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graph of f(x) is shown below. Find the graph of f'(x). HI -6-5-4-3-2- 5 -2 I N I T m A 10 16 1 Im 14 5

graph of f(x) is shown below. Find the graph of f'(x). HI -6-5-4-3-2- 5 -2 I N I T m A 10 16 1 Im 14 5

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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**Question** The graph of \(f(x)\) is shown below. Find the graph of \(f'(x)\). **Graph Description** The graph represents a function \(f(x)\) and appears as a smooth curve plotted on a coordinate system with \(x\)-axis ranging from \(-6\) to \(6\) and \(y\)-axis ranging from \(-6\) to \(6\). Key features of the graph include: - The curve begins from the left, decreasing initially until approximately \(x = -5.5\), where it reaches a local minimum. - It then increases, reaching a local maximum slightly above \(y = -2\) around \(x = -3.5\). - Following the peak, the curve decreases again, crossing the x-axis near \(x = -2\). - The function again rises, reaching a local maximum around \(x = 0.5\) with \(y\) slightly above \(4\). - Finally, the curve descends as it moves to the right, crossing below the x-axis around \(x = 2.5\). Your task is to find the derivative \(f'(x)\), which represents the slope of the tangent line to the curve at each point. Identifying the points of minimum and maximum on the original graph helps to determine where \(f'(x)\) would cross the x-axis (i.e., where the slope is zero).
Transcribed Image Text:**Question** The graph of \(f(x)\) is shown below. Find the graph of \(f'(x)\). **Graph Description** The graph represents a function \(f(x)\) and appears as a smooth curve plotted on a coordinate system with \(x\)-axis ranging from \(-6\) to \(6\) and \(y\)-axis ranging from \(-6\) to \(6\). Key features of the graph include: - The curve begins from the left, decreasing initially until approximately \(x = -5.5\), where it reaches a local minimum. - It then increases, reaching a local maximum slightly above \(y = -2\) around \(x = -3.5\). - Following the peak, the curve decreases again, crossing the x-axis near \(x = -2\). - The function again rises, reaching a local maximum around \(x = 0.5\) with \(y\) slightly above \(4\). - Finally, the curve descends as it moves to the right, crossing below the x-axis around \(x = 2.5\). Your task is to find the derivative \(f'(x)\), which represents the slope of the tangent line to the curve at each point. Identifying the points of minimum and maximum on the original graph helps to determine where \(f'(x)\) would cross the x-axis (i.e., where the slope is zero).
### Graph Analysis of Polynomial Functions In the image, there are five graphs representing various polynomial functions. Here’s a detailed analysis of each: #### Graph 1 - **Type**: Quadratic Function - **Shape**: Upside-down parabola (concave down). - **Vertex**: The highest point of the curve, located at (0, 4). - **X-Intercepts**: Approximately -2.3 and 2.3. - **Y-Intercept**: Located at (0, 4). - **Symmetry**: Symmetrical around the vertical axis (y-axis). #### Graph 2 - **Type**: Cubic Function - **Shape**: S-curve, exhibiting one local maximum and one local minimum. - **Turning Points**: The local maximum and minimum indicate changes in direction. - **X-Intercepts**: Approximately -2.8 and 1. - **Y-Intercept**: Crosses the y-axis at around (0, -1). - **Symmetry**: Not symmetrical. #### Graph 3 - **Type**: Cubic Function - **Shape**: An inverted S-curve with two turning points. - **Turning Points**: Local maximum around (-2, 5) and local minimum around (2, -2). - **X-Intercepts**: No apparent intercept on the visible portion. - **Y-Intercept**: Appears to pass through (0, 2). - **Symmetry**: Not symmetrical. #### Graph 4 - **Type**: Quadratic Function - **Shape**: Upside-down parabola (concave down). - **Vertex**: Highest point located around (1, 5). - **X-Intercepts**: Approximately -1.6 and 3.6. - **Y-Intercept**: Crosses y-axis around (0, 3). - **Symmetry**: Symmetrical about the vertical axis through its vertex. #### Graph 5 - **Type**: Cubic Function - **Shape**: S-curve with gradual changes in slope. - **Turning Points**: These occur where the function changes direction. - **X-Intercepts**: Approximately -3.7 and 2.7. - **Y-Intercept**: Appears to cross y-axis near (0, -3). - **Symmetry
Transcribed Image Text:### Graph Analysis of Polynomial Functions In the image, there are five graphs representing various polynomial functions. Here’s a detailed analysis of each: #### Graph 1 - **Type**: Quadratic Function - **Shape**: Upside-down parabola (concave down). - **Vertex**: The highest point of the curve, located at (0, 4). - **X-Intercepts**: Approximately -2.3 and 2.3. - **Y-Intercept**: Located at (0, 4). - **Symmetry**: Symmetrical around the vertical axis (y-axis). #### Graph 2 - **Type**: Cubic Function - **Shape**: S-curve, exhibiting one local maximum and one local minimum. - **Turning Points**: The local maximum and minimum indicate changes in direction. - **X-Intercepts**: Approximately -2.8 and 1. - **Y-Intercept**: Crosses the y-axis at around (0, -1). - **Symmetry**: Not symmetrical. #### Graph 3 - **Type**: Cubic Function - **Shape**: An inverted S-curve with two turning points. - **Turning Points**: Local maximum around (-2, 5) and local minimum around (2, -2). - **X-Intercepts**: No apparent intercept on the visible portion. - **Y-Intercept**: Appears to pass through (0, 2). - **Symmetry**: Not symmetrical. #### Graph 4 - **Type**: Quadratic Function - **Shape**: Upside-down parabola (concave down). - **Vertex**: Highest point located around (1, 5). - **X-Intercepts**: Approximately -1.6 and 3.6. - **Y-Intercept**: Crosses y-axis around (0, 3). - **Symmetry**: Symmetrical about the vertical axis through its vertex. #### Graph 5 - **Type**: Cubic Function - **Shape**: S-curve with gradual changes in slope. - **Turning Points**: These occur where the function changes direction. - **X-Intercepts**: Approximately -3.7 and 2.7. - **Y-Intercept**: Appears to cross y-axis near (0, -3). - **Symmetry
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