One notebook sized page of notes will be allowed on the final. You may write on both sides of the page. No calculators will be allowed. This set of problems only covers material since the midterm. The final will be comprehensive. There will be questions covering the first half and second half of the course. The exam is at 8:30 a.m. on Wednesday, December 15.
1. Let Q(f) = af( 1 6 ) + bf( 1 2 ) + cf( 5 6 ) be a quadrature formula on [0, 1].
a) If Q is to have precision at least 2, what are the values of a, b, c?
b) What is the exact precision of the resulting quadrature?
2. Let p(x) be the linear function that interpolates sin(x) at 0 and π/2. Prove that |p(x) − sin(x)| ≤ 1 2 (π/4)2 on [0, π/2].
3. Let A be a nonsingular 2×2 matrix. Prove that the ∞-norm condition number and 1-norm condition number are equal.
4. Find the polynomial p of degree 3 that satisfies the following conditions: p(1) = 1, p0 (1) = 2, p(2) = 1, p0 (2) = 3.
5. a) What points x0, x1, x2, x3 should be used in [0, 1] to minimize the error bound in polynomial interpolation at four points? Express your answer using the symbols Hint: Use cos x = r 1 + cos 2x 2 .
b) What is the largest value attained by (x−x0)(x−x1)(x−x2)(x−x3) in [0, 1], where x0, x1, x2, x3 are the points chosen in part a)? Be careful in this problem, [0, 1] is not the interval [−1, 1].
6. a) Let p(x) = 2x 4 − 3x 3 + 4x 2 − 2x + 1. Use nested multiplication to evaluate p(2). Show the multiplications. b) Use nested multiplication to evaluate p 0 (2). Show the multiplications. Do not take the derivative of p(x) and then use the polynomial p 0 (x) to evaluate p 0 (2).
7. Let p3(x) be the unique polynomial of degree at most 3 that interpolates f(x) = 2x+2 at the points (x0, x1, x2, x3) = (−2, −1, 1, 2). Notice that 0 is not one of the points. a) Form the divided difference table for f at these points and write p3(x) in its Newton form. b) Give the best upper bound for kf − p3k∞ on [−2, 2].
