A function f(x) is big-O of g(x) if and only if ………. A function f(x) is big-Ω of g(x) if and only if …………. A function f(x) is big-ϴ of g(x) if and only if …………. 1. Convert 12021 to duodecimal (base 12) using repeated application of the division algorithm. Show all your steps. 2

MAT 243 Online Written Homework Assignments for Week 4 (units 10-13) Free Response Questions 0. A function f(x) is big-O of g(x) if and only if ………. A function f(x) is big-Ω of g(x) if and only if …………. A function f(x) is big-ϴ of g(x) if and only if …………. 1. Convert 12021 to duodecimal (base 12) using repeated application of the division algorithm. Show all your steps. 2. Determine exactly how many digits the number 2 144,000 has in sexagesimal (base 60) . Show all work. 3. Carry out the hexadecimal addition FA1 + 9BC using the pencil and paper method of addition. DO NOT convert the hex numbers to decimal. 4. Carry out the binary multiplication 110001001 x 1001 efficiently using the pencil and paper method of multiplication. Show all work. DO NOT convert the binary numbers to decimal. 5. Use fast modular exponentiation to evaluate 7 33554432 mod 11 and show all your steps. Take advantage of repetition of remainders. 6. Find a big-O estimate for the following function such that 𝑓(𝑛) is 𝑂(𝑔(𝑛)) and 𝑔(𝑛) is a simple function of the smallest order. Show your work. 𝑓(𝑛) = (n 2 + 𝑛 log(𝑛 3 ))(𝑛 3 + 𝜋·3 𝑛) + (5·2𝑛 + 1.9 𝑛+1 )(3 𝑛 2 + 5𝑛 log 𝑛) 7. Use the definition of “𝑓(𝑥) 𝑖𝑠 𝑂(𝑔(𝑥))” to show that 𝑓(𝑥) = 3𝑥 5 − 2𝑥 3 + 89 is 𝑂(𝑥 5 ). You have to find constants 𝐶 and 𝑘 such that |3𝑥 5 − 2𝑥 3 + 89| ≤ 𝐶|𝑥 5 |, when 𝑥 > 𝑘. 8. Give a formula for the value of the octal number 7676 ⏟ … 76 76 𝑛 blocks 76 in decimal, in terms of 𝑛.

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