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.
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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