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) = 2x +
2x2 on the interval [0, 0.2]. Compare the two estimates
with the exact range -- which you should compute by using
3. If use the simplest "and"- and "or"-operations min and max, and our degrees
of belief in A, B, and C are, correspondingly, 0.5, 0.6, and 0.7,
what is our degree of belief in (A \/ B) & C?
4. For a membership function μ(x) = 1 − |1 − 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 − |1 −
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 that, given a function f, values x1
, ..., xn
, and a number h,
computes the array of values
ci = (f(x1, ..., xi − 1,
xi + h, xi + 1, ..., xn) +
f(x1, ..., xi − 1, xi −
h, xi + 1, ..., xn) −
..., xi − 1, xi, xi + 1, ...,
corresponding to i = 1, ..., n.
9. Suppose that we have three alternatives, for which the gains are
in the intervals [0, 5], [−4, −1], and [−3, 10].
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,
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 the estimates provided by
straightforward interval computations.
These expressions should be different from
the examples that I gave; minor difference is OK.
11. If we use 2-digit decimal numbers, what will be the result of
multiplying the intervals [0.30, 0.65] and [0.77, 1.30]?