### Continuity and Force Analysis**Problem 16:**Suppose a force exerted by an object on another mass at a distance \( r \) from the center of a planet is given by the function:\[ F(r) = \begin{cases} \frac{2.8r}{R^6}, & \text{if } r < R \\\frac{2.8R}{r^4}, & \text{if } r \ge R \end{cases} \]**Task:**Use the definition of continuity at a point to determine if \( F \) is continuous at \( r = R \).### Explanation:1. **Definition of Continuity:** A function \( f(x) \) is continuous at \( x = c \) if the following three conditions are satisfied: - The function \( f \) is defined at \( x = c \); \( f(c) \) exists. - The limit of \( f(x) \) as \( x \) approaches \( c \) exists. - The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).2. **Checking the Conditions for Continuity at \( r = R \):** - **Existence of \( F(R) \):** \[ F(R) = \frac{2.8R}{R^4} = \frac{2.8R}{R^4} = \frac{2.8}{R^3} \] Thus, \( F(R) \) is defined and equals \( \frac{2.8}{R^3} \). - **Existence of the Limit of \( F(r) \) as \( r \) Approaches \( R \):** - From the left (\( r \to R^- \)): \[ \lim_{r \to R^-} F(r) = \lim_{r \to R^-} \frac{2.8r}{R^6} = \frac{2.8R}{R^6} = \frac{2.8}{R^5} \] - From the right (\( r \to R^+ \)): \[ \lim_{r \to R^+} F(r) = \lim_{r \to R

For any function to continuous it’s LHL, RHL, and function at that point must be equal. The LHL of…

16. Suppose a force exerted by an object on another mass at a distance r from the center of the planet is 2.8r ', if r < R R6 2.8R , if r> R F(r) = r4 at a point to determine if F is continuous at r=R. Use the definition of continuity

16. Suppose a force exerted by an object on another mass at a distance r from the center of the planet is 2.8r ', if r < R R6 2.8R , if r> R F(r) = r4 at a point to determine if F is continuous at r=R. Use the definition of continuity

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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