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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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### Graph and Limits of \( y = f(x) \) #### The graph of \( y = f(x) \) is pictured below: (Include the provided graph image here on the website for reference) **Graph Description:** The graph shows a function \( f(x) \) plotted on the Cartesian plane with: - The x-axis ranging from -5 to 5. - The y-axis ranging from -5 to 5. - Key points and behaviors of the function: - There is a jump discontinuity at \( x = 3 \): - As \( x \) approaches 3 from the left, \( f(x) \) is approaching 4. - As \( x \) approaches 3 from the right, \( f(x) \) is approaching -4. - There is an asymptote at \( x = -1 \): - The function approaches \( \infty \) as \( x \) approaches -1 from both the left and the right. **Problems:** (a) Find \( \lim_{x \to -1} f(x) \). (b) Find \( \lim_{x \to 3^-} f(x) \). (c) Find \( \lim_{x \to 4} f(x) \). (d) Find \( \lim_{x \to -4} f(x) \). (e) Find \( \lim_{x \to -1^-} f(x) \). (f) Find \( \lim_{x \to -1^+} f(x) \). (g) Sketch the graph of the derivative \( y = f'(x) \). **Visual Details of the Graph:** 1. For \( x < -1 \), the function \( f(x) \) is increasing until it reaches an asymptotic behavior at \( x = -1 \). 2. For \( -1 < x < 3 \), the function shows different behaviors. Notably: - Near \( x = 0 \), \( f(x) \) reaches a local maximum approximately at \( y = 0 \). - Between \( x = 1 \) and \( x = 2 \), the function exhibits a steep descent and then rises again. 3. At \( x = 3 \), there is a distinct jump discontinuity: - The left-hand limit (\( x \to 3^- \)) reaches
Transcribed Image Text:### Graph and Limits of \( y = f(x) \) #### The graph of \( y = f(x) \) is pictured below: (Include the provided graph image here on the website for reference) **Graph Description:** The graph shows a function \( f(x) \) plotted on the Cartesian plane with: - The x-axis ranging from -5 to 5. - The y-axis ranging from -5 to 5. - Key points and behaviors of the function: - There is a jump discontinuity at \( x = 3 \): - As \( x \) approaches 3 from the left, \( f(x) \) is approaching 4. - As \( x \) approaches 3 from the right, \( f(x) \) is approaching -4. - There is an asymptote at \( x = -1 \): - The function approaches \( \infty \) as \( x \) approaches -1 from both the left and the right. **Problems:** (a) Find \( \lim_{x \to -1} f(x) \). (b) Find \( \lim_{x \to 3^-} f(x) \). (c) Find \( \lim_{x \to 4} f(x) \). (d) Find \( \lim_{x \to -4} f(x) \). (e) Find \( \lim_{x \to -1^-} f(x) \). (f) Find \( \lim_{x \to -1^+} f(x) \). (g) Sketch the graph of the derivative \( y = f'(x) \). **Visual Details of the Graph:** 1. For \( x < -1 \), the function \( f(x) \) is increasing until it reaches an asymptotic behavior at \( x = -1 \). 2. For \( -1 < x < 3 \), the function shows different behaviors. Notably: - Near \( x = 0 \), \( f(x) \) reaches a local maximum approximately at \( y = 0 \). - Between \( x = 1 \) and \( x = 2 \), the function exhibits a steep descent and then rises again. 3. At \( x = 3 \), there is a distinct jump discontinuity: - The left-hand limit (\( x \to 3^- \)) reaches
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