The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimat 2 [f(x) dx, where fis the function whose graph is shown. The estimates were 0.3125, 0.6444, 0.2709, and 0.5241, and the same number of subintervals were used in eac case. N
The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimat 2 [f(x) dx, where fis the function whose graph is shown. The estimates were 0.3125, 0.6444, 0.2709, and 0.5241, and the same number of subintervals were used in eac case. N
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate the integral
\[ \int_{0}^{2} f(x) \, dx, \]
where \( f \) is the function whose graph is shown below. The estimates were 0.3125, 0.6444, 0.2709, and 0.5241, and the same number of subintervals were used in each case.
[Graph Description]
The graph shows a decreasing function. The curve is concave upwards and appears to be continuously decreasing from left to right over the interval \([0, 2]\).
Choose the correct statement from the following options:
(a) \( L_n = 0.6444, \, T_n = 0.2709, \, M_n = 0.3125 \) and \( R_n = 0.5241 \)
(b) \( L_n = 0.3125, \, T_n = 0.2709, \, M_n = 0.6444 \) and \( R_n = 0.5241 \)
(c) \( L_n = 0.5241, \, T_n = 0.6444, \, M_n = 0.2709 \) and \( R_n = 0.3125 \)
(d) \( L_n = 0.6444, \, T_n = 0.5241, \, M_n = 0.3125 \) and \( R_n = 0.2709 \)
(e) \( L_n = 0.3125, \, T_n = 0.5241, \, M_n = 0.6444 \) and \( R_n = 0.2709 \)
Here, \( L_n \) represents the approximation using the Left Rule, \( R_n \) the approximation using the Right Rule, \( T_n \) the approximation using the Trapezoidal Rule, and \( M_n \) the approximation using the Midpoint Rule.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2F2cbdcb09-7ab0-432d-a74e-3fed0ec5c942%2F7j21yi8_processed.png&w=3840&q=75)
Transcribed Image Text:The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate the integral
\[ \int_{0}^{2} f(x) \, dx, \]
where \( f \) is the function whose graph is shown below. The estimates were 0.3125, 0.6444, 0.2709, and 0.5241, and the same number of subintervals were used in each case.
[Graph Description]
The graph shows a decreasing function. The curve is concave upwards and appears to be continuously decreasing from left to right over the interval \([0, 2]\).
Choose the correct statement from the following options:
(a) \( L_n = 0.6444, \, T_n = 0.2709, \, M_n = 0.3125 \) and \( R_n = 0.5241 \)
(b) \( L_n = 0.3125, \, T_n = 0.2709, \, M_n = 0.6444 \) and \( R_n = 0.5241 \)
(c) \( L_n = 0.5241, \, T_n = 0.6444, \, M_n = 0.2709 \) and \( R_n = 0.3125 \)
(d) \( L_n = 0.6444, \, T_n = 0.5241, \, M_n = 0.3125 \) and \( R_n = 0.2709 \)
(e) \( L_n = 0.3125, \, T_n = 0.5241, \, M_n = 0.6444 \) and \( R_n = 0.2709 \)
Here, \( L_n \) represents the approximation using the Left Rule, \( R_n \) the approximation using the Right Rule, \( T_n \) the approximation using the Trapezoidal Rule, and \( M_n \) the approximation using the Midpoint Rule.
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