Math Problem Statement
Suppose that |f ''(x)| ≤ 44 and |f (4)(x)| ≤ 44 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
Error Bound Estimation
Calculus
Formulas
Trapezoidal Rule Error Bound: E_T ≤ (b-a)^3 / (12n^2) * max |f''(x)|
Midpoint Rule Error Bound: E_M ≤ (b-a)^3 / (24n^2) * max |f''(x)|
Simpson's Rule Error Bound: E_S ≤ (b-a)^5 / (180n^4) * max |f^(4)(x)|
Theorems
Error Bounds for Trapezoidal, Midpoint, and Simpson's Rules
Suitable Grade Level
College Level (Calculus II or Numerical Methods)
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