Test 1 for the course

CS 5391/CS 6391, Fall 2015

Name: ___________________________________________________________

1-4. We want to estimate the range of a function (1 − x) * (2 + x) on the interval [−2, 0]. Do the following:

- find the exact range by using calculus;
- estimate the range by applying naive (straightforward) interval computations to the original interval;
- use bisection and check monotonicity of the function on each of the sub-intervals, applying naive interval computations only if the function is not monotonic;
- use bisection and apply the centered form (with checking monotonicity first) to both sub-intervals.

1.

2.

3.

4.

5-6. Find the range of a function f(x_{1},
x_{2}) = (x_{1})^{2} + x_{1}
* x_{2} − (1/2) * (x_{2})^{2} −
x_{2} when x_{1} is in the interval [−1, 1],
and x_{2} is in the interval [−2, 2]:

- by using calculus, and
- by using naive interval computations.

5.

6.

7. Use the interval-based optimization algorithm to locate the
maximum of the function f(x) = (1 − x) * (2 +
x) on the interval [−0.8, 0]. Divide this interval into
two, then divide each of the remaining intervals into two again,
etc. Stop when you get intervals of width 0.1.

8-9. Use the constraints method to solve the following two
problems:

- find x
_{1}and x_{2}, both from the interval [0, 1], that satisfy the system of equations x_{1}+ x_{2}= 1 and

x_{1}* x_{2}= 0.21; - find x
_{1}and x_{2}, both from the interval [0, 1], that satisfy the system of equations x_{1}+ x_{2}= 1 and

x_{1}* x_{2}= 0.9.

8.

9.

10. Briefly describe the topic of your class project and what you
have done so far.