Assignment #2 1) Improve the result from problem 4 of the previous assignment by showing that for every e> 0, no matter how small, given n real numbers x1,...,xn where each xi is a real number in the interval [0, 1], there exists an algorithm that runs in linear time and that will output a permutation of the numbers, say y1, ...., yn, such that ∑ ni=2 |yi - yi-1| < 1 + e. (Hint: use buckets of size smaller than 1/n; you might also need the solution to problem 3 from the first assignment!) 2) To evaluate FFT(a0,a1,a2,a3,a4,a5,a6,a7) we apply recursively FFT and obtain FFT( a0,a2,a4,a6) and FFT(a1,a3,a5,a7). Proceeding further with recursion, we obtain FFT(a0,a4) and FFT(a2,a6) as well as FFT(a1,a5) and FFT(a3,a7). Thus, from bottom up, FFT(a0,a1,a2,a3,a4,a5,a6,a7)
The common difference is 3. 4. 1, −1, 1, −1, … SOLUTION: Since the ratios are constant, the sequence is geometric. The common ratio is –1. Find the next three terms in each geometric sequence.
To multiply and divide radicals, perform the indicated operation on the radicands and then simplify: (You do not need radicand to be the same in order to multiply or divide) Example 1: (85) (42) = (8)(4) 5 2 = 3210 Example 2: (105) ÷ (25)= (10 ÷ 2) (5 ÷ 5) = 5 (1) = 5 Example 3: (320 ) (32) sometimes it is easier to simplify first (64 ∙ 5) (16 ∙ 2 (85) (42) Answer = 3210 Example 4: (6500) ÷ 220 ← Rewrite as a fraction and divide =
Unit 3 – Algebra Basics Module 3C Sections 10.3 – 10.8 3C Addition of Real Numbers Addition on the Number Line To do the addition a + b on the number line, start at 0, move to a, and then move according to b. a) If b is positive, move from a to the right. b) If b is negative, move from a to the left. c) If b is 0, stay at a. Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Slide 2 3C Addition of Real Numbers Add real numbers without using the number line. Add: –4 + 9.
and a width of (2x + 3) in. Find an expression that represents the area of the rectangle. Write the expression in simplified form. Simplify each product using the FOIL method. 24.
They are equivalent because they represent the powers of 10 3: Based on the breakdown of the decimal and binary systems in this lab, describe the available digit values and the first four digits of a base 5 numbering system. You can use the binary system as a reference, where the available digit values are 0 and 1 and the first four digits are 1, 2, 4, and 8. The digit values in a base numbering system are 1s and 0s. You are using 16, 8, 4, 2, 1 instead of 128, 64, 32, 16, 8, 4, 2, and 1. 4: Using the Internet and the Help files in Excel, explain why creating a converter from decimal to binary would be more difficult to construct.
Math Background Simplify the following expressions (ln is the natural logarithm): (a) ln (b) b − ln(ex ) (a) ln(a) + ln(b) (c) eln(a)+ln(b)−ln(c) (d) ln ea × eb × ec (e) ex × e−x (f) ex+y − ex × ey (g) eln(x) 1 (h) ln(x) + ln( x ) ( ) 2. Math Background Sketch functions with (a) a positive first derivative and a positive second derivative over the range x in (0, 1). (b) a negative first derivative and a negative second derivative over the range x in (0, 1). 3. Math Background Consider the following function: 1 f (K, L) = K 1/2 × L1/2 2 This
In this figure <1 and <8 are alternate exterior angles, as are <2 and <7. Same-side interior angles are in between the two lines that are not the transversal and are on the same side of the transversal. In this figure <3 and <5 are same-side interior angles, as are <4 and <6. Polygon interior angle sum formula S=180 (n-2) ( polygon= 180 (number of sides -2) S=180 (4-2) =
There are two types of special right triangles: 45-45-90 and 30-60-90. The legs on a 45-45-90 triangle are 1 and 1 and the hypotenuse is the square root of 2. The legs on a 30-60-90 triangle are 1 and the square root of 3 and the hypotenuse is 2. If you were to take the three trigonometric functions of either 45 degree angle, you would get the (square root of 2)/2 for both cosine (x) and sine (y) and 1 for tangent (y/x). If you were to take the three trigonometric functions of the 30 degree angle, you would get the (square root of 3)/2 for cosine, ½ for sine and the (square root of 3)/3 for tangent.
Set up the matrix equation to solve this system. 2. Given the inequality y < x2 + 2x – 3, is the point (0, -3) part of the solution? Name a point that is part of the solution and one that is not. 3.