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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2012 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Spring 2012 • Question 3

Midpoint Rule

Derive error estimate for the midpoint rule in the form: \[ |E_n^M|\leq \frac {h^2(b-a)}{24} \max _{a\leq x\leq b}|f''(x)| \] Midpoint rule: \[ M_n(f)=h(f(x_1)+f(x_2)+\cdots +f(x_n)) \] where \(h=(b-a)/n\) and \[ x_j=a+\left (j-\frac {1}{2}\right )h,\qquad j=1,\ldots ,n \]

Raw LaTeX

{Midpoint Rule}

Derive error estimate for the midpoint rule in the form:
\[
  |E_n^M|\leq \frac{h^2(b-a)}{24}
  \max_{a\leq x\leq b}|f''(x)|
\]
Midpoint rule:
\[
  M_n(f)=h(f(x_1)+f(x_2)+\cdots+f(x_n))
\]
where \(h=(b-a)/n\) and
\[
  x_j=a+\left(j-\frac{1}{2}\right)h,\qquad j=1,\ldots,n
\]

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Topic Tags

Linear Programming and Optimization Numerical Quadrature Quadrature Rules