Sample Problems

Math 464

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 16.

1. Let Q(f) = af(\frac{1}{4}) + bf(\frac{1}{2}) + cf(\frac{3}{4}) 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)| ≤ \frac{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, p′(1) = 2, p′′(1) = 1, p(2) = 3. Prove that the polynomial is unique.

5. a) What points x_{0}, x_{1}, x_{2}, x_{3} should be used in [0, 1] to minimize the error bound in polynomial in-

terpolation at four points? Express your answer using the symbols Hint: Use cos x = \sqrt{\frac{1 + cos 2x}{2}}.

b) What is the largest value attained by (x−x_{0})(x−x_{1})(x−x_{2})(x−x_{3}) in [0, 1], where x_{0}, x_{1}, x_{2}, x_{3} are the points chosen in part a)?

In this problem, [0, 1] is \underline{not} the interval [−1, 1].

6. Prove that

2T_{n}(x) = (x + i\sqrt{1 − x^{2}})^{n} + (x − i\sqrt{1 − x^{2}})^{n} ,

for n = 0, 1, 2, where i = \sqrt{−1} and T_{n} is the n^{th} Chebyshev polynomial. (This is formula is true for all n.)

7. The Chebyshev polynomial of degree 3 is T_{3}(x) = 4x^{3} − 3x.

(a) What is the interpolatory quadrature rule on the interval [−1, 1] that uses the roots of T_{3} as nodes?

(b) What is the corresponding rule on any interval [a, b]? (”Translate” the interval.)

8. a) Let p(x) = 2x^{4} − 3x^{3} + 5x^{2} − 2x + 7. Use nested multiplication to evaluate p(2). Show the multiplications.

b) Use nested multiplication to evaluate p′(2). Show the multiplications. Do \underline{not} take the derivative of p(x) and then use the polynomial p′(x) to evaluate p′(2).

c) Use nested multiplication again to evaluate p′′(2).

9. Let p_{3}(x) be the unique polynomial of degree at most 3 that interpolates f(x) = 2^{x+2} at the points (x_{0}, x_{1}, x_{2}, x_{3}) = (−2, −1, 1, 2). Notice that 0 is \underline{not} one of the points.

a) Form the divided difference table for f at these points and write p_{3}(x) in its Newton form.

b) Give the best upper bound for ∥f − p_{3}∥_{∞} on [−2, 2].