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(a) Graph the function f(x) = x + 7/x and the secant line that passes through the points (1, 8) and (14, 14.5) in the viewing rectangle [0, 16] by [0, y y 15 15 10 10 X 10 15 5 10 15 y 15 15 10 10 5 10 15 10 15 (b) Find the number c that satisfies the conclusion of the Mean Value Theorem for this function f and the interval [1, 14]. C =

(a) Graph the function f(x) = x + 7/x and the secant line that passes through the points (1, 8) and (14, 14.5) in the viewing rectangle [0, 16] by [0, y y 15 15 10 10 X 10 15 5 10 15 y 15 15 10 10 5 10 15 10 15 (b) Find the number c that satisfies the conclusion of the Mean Value Theorem for this function f and the interval [1, 14]. C =

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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**Educational Content: Graphing Functions and Applying the Mean Value Theorem** **Task (a): Graphing the Function** We are given the function \( f(x) = x + \frac{7}{x} \). The task is to graph this function along with the secant line that passes through the points (1, 8) and (14, 14.5). The viewing rectangle is defined by the intervals [0, 16] on the x-axis and [0, 16] on the y-axis. **Description of Graphs:** 1. **First Graph (Top Left):** - The function \( f(x) = x + \frac{7}{x} \) is depicted as a curve that dips and rises sharply. - A straight line representing the secant line intersects the curve twice. 2. **Second Graph (Top Right):** - The function curve and secant line are similar, but the secant does not intersect the function curve properly. 3. **Third Graph (Bottom Left):** - The depiction is again similar, with attempts to align the secant line more visibly with the curve, but it does not align with the description. 4. **Fourth Graph (Bottom Right):** - Correct graph representing both the function and the secant line. The secant aligns between the points given (1, 8) and (14, 14.5). **Task (b): Applying the Mean Value Theorem** We are tasked to find the number \( c \) that satisfies the conclusion of the Mean Value Theorem (MVT) for the interval [1, 14]. The Mean Value Theorem essentially states that for a continuous and differentiable function, there exists a point \( c \) within (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over [a, b]. - **Conclusion:** - The correct graph, marked with a blue check, indicates the function and secant line implementation. - After computing, the necessary value \( c \) is expected to be listed in the input box. This lesson covers graphing techniques, visualization of functions and secants, and the application of fundamental calculus principles through the Mean Value Theorem.
Transcribed Image Text:**Educational Content: Graphing Functions and Applying the Mean Value Theorem** **Task (a): Graphing the Function** We are given the function \( f(x) = x + \frac{7}{x} \). The task is to graph this function along with the secant line that passes through the points (1, 8) and (14, 14.5). The viewing rectangle is defined by the intervals [0, 16] on the x-axis and [0, 16] on the y-axis. **Description of Graphs:** 1. **First Graph (Top Left):** - The function \( f(x) = x + \frac{7}{x} \) is depicted as a curve that dips and rises sharply. - A straight line representing the secant line intersects the curve twice. 2. **Second Graph (Top Right):** - The function curve and secant line are similar, but the secant does not intersect the function curve properly. 3. **Third Graph (Bottom Left):** - The depiction is again similar, with attempts to align the secant line more visibly with the curve, but it does not align with the description. 4. **Fourth Graph (Bottom Right):** - Correct graph representing both the function and the secant line. The secant aligns between the points given (1, 8) and (14, 14.5). **Task (b): Applying the Mean Value Theorem** We are tasked to find the number \( c \) that satisfies the conclusion of the Mean Value Theorem (MVT) for the interval [1, 14]. The Mean Value Theorem essentially states that for a continuous and differentiable function, there exists a point \( c \) within (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over [a, b]. - **Conclusion:** - The correct graph, marked with a blue check, indicates the function and secant line implementation. - After computing, the necessary value \( c \) is expected to be listed in the input box. This lesson covers graphing techniques, visualization of functions and secants, and the application of fundamental calculus principles through the Mean Value Theorem.
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