Math Problem Statement
Suppose that |fβ''(x)| β€ 39 and |f (4)(x)| β€ 39 for all x in [0, π] . How large do we have to choose n so that the approximations Tn, Mn, and Sn to the integral π f(x) dx
0 are accurate to within 0.00001?
Solution
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Math Problem Analysis
Mathematical Concepts
Numerical Integration
Trapezoidal Rule
Midpoint Rule
Simpson's Rule
Error Bounds
Formulas
Trapezoidal Rule Error: E_T = (b - a)^3 / (12n^2) * max |f''(x)|
Midpoint Rule Error: E_M = (b - a)^3 / (24n^2) * max |f''(x)|
Simpson's Rule Error: E_S = (b - a)^5 / (180n^4) * max |f^{(4)}(x)|
Theorems
Error Bound for the Trapezoidal Rule
Error Bound for the Midpoint Rule
Error Bound for Simpson's Rule
Suitable Grade Level
University Level (Calculus II or Numerical Analysis)
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