Sample Test 3

# Sample Test 3 - wEgg/62m @ﬁ 5 MATH 2600 - 01 TEST 3 1.00...

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Unformatted text preview: wEgg/62m @ﬁ 5 MATH 2600 - 01 TEST 3 1.00 pts.) Complete the following deﬁnitions: { l) If f : N -—> N and x is an element of N, then the orbit of x equals: (2) We say an element is geriodic of period 11 for a ﬁmction ﬁx) provided: In 10 pts.) Let f: N a N and let f(a1 ...a,,) = 3.12 + ...+a,,2 +1. (1)};‘indtheorbitof 3; {I12} 67 3’7! 5’?) {63?}, 57,. 29’ 5’77“! 42,, 27f} (2) Find two different periodic points for this function. 4 3 7/ ff}, aft.» I llI.(5 pts.) State the Collatz conjecture. IV.(20pts.) Let ﬁx) be the sawtooth function with 5 teeth. 1 ‘ g2§5”7"¥ “3’79 (1) Express f6(x) as an repeating decimal where: (a) x = 3/14 , (b) x = .1579 133?. 7(b)xm.113 57 21. - é ._ ZQO7000O7GOOOO7. .. (2)Find the number of elements in the orbit of x where: (a) x (3)Give three different limit points for the orbit of x = 1 ﬂag. ,‘76’ ,w3 '0‘”? V. (12 pts.) Let ﬁx) be the tent function. ' (1) Express 5‘ (3:14) in base 2. \$Z’y = . 0 3'57 3;. ‘7 53,4} =‘ , m a (2) Find the number of elements in the orbit of .1601 W @ (3) Determine four diﬂ‘erent values ofx that satisfy f2 (x) = x. .. 5'5 .. F5- . W i, t ’0 0 V1.35 pts.) Complete the following deﬁnition: A function f(x) from [0,1] to [0,1] exhibits Mathematical Chaos provided: V1160 pts.) For the mnotion ﬁx) =3 x3 + 1/4 x, determine the ﬁxed points and determine which of these ﬁxed points is attracting, repelling or neither attracting nor repelling. - 2 O i- i; VIE.(8 pts.) Let ﬁx) = x + ox + e (1) Determine all values ofo that make { 0 , e } a Z-cyeie for ﬁx). (2) Are the 2-cycles in Part ( l) attracting, repelling or neither attracting nor repelling? 1546c) : a; +- £1~+ c : G :5? c; w m i {are} mi; - , 2‘ g i i (7"!) e‘ x ‘2 PX 1. VILLANOVA UNIVERS IV till it lilllllllllilllml a 9346 00890175 5 l“. ‘0; En; Mn IX.(9 pts.) Answer True or False. ‘T’ (1) Iff-(x) = ax(1 - x) and a is greater than 3, then all the fixed points for ﬁx) are repelling. ,‘r’ (2) Iff(x) is the sawtooth funct ion with 5 teeth and one. element in tl‘ then all elements of the orbit to orbit ofa value x is irrational, of x must be irrational. L § (3) A. ﬁmction can not have exactly two pointsof period 3. X.(10 pts.) Given below is the graph of a ﬁmctioo f(x). (1) Determine the integer closest to f 3(3). < -- l ‘9 (2)7 Detennine the number of attracting ﬁxed points for f(x). l ...
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## This note was uploaded on 05/31/2008 for the course MATH 2600 taught by Professor Sprows during the Spring '08 term at Villanova.

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Sample Test 3 - wEgg/62m @ﬁ 5 MATH 2600 - 01 TEST 3 1.00...

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