## Interval Computations, 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(x1, x2) = (x1)2 − x1 * x2 − (1/2) * (x2)2 + x2 when x1 is in the interval [−1, 1], and x2 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

f(x) = (1 + x) * (2 − x)
on the interval [0, 0.8]. 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 x1 and x2, both from the interval [0, 1], that satisfy the system of equations x1 + x2 = 1 and
x1 * x2 = 0.25;
• find x1 and x2, both from the interval [0, 1], that satisfy the system of equations x1 + x2 = 1 and
x1 * x2 = 1.0.

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