Suppose that N(h) is an approximation to M for every h > 0 and...

(See attached file for full problem description with proper equations and exponents) --- 1. Suppose that N(h) is an approximation to M for every h > 0 and that M = N(h)+K1h2+K2h4+K3h6+….. For some value K1, K2, K3… Use the values N(h), N(h/3), and N(h/9) to produce an O(h6) approximation to M. ---

Approximate the integrals using the Trapezoidal rule.

(See attached file for full problem description with proper equations and exponents) --- 1. Approximate the integrals using the Trapezoidal rule. a) Integral from -0.5 to 0 x ln(x+1) dx b) Integral from 0.75 to 1.3 ((sin x)2 - 2x sin x +1) dx 2. Find a bound for error in question 1. using the error formula, and compare this to the actual error. 3. Repeat question 1. using Simpson's rule 4. Repeat question 2. using Simpson's rule and the result of question 3

1. The trapezoid rule applied to the integral from 0 to 2 f(x) dx gives the value 5, and the midpoint rule gives the value 4. What value does Simpson's rule give?

1. The trapezoid rule applied to the integral from 0 to 2 f(x) dx gives the value 5, and the midpoint rule gives the value 4. What value does Simpson's rule give?

Fixed Point Iteration Problem

(See attached file for full problem description with proper symbols) --- Given the function , show that it has a unique fixed point in the interval and that the iterations converge to the fixed point for any . Also, find the number of iterations necessary to guarantee that . Write a short Matlab code to find the fixed point. ---

Proving an Inequality

(See attached file for full problem description with equation and symbols) --- Let be an increasing sequence of positive real numbers. Prove that Can anything be said about the constant 4? ---