Test 1 for the course

CS 5351/CS 4365, Fall 2013

Name: ___________________________________________________________

1. What is the main problem of interval computations? Why is this problem useful in practice?

2-5. We want to estimate the range of a function (1 + x) * (2 − x) on the interval [0,2]. 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 mean value form (with checking monotonicity first) to both sub-intervals.

6. Use calculus to 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]. *For extra
credit:* use naive interval computations to estimate this
range.

7. Use the interval-based optimization algorithm to locate the maximum of the function

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}= 0.25; - 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}= 1.0.

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