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f f' "1 av by ? ♥ f" ? ♥ a b c d

f f' "1 av by ? ♥ f" ? ♥ a b c d

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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### Analyzing Graphs of Functions and Their Derivatives The figure illustrates the graphs of four functions: f, f', f'', and f'''. The objective is to correctly identify each curve. **Graph Details:** 1. **Labeling the Curves (a, b, c, d):** - There are four curves labeled as a, b, c, and d, each representing one of the functions f, f', f'', and f'''. 2. **Color Coding and Shape Analysis:** - The curves are multi-colored and shaped differently to represent the various functions. - Curve **a** is in magenta. - Curve **b** is in cyan. - Curve **c** is in green. - Curve **d** is in black. 3. **Behavior of the Functions:** - The graph's behavior at various points can be observed: - **f (original function)** - Expected to have a continuous, possibly oscillating curve. - **f' (first derivative)** - Expected to indicate the rate of change (slope) of f. - **f'' (second derivative)** - Expected to indicate the concavity (curvature) of f. - **f''' (third derivative)** - Expected to represent the rate of change of the second derivative (inflection points). **Task:** - Identify which curve corresponds to which function by analyzing their behavior. Based on the visual representation and typical characteristics of polynomial functions and their derivatives: **Dropdown Choices:** - f: ? - f': ? - f'': ? - f''': ? **Explanation:** - **Curve a (Magenta)**: Typically represents a smoothly curved line that could either be f or g. - **Curve b (Cyan)**: Exhibits behavior of a derivative, higher slopes and crossings; possibility of f' or f''. - **Curve c (Green)**: Indicates changes in concavity, fitting for f''. - **Curve d (Black)**: A broader curve indicating the original function f. By analyzing the curves' behavior: - Look at the points where the curves cross the x-axis. - The smoothness and changes in the slopes would help in identifying f, f', f'', and f'''. With practice, the identification of such graphs becomes more intuitive as the student learns to associate the visual cues from the derivatives with the corresponding functions. **
Transcribed Image Text:### Analyzing Graphs of Functions and Their Derivatives The figure illustrates the graphs of four functions: f, f', f'', and f'''. The objective is to correctly identify each curve. **Graph Details:** 1. **Labeling the Curves (a, b, c, d):** - There are four curves labeled as a, b, c, and d, each representing one of the functions f, f', f'', and f'''. 2. **Color Coding and Shape Analysis:** - The curves are multi-colored and shaped differently to represent the various functions. - Curve **a** is in magenta. - Curve **b** is in cyan. - Curve **c** is in green. - Curve **d** is in black. 3. **Behavior of the Functions:** - The graph's behavior at various points can be observed: - **f (original function)** - Expected to have a continuous, possibly oscillating curve. - **f' (first derivative)** - Expected to indicate the rate of change (slope) of f. - **f'' (second derivative)** - Expected to indicate the concavity (curvature) of f. - **f''' (third derivative)** - Expected to represent the rate of change of the second derivative (inflection points). **Task:** - Identify which curve corresponds to which function by analyzing their behavior. Based on the visual representation and typical characteristics of polynomial functions and their derivatives: **Dropdown Choices:** - f: ? - f': ? - f'': ? - f''': ? **Explanation:** - **Curve a (Magenta)**: Typically represents a smoothly curved line that could either be f or g. - **Curve b (Cyan)**: Exhibits behavior of a derivative, higher slopes and crossings; possibility of f' or f''. - **Curve c (Green)**: Indicates changes in concavity, fitting for f''. - **Curve d (Black)**: A broader curve indicating the original function f. By analyzing the curves' behavior: - Look at the points where the curves cross the x-axis. - The smoothness and changes in the slopes would help in identifying f, f', f'', and f'''. With practice, the identification of such graphs becomes more intuitive as the student learns to associate the visual cues from the derivatives with the corresponding functions. **
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