1. Answers for Review Questions for Lectures 1-4

2. Review Lectures 1-4 Problems Question 2. Derive a closed form for the estimate of the solution of the equation below obtained by applying 2 iterations of Newton’s Method with the initial estimate 0. Question 4. Compute the quadratic least squares that interpolate Question 3. Compute a linear combination of functions at Question 1. Do problem 13 on page 79 of the textbook. approximation to the functionover Question 5. Use Gram-Schmidt to compute orthonormal functions from the sequence over Question 6. Do problem 1 on page 215 of the textbook.

3. Question 1 Do problem 13 on page 79 of the textbook. Let Find an interval [a,b] and for whichcontaining be the largest root of the bisection method will converge toThen estimate The number of iterates needed to findwithin an accuracy of Answer: [a,b] = [1,2]. if

4. Question 2 Derive a closed form for the estimate of the solution of the equation below obtained by applying 2 iterations of Newton’s Method with the initial estimate 0. Answer. Define Initial estimate then it derivative

5. Question 3 Answer Let the interpolant have the form that interpolate Compute a linear combination of functions at Then determine the coefficientsso that The solution is:

6. Question 4 Compute the quadratic least squares approximation to the functionover Answer. Since ‘quadratic’ means quadratic polynomial, the approximant has the form to minimize where the coefficients are chosen 

7. Question 4

8. Question 5 Use Gram-Schmidt to compute orthonormal functions from the sequence over Answer. Here the inner product onis therefore our sequence of functions satisfies

9. Question 5 The Gram-Schmidt orthonormalization rule gives

10. Question 6 Using the error formula (5.23), bound the error in Do problem 1 on page 215 of the textbook. for the following integrals: Answer Error formula (5.23) on page 204 gives where are: subinterval lengths, interval [a.b], maximum absolute value of integrand Therefore, for integral: