Test 2 for the course

CS 5351/CS 4365, Fall 2013

Name: ___________________________________________________________

1-2. Use both linearization algorithms that we studied in class
(Algorithm 1 that uses partial derivatives and Algorithm 2 which
does not) to estimate the range of the function f(x) = x +
2x^{2} on the interval [0, 0.2]. Compare the two estimates
with the exact range -- which you should compute by using
calculus.

3. If use the simplest "and"- and "or"-operations, and our degrees of belief in A, B, and C are, correspondingly, 0.7, 0.6, and 0.8, what is our degree of belief in (A \/ B) & C?

4. For a membership function μ(x) = 1 − |2 − x|, what are the α-cuts corresponding to α = 0.6? to α = 0.7?

5-6. Let us assume that the quantity x is described by the membership function μ(x) = 1 − |x|, and the quantity y is described by the membership function μ(y) = 1 − |2 − y|. Use the values α = 0.2, 0.4, 0.6, 0.8, and 1.0 to form membership functions for z = x − y and t = x * y.

7. Follow the first three steps of bisection to locate the square
root of 5, i.e., the solution to the equation x^{2} −
5 = 0 on the interval [0, 4].

8. Write down a Java method which computes the sum c_{1} +
c_{2} + ... + c_{n}, where

9. Suppose that we have three alternatives, for which the gains are in the intervals [10, 15], [4, 9], and [7, 20];

- is there an alternative which is guaranteed to be optimal (i.e., for which the gain is the largest)?
- list all alternatives which can be optimal;
- which alternative(s) should we select if we use Hurwicz optimism-pessimism criterion with α = 0, 0.5, and 1.

10. Write down:

- an expression for which an optimizing compiler improves the estimates provided by straightforward interval computations, and
- an expression for which an optimizing compiler worsens improves the estimates provided by straightforward interval computations.

11. If we use 2-digit decimal numbers, what will be the result of multiplying the intervals [0.40, 0.75] and [0.77, 1.30]?