Контрольная по математике

Tasks

1. Consider a number that is equal to th following infinite sum s = £n>0 1/2(2")-ls a rat'°nal number? (Hint: consider the number in the binary notation.)

2. Let day, month, and year be the day, the month, and the year of your birth. (For example, if you are born on April 3, 2005, then day = 3, month = 4, and year = 2005.) Lety = day + month + year. Using Newton-Raphson method compute an approximation for Jy with accuracy 0.01 and represent steps of your computations in the table like a depicted below (fory = 2012):

n xn dn 0 2012 -1005.5 1 1006.5 -502.25

3. Consider equation x = cos(x). Does it have a positive root? Is it

possible to find approximation of the root using Newton-Raphson r A

algorithm from lecture notes? (Explain your answer based on pre- i \

condition and post-condition of the algorithm.) 5

4. Recall from lecture notes for week 9, that pn and Pn are perimeters ,

of right 2n+1-polygons inscribed and circumscribed in/around the

unit circle. Evaluate |P2 — P21• ~

5. Let day and month be the day and the month of your birth. (For example, if you are born on April 3, then day = 3 and month = 4.) Find area between graphs of functions f (x) = x2 and g(x) = x on the interval [—day, +month].