Summer 2016, Test 1, version 2

1. Similarly to how we used Newton's method to design algorithms
for computing square root and cubic root, design an algorithm for
solving the equation e^{x} = r * x, where r is
given.

2. Use the algorithm for computing 1/b that we had in class (and
that is implemented in the computers) to perform the few first
steps of computing the ratio 1 / 0.9.

3-6.

3. Use numerical differentiation to compute the derivative of the
function x^{2} + 2 * x + 1 when x = 1.

4. Use linearization technique and your estimate for the derivative to estimate the range of this function when x is in the interval [0.8, 1.2].

5. Use naive interval computations to estimate the same range.

6. Use mean value form to estimate the same range.

7. Use Newton's method to solve the following system of non-linear
equations: x_{1} * x_{2} = 3,
x_{1} + x_{2} = 3.5. Start with the first
approximation x_{1} = 1 and x_{2} = 2. One
iteration is good enough.

8. Find the point closest the origin on the line 2 * x_{1}
+ x_{2} = 1. In other words, find the values x_{1}
and x_{2} for which the sum (x_{1})^{2} +
(x_{2})^{2} attains the smallest possible value
under the constraint 2 * x_{1} + x_{2} = 1.

9. Explain what is l-diversity, and why it is important. If l
increases, will we get more or less privacy protection? Explain
your answer.