Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Understanding the Safety in Vehicle Race Tracks: A Calculus Application** **Graphs and Diagrams Analysis:** - **Figure 3.21** - **Part (a)**: This graph depicts the function \( f(x) = x^3 + 3x^2 + x \). - The graph exhibits a cubic curve with several turning points, located on a standard Cartesian coordinate plane with x-axis ranging from -5 to 5 and y-axis ranging from -5 to 5. - **Part (b)**: This graph represents the same cubic function, \( f(x) = x^3 + 3x^2 + x \), but includes additional elements. - A visual representation of the grandstand is illustrated at the coordinates (-1.9, 2.8). - This diagram particularly highlights the section of the racetrack near the point where the turning point of the function—the slope of the tangent line—is 1. **Educational Problems and Questions:** 1. **Problem Statement**: - Physicists have observed that drivers are most likely to lose control of their vehicles when navigating a turn, specifically at the juncture where the slope of the tangent line to the curve is 1. First, determine the \((x, y)\) coordinates of this key point on the racetrack. 2. **Equation of Tangent Line**: - Derive the equation for the tangent line to the curve at the identified point. 3. **Spectator Safety Analysis**: - Assess whether the spectators, located at \((x,y) = (-1.9, 2.8)\), are in a potential danger zone. Determine this by finding the x-coordinate where the tangent line intersects \(y = 2.8\) and decide if this point is safely positioned relative to the grandstand. 4. **Additional Safety Scenario**: - Consider an alternative scenario where a driver loses control earlier than anticipated, at the coordinates \((-2.5, 0.625)\). Calculate the slope of the tangent line at this point. 5. **Safety Confirmation**: - Based on the tangent line slope derived in part 4, analyze whether the spectators remain safe in this scenario. 6. **Final Decision Making**: - Conclude if the current grandstand placement is adequate for ensuring spectator safety, or if a redesign or relocation is warranted based on the mathematical insights gained
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Transcribed Image Text:**Understanding the Safety in Vehicle Race Tracks: A Calculus Application** **Graphs and Diagrams Analysis:** - **Figure 3.21** - **Part (a)**: This graph depicts the function \( f(x) = x^3 + 3x^2 + x \). - The graph exhibits a cubic curve with several turning points, located on a standard Cartesian coordinate plane with x-axis ranging from -5 to 5 and y-axis ranging from -5 to 5. - **Part (b)**: This graph represents the same cubic function, \( f(x) = x^3 + 3x^2 + x \), but includes additional elements. - A visual representation of the grandstand is illustrated at the coordinates (-1.9, 2.8). - This diagram particularly highlights the section of the racetrack near the point where the turning point of the function—the slope of the tangent line—is 1. **Educational Problems and Questions:** 1. **Problem Statement**: - Physicists have observed that drivers are most likely to lose control of their vehicles when navigating a turn, specifically at the juncture where the slope of the tangent line to the curve is 1. First, determine the \((x, y)\) coordinates of this key point on the racetrack. 2. **Equation of Tangent Line**: - Derive the equation for the tangent line to the curve at the identified point. 3. **Spectator Safety Analysis**: - Assess whether the spectators, located at \((x,y) = (-1.9, 2.8)\), are in a potential danger zone. Determine this by finding the x-coordinate where the tangent line intersects \(y = 2.8\) and decide if this point is safely positioned relative to the grandstand. 4. **Additional Safety Scenario**: - Consider an alternative scenario where a driver loses control earlier than anticipated, at the coordinates \((-2.5, 0.625)\). Calculate the slope of the tangent line at this point. 5. **Safety Confirmation**: - Based on the tangent line slope derived in part 4, analyze whether the spectators remain safe in this scenario. 6. **Final Decision Making**: - Conclude if the current grandstand placement is adequate for ensuring spectator safety, or if a redesign or relocation is warranted based on the mathematical insights gained
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