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Given that f'(a) - which of the following graphs could be the graph of f(z)? (a- 1)" -10 -5 5 a) O 10

Given that f'(a) - which of the following graphs could be the graph of f(z)? (a- 1)" -10 -5 5 a) O 10

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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The image presents two graphs labeled b) and c), each depicting mathematical functions with vertical asymptotes. ### Graph b) Description: - **Axes:** The horizontal axis (x-axis) ranges from -10 to 10, and the vertical axis (y-axis) appears unmarked. - **Function Behavior:** The graph shows a curve that approaches a vertical line (asymptote) near x = 0 from both sides. As x approaches zero from the left, the function decreases without bound. As x approaches zero from the right, the function increases without bound. The function appears to have a discontinuity at x = 0. ### Graph c) Description: - **Axes:** The horizontal axis (x-axis) also ranges from -10 to 10, similar to graph b). - **Function Behavior:** This graph shows two vertical asymptotes, one near x = -5 and another near x = 5. As x approaches -5 from the left, the function increases steeply, and as x approaches -5 from the right, it decreases steeply. Similarly, near x = 5, the function decreases steeply as x approaches from the left and increases steeply from the right. There is a marked trough between these asymptotes. Both graphs illustrate functions with distinct vertical asymptotes and overall behavior at those asymptotes.
Transcribed Image Text:The image presents two graphs labeled b) and c), each depicting mathematical functions with vertical asymptotes. ### Graph b) Description: - **Axes:** The horizontal axis (x-axis) ranges from -10 to 10, and the vertical axis (y-axis) appears unmarked. - **Function Behavior:** The graph shows a curve that approaches a vertical line (asymptote) near x = 0 from both sides. As x approaches zero from the left, the function decreases without bound. As x approaches zero from the right, the function increases without bound. The function appears to have a discontinuity at x = 0. ### Graph c) Description: - **Axes:** The horizontal axis (x-axis) also ranges from -10 to 10, similar to graph b). - **Function Behavior:** This graph shows two vertical asymptotes, one near x = -5 and another near x = 5. As x approaches -5 from the left, the function increases steeply, and as x approaches -5 from the right, it decreases steeply. Similarly, near x = 5, the function decreases steeply as x approaches from the left and increases steeply from the right. There is a marked trough between these asymptotes. Both graphs illustrate functions with distinct vertical asymptotes and overall behavior at those asymptotes.
The text in the image is a mathematical problem that reads: "Given that \( f'(x) = -\frac{8}{(x-1)^2} \), which of the following graphs could be the graph of \( f(x) \)?" The image shows one graph labeled "a)". **Graph Description:** - The graph is of a function that involves two asymptotic behaviors. - There is a vertical asymptote at \( x=1 \), where the curve appears to approach but never touches this line. The curve dips steeply downwards on both sides of this asymptote. - To the left of the asymptote (for \( x < 1 \)), the graph drops steeply and continues decreasing as it approaches from the left. - To the right of the asymptote (for \( x > 1 \)), the graph rises steeply and then continues upward as it moves to the right. - The function appears to resemble a hyperbola centered around the vertical asymptote. - The horizontal asymptote seems to be at \( y = 0 \) as the function heads towards both the negative and positive directions of the x-axis.
Transcribed Image Text:The text in the image is a mathematical problem that reads: "Given that \( f'(x) = -\frac{8}{(x-1)^2} \), which of the following graphs could be the graph of \( f(x) \)?" The image shows one graph labeled "a)". **Graph Description:** - The graph is of a function that involves two asymptotic behaviors. - There is a vertical asymptote at \( x=1 \), where the curve appears to approach but never touches this line. The curve dips steeply downwards on both sides of this asymptote. - To the left of the asymptote (for \( x < 1 \)), the graph drops steeply and continues decreasing as it approaches from the left. - To the right of the asymptote (for \( x > 1 \)), the graph rises steeply and then continues upward as it moves to the right. - The function appears to resemble a hyperbola centered around the vertical asymptote. - The horizontal asymptote seems to be at \( y = 0 \) as the function heads towards both the negative and positive directions of the x-axis.
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