Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Function Analysis: Exploring \( g(x) = \cos(2x) - \cos(x) \)** To understand the behavior of a given function, it is crucial to consider several aspects such as domain, critical points, range, and graphical representation. Let's dive into each of these for the function \( g(x) = \cos(2x) - \cos(x) \). --- ### **1. Domain of the Function \( g \)** - **Question:** What is the domain of the function \( g \)? Explain. - **Discussion:** The domain of a function refers to all the possible input values (x-values) that the function can accept. Since \( \cos(2x) \) and \( \cos(x) \) are defined for all real numbers, the domain of the function \( g(x) = \cos(2x) - \cos(x) \) is all real numbers. In interval notation, this is expressed as \((-\infty, \infty)\). --- ### **2. Critical Points of the Function \( g \)** - **Question:** Find and classify all critical points of the function \( g \). - **Discussion:** Critical points occur where the derivative of the function is zero or undefined. To find these, calculate \( g'(x) \) and solve \( g'(x) = 0 \). Once the critical points are found, use the second derivative test or first derivative test to classify them as local maxima, minima, or points of inflection. --- ### **3. Range of the Function \( g \)** - **Question:** Find, with explanation, the range of the function \( g \). - **Discussion:** The range refers to the possible output values (y-values) of the function. Analyze the behavior of \( g(x) \) over one period of the cosine function and determine how the values of \( \cos(2x) \) and \( \cos(x) \) combine to establish the range. --- ### **4. Graph of the Function \( g \)** - **Question:** Plot the graph of the function \( g \). - **Discussion:** Graphing \( g(x) \) helps visualize its behavior. Use graphing software or manual plotting to detail how the function changes over different intervals. Pay attention to the periodic nature of the cosine function and illustrate any critical points identified earlier.
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Transcribed Image Text:**Function Analysis: Exploring \( g(x) = \cos(2x) - \cos(x) \)** To understand the behavior of a given function, it is crucial to consider several aspects such as domain, critical points, range, and graphical representation. Let's dive into each of these for the function \( g(x) = \cos(2x) - \cos(x) \). --- ### **1. Domain of the Function \( g \)** - **Question:** What is the domain of the function \( g \)? Explain. - **Discussion:** The domain of a function refers to all the possible input values (x-values) that the function can accept. Since \( \cos(2x) \) and \( \cos(x) \) are defined for all real numbers, the domain of the function \( g(x) = \cos(2x) - \cos(x) \) is all real numbers. In interval notation, this is expressed as \((-\infty, \infty)\). --- ### **2. Critical Points of the Function \( g \)** - **Question:** Find and classify all critical points of the function \( g \). - **Discussion:** Critical points occur where the derivative of the function is zero or undefined. To find these, calculate \( g'(x) \) and solve \( g'(x) = 0 \). Once the critical points are found, use the second derivative test or first derivative test to classify them as local maxima, minima, or points of inflection. --- ### **3. Range of the Function \( g \)** - **Question:** Find, with explanation, the range of the function \( g \). - **Discussion:** The range refers to the possible output values (y-values) of the function. Analyze the behavior of \( g(x) \) over one period of the cosine function and determine how the values of \( \cos(2x) \) and \( \cos(x) \) combine to establish the range. --- ### **4. Graph of the Function \( g \)** - **Question:** Plot the graph of the function \( g \). - **Discussion:** Graphing \( g(x) \) helps visualize its behavior. Use graphing software or manual plotting to detail how the function changes over different intervals. Pay attention to the periodic nature of the cosine function and illustrate any critical points identified earlier.
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