graphs of the piecewise linear functions \( f \) and \( g \) are shown. Let \( h \) be a function such that \( f(x) \leq h(x) \leq g(x) \) for all \( x \). What is \( \lim_{{x \to 2}} h(x) \)? **Options:** - A. \( 1 \) - B. \( 2 \) - C. \( 3 \) - D. The limit does not exist. --- ***Explanation of Graphs:*** *Note: In the original image, there is no actual graph provided, so this explanation assumes one has graphs from another source.* **Graphs Interpretation:** - The hypothetical graphs illustrate the piecewise linear functions \( f(x) \) and \( g(x) \). Assume that at \( x = 2 \), the graphs of \( f \) and \( g \) intersect or are very close. Analyze their behavior around \( x = 2 \). **Solution Approach:** 1. **Understanding the given condition:** - The function \( h(x) \) is always sandwiched between \( f(x) \) and \( g(x) \): \( f(x) \leq h(x) \leq g(x) \). 2. **Limit Definitions:** - \( \lim_{{x \to 2}} f(x) \) denotes the value that \( f \) approaches as \( x \) approaches 2. - \( \lim_{{x \to 2}} g(x) \) denotes the value that \( g \) approaches as \( x \) approaches 2. 3. **Using the Squeeze Theorem:** - Given \( f(x) \leq h(x) \leq g(x) \) for all \( x \), by the Squeeze Theorem, if \( \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} g(x) = L \), then \( \lim_{{x \to 2}} h(x) = L \). 4. **Calculating the Limits:** - Assume that the limits of \( f(x) \) and \( g(x) \) as \( x
**AP Calculus AB - Graph Interpretation** In this section, we explore the behavior of two functions, f and g, through their graphical representations. ### Graph of \(f\) The graph of \(f\) is presented within a standard Cartesian plane. Key features include: - The y-axis spans from -1 to 4. - The x-axis spans from -1 to 4. - The graph consists of two distinct linear segments: 1. The first segment runs from the point \((0, 1)\) to the open circle at \((2, 3)\). 2. The second segment runs from the filled circle at \((2, 3)\) to the point \((4, 1)\). This plot indicates that the graph of \(f\) is piecewise linear, with a possible discontinuity or change in slope at \(x = 2\). ### Graph of \(g\) The graph of \(g\) is also displayed on a Cartesian plane with the following details: - The y-axis spans from -1 to 4. - The x-axis spans from -1 to 4. - The graph consists of two distinct linear segments: 1. The first segment runs from the point \((0, 3)\) to the open circle at \((2, 1)\). 2. The second segment runs from the filled circle at \((2, 1)\) to the point \((4, 3)\). This plot suggests that the graph of \(g\) is another piecewise linear function with a slope change at \(x = 2\). In both graphs, the filled circles indicate included endpoints on the segments, whereas open circles show points that are not included. This notation is crucial for understanding the continuity and domain of the functions \(f\) and \(g\). ### Conclusion Understanding how to interpret these graphs is vital for analyzing and predicting the behavior of piecewise functions in calculus. By studying the points of connection and discontinuity, students can glean deeper insights into function behavior and continuity.
Transcribed Image Text:**AP Calculus AB - Graph Interpretation** In this section, we explore the behavior of two functions, f and g, through their graphical representations. ### Graph of \(f\) The graph of \(f\) is presented within a standard Cartesian plane. Key features include: - The y-axis spans from -1 to 4. - The x-axis spans from -1 to 4. - The graph consists of two distinct linear segments: 1. The first segment runs from the point \((0, 1)\) to the open circle at \((2, 3)\). 2. The second segment runs from the filled circle at \((2, 3)\) to the point \((4, 1)\). This plot indicates that the graph of \(f\) is piecewise linear, with a possible discontinuity or change in slope at \(x = 2\). ### Graph of \(g\) The graph of \(g\) is also displayed on a Cartesian plane with the following details: - The y-axis spans from -1 to 4. - The x-axis spans from -1 to 4. - The graph consists of two distinct linear segments: 1. The first segment runs from the point \((0, 3)\) to the open circle at \((2, 1)\). 2. The second segment runs from the filled circle at \((2, 1)\) to the point \((4, 3)\). This plot suggests that the graph of \(g\) is another piecewise linear function with a slope change at \(x = 2\). In both graphs, the filled circles indicate included endpoints on the segments, whereas open circles show points that are not included. This notation is crucial for understanding the continuity and domain of the functions \(f\) and \(g\). ### Conclusion Understanding how to interpret these graphs is vital for analyzing and predicting the behavior of piecewise functions in calculus. By studying the points of connection and discontinuity, students can glean deeper insights into function behavior and continuity.
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