Inversions Let A1 : : n be an array of n distinct numbers. If i Aj , then the pair .i; j / is called an inversion of A. a. List the five inversions of the array h2; 3; 8; 6; 1i. b. What array with elements from the set f1; 2; : : : ; ng has the most inversions? How many does it have? c. What is the relationship between the running time of insertion sort and the number of inversions in the input array? Justify your answer. d. Give an algorithm that determines the number of inversions in any permutation on n elements in .n lg n/ worst-case time. (Hint: Modify merge sort.)

MTH 132 - Lecture 23 - Graphing How to use f’ and f’’ to graph 1. Determine domain 2. X intercept a. Solve for f(x) = 0 if possible 3. Y intercept a. Solve for f(0) 4. Critical points a. When derivative f’(x) = 0 5. Increasing or decreasing on the interval f(x) 6. Inflection points 7. Concavity f” 8. Vertical and horizontal asymptotes. 9. Use combined information to graph. Example: ● x − 3x 1. Domain: (-infinity, infinity) 2. X intercept: a. x(x^2 -3) = 0 √ √ b. X1 = 0, x2 = 3, x3 = − 3 3. Y intercept: a. f(0) = 0 4. Critical Points: a. 3x^2 - 3 b. 3(x^2-1) c. (x+1)(x-1) d. Cr