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12. f(x) = x³ + 3x² - 9x - 12, g(x) = 4x + 3 УЛ (-1,-1), 20 15 10- 5 L -64-2 1 2/3 4 х -10 -15 -20 | (3, 15) (-5, -17)

12. f(x) = x³ + 3x² - 9x - 12, g(x) = 4x + 3 УЛ (-1,-1), 20 15 10- 5 L -64-2 1 2/3 4 х -10 -15 -20 | (3, 15) (-5, -17)

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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How would you solve problem 12?
### Transcription for an Educational Website #### Calculating Definite Integrals **Exercises:** 8. Evaluate the integral \( \int_{0}^{2} |x^3 - 1| \, dx \). 10. Evaluate the integral \( \int_{-1}^{1} |3x - 2| \, dx \). #### Solving Equations for Shaded Regions Given Functions: - \( f(x) = x^3 + 3x^2 - 9x - 12 \) - \( g(x) = 4x + 3 \) ### Detailed Description of Graphs The graph features two functions, \( f(x) \) and \( g(x) \), plotted on a coordinate system. The function \( f(x) = x^3 + 3x^2 - 9x - 12 \) appears as a cubic curve, while \( g(x) = 4x + 3 \) is a straight line. The graph displays the following key points: - Intersection Points: The graph shows points where the functions intersect, located at approximately \((-5, -17)\) and \((3, 15)\). - Axes: The x-axis ranges from -6 to 4 and the y-axis from -20 to 20. - Shaded Region: There is a shaded area between the two curves, indicating the region between the functions \( f(x) \) and \( g(x) \). Find the solutions for these intersecting equations: 15. \( y = \) 16. \( y = \) 17. \( y = \) 19. \( y = \) 21. \( y = \) 23. \( y = \) 25. \( y = \) 27. \( y = \) 28. \( y = \) 29. \( y = \) 30. \( f(x) \) 31. \( f(x) \) 32. \( y = \) 33. \( y = \) The graph helps visualize the intersection and the areas under discussion, providing insight into solving the integrals and analyzing the relationship between the two functions.
Transcribed Image Text:### Transcription for an Educational Website #### Calculating Definite Integrals **Exercises:** 8. Evaluate the integral \( \int_{0}^{2} |x^3 - 1| \, dx \). 10. Evaluate the integral \( \int_{-1}^{1} |3x - 2| \, dx \). #### Solving Equations for Shaded Regions Given Functions: - \( f(x) = x^3 + 3x^2 - 9x - 12 \) - \( g(x) = 4x + 3 \) ### Detailed Description of Graphs The graph features two functions, \( f(x) \) and \( g(x) \), plotted on a coordinate system. The function \( f(x) = x^3 + 3x^2 - 9x - 12 \) appears as a cubic curve, while \( g(x) = 4x + 3 \) is a straight line. The graph displays the following key points: - Intersection Points: The graph shows points where the functions intersect, located at approximately \((-5, -17)\) and \((3, 15)\). - Axes: The x-axis ranges from -6 to 4 and the y-axis from -20 to 20. - Shaded Region: There is a shaded area between the two curves, indicating the region between the functions \( f(x) \) and \( g(x) \). Find the solutions for these intersecting equations: 15. \( y = \) 16. \( y = \) 17. \( y = \) 19. \( y = \) 21. \( y = \) 23. \( y = \) 25. \( y = \) 27. \( y = \) 28. \( y = \) 29. \( y = \) 30. \( f(x) \) 31. \( f(x) \) 32. \( y = \) 33. \( y = \) The graph helps visualize the intersection and the areas under discussion, providing insight into solving the integrals and analyzing the relationship between the two functions.
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