Interval Computations,
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 + 2x2 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 x2 − 5 = 0 on the interval [0, 4].

8. Write down a Java method which computes the sum c1 + c2 + ... + cn, where

ci = (f(x1, ..., xi−1, xi + h, xi+1, ..., xn) + f(x1, ..., xi−1, xi − h, xi+1, ..., xn) − 2f(x1, ..., xi−1, xi, xi+1, ..., xn))/h2.

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

10. Write down:

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]?