## Interval Computations, Test 1 for the course CS 5351/CS 4365, Fall 2017

Name: ___________________________________________________________

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

• find the exact range by using calculus;
• estimate the range by using linearization;
• estimate the range by applying naive (straightforward) interval computations to the original interval;
• use the general interval computations algorithm: check monotonicity, apply centered one if needed, apply bisection if the function is not monotonic.

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5-6. Find the range of a function f(x1, x2) = (x1 − 1)2 + (x1 − 1) * x2 − (1/2) * (x2)2 − x2 when both x1 is in the interval [0, 2], and x2 is in the interval [−2, 2]:
• by using calculus, and
• by using naive interval computations.

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7. Use the interval-based optimization algorithm to locate the maximum of the function
f(x) = (1 − 2x) * (2 + 2x)
on the interval [−0.4, 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 x1 and x2, both from the interval [0, 2], that satisfy the system of equations x1 + x2 = 1 and
x1 * x2 = 0.84;
• find x1 and x2, both from the interval [0, 2], that satisfy the system of equations x1 + x2 = 1 and
x1 * x2 = 3.6.

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