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. 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.

1. Let Q(f) = af(\frac{1}{6}) + bf(\frac{1}{2}) + cf(\frac{5}{6}) be a quadrature formula on [0, 1].

a) If Q is to have precision 2, what are the values of a, b, c?

b) What is the maximum precision of the resulting quadrature?

2. Let f(x) = sin(x) and let T_{N}(f), S_{N}(f) be the composite trapezoid and Simpson rules on N panels on [0, 2].

a) Compute S_{2}(f) and T_{2}(f).

b) Use the error bound to find the smallest N so that |e_{N}^{S}(f)| ≤ 10^{−6}.

3. Let

g(x) = \begin{cases}x^{3} & if 0 ≤ x ≤ 1 \\ a(x − 1)^{3} + 3(x − 1)^{2} + 3(x − 1) + 1 & if 1 ≤ x ≤ 2 \end{cases}

a) Is g(x) a cubic spline on [0, 2]? Justify your answer.

b) Is xg(x) a cubic spline on [0, 2]? Justify your answer.

c) What should the value of a be if g is to be a \underline{natural} cubic spline?

4. a) What points x_{0}, x_{1}, x_{2}, x_{3} should be used in [0, 1] to minimize the error bound in polynomial interpolation at four points? Express your answer using the symbols 1, 2, ±,^{√} . 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)?

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

5. a) Using your calculator, solve e^{−x} = x by means of Newton’s method. Let x_{0} = 0. Compute x_{1}, x_{2}, x_{3}, x_{4}, x_{5}.

b) If you were able to compute in infinite precision, for which n would x_{n} have 100 correct digits?

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

7. 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].