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3.2 The Growth of Functions 1. Determine whether each of these functions is O(x). a) f (x) = 10 b) f (x) = 3x + 7 c) f (x) = x2 + x +1 d) f (x) = 5 log x

3.2 The Growth of Functions 1. Determine whether each of these functions is O(x). a) f (x) = 10 b) f (x) = 3x + 7 c) f (x) = x2 + x +1 d) f (x) = 5 log x

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
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3.2 The Growth of Functions 1. Determine whether each of these functions is O(x). a) f(x) = 10 b) f (x) = 3x + 7 c) f (x) = x2 + x + 1 d) f (x) = 5 log x 2. Determine whether each of these functions is O(x2). a) f (x) = 17x + 11 b) f(x) = x² + 1000 c) f (x) = x log x x* d) f(x)= e) f (x)=2* f) ) f (x) = (x³ + 2x)/(2x + 1) 3. Find the least integer n such that f(x) is O(x") for each of these functions. a) f (x) = 2x³ + x² log x b) f (x) = 3x³ + (log x)* c) f (x) = (x* + x' + 1/(x³ + 1) d) f (x) = (x* + 5 log x)/(x* + 1) 4. Determine whether x³ is O(g(x)) for each of these functions g(x). a) g(x) = x? b) g(x) = x³ c) g(x) = x² + x³ d) g(x) = x² + x* e) g(x) = 3* f) g(x) = x³/2 5. Arrange the functions n, 1000 log n, n log n, 2n!, 2", 3ª, and n²/1,000,000 in a list so that each function is big-O of the next function. 6. Give as good a big-O estimate as possible for each of these functions. a) (n² + 8)(n + 1) b) (n log n + n²)(n³ + 2) c) (n! + 2")(n³ + log(n² + 1))
Transcribed Image Text:3.2 The Growth of Functions 1. Determine whether each of these functions is O(x). a) f(x) = 10 b) f (x) = 3x + 7 c) f (x) = x2 + x + 1 d) f (x) = 5 log x 2. Determine whether each of these functions is O(x2). a) f (x) = 17x + 11 b) f(x) = x² + 1000 c) f (x) = x log x x* d) f(x)= e) f (x)=2* f) ) f (x) = (x³ + 2x)/(2x + 1) 3. Find the least integer n such that f(x) is O(x") for each of these functions. a) f (x) = 2x³ + x² log x b) f (x) = 3x³ + (log x)* c) f (x) = (x* + x' + 1/(x³ + 1) d) f (x) = (x* + 5 log x)/(x* + 1) 4. Determine whether x³ is O(g(x)) for each of these functions g(x). a) g(x) = x? b) g(x) = x³ c) g(x) = x² + x³ d) g(x) = x² + x* e) g(x) = 3* f) g(x) = x³/2 5. Arrange the functions n, 1000 log n, n log n, 2n!, 2", 3ª, and n²/1,000,000 in a list so that each function is big-O of the next function. 6. Give as good a big-O estimate as possible for each of these functions. a) (n² + 8)(n + 1) b) (n log n + n²)(n³ + 2) c) (n! + 2")(n³ + log(n² + 1))
3.6 Integers and Algorithms 1. Suppose pseudo-random numbers are produced by using: xn-+1 = (3xa + 11) mod 13. If x3= 5, find x2 and x4. 2. Suppose pseudo-random numbers are produced by using: Xp+1 = (2xn + 7) mod 9. a) If xo = 1, find x2 and x3 b) If x3 = 3, find X2 and x4. 3. Using the function f(x) = (x+ 10) mod 26 to encrypt messages. Answer each of these questions. a) Encrypt the message STOP b) Decrypt the message LEI 4. Which memory locations are assigned by the hashing function h(k) = k mod 101 to the records of insurance company customers with these Social Security Numbers? a) 104578690 b) 432222187 5. Use the Euclidean algorithm to find a) gcd(14, 28) b) gcd(8, 28) c) gcd(100, 101) d) gcd(28,35) e) lcm(7, 28) f) lem(12, 28) g) lem(100, 101) h) lcm(28,35)
Transcribed Image Text:3.6 Integers and Algorithms 1. Suppose pseudo-random numbers are produced by using: xn-+1 = (3xa + 11) mod 13. If x3= 5, find x2 and x4. 2. Suppose pseudo-random numbers are produced by using: Xp+1 = (2xn + 7) mod 9. a) If xo = 1, find x2 and x3 b) If x3 = 3, find X2 and x4. 3. Using the function f(x) = (x+ 10) mod 26 to encrypt messages. Answer each of these questions. a) Encrypt the message STOP b) Decrypt the message LEI 4. Which memory locations are assigned by the hashing function h(k) = k mod 101 to the records of insurance company customers with these Social Security Numbers? a) 104578690 b) 432222187 5. Use the Euclidean algorithm to find a) gcd(14, 28) b) gcd(8, 28) c) gcd(100, 101) d) gcd(28,35) e) lcm(7, 28) f) lem(12, 28) g) lem(100, 101) h) lcm(28,35)
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