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3 Sketch each of the following sets on a separate real number line (a) S = {x R : |x 1| < 2} (b) T = {t R : t 2 + 2 = 4} (c) U = {s R : 2s 3 5} (d) V = {y R : |3y + 1| > 1} (e) S = {x R : x 2 + 1 < 2} (f) T = {s R : |s| = |s 1|} 4 Plot each of the points (1, 3), ( 2, 1), ( 2 , ), ( 2, 3), ( 3 , 1), and (1/2, 7/4) on a pair of cartesian coordinate axes Label each point 5 Plot each of these planar loci on a separate set of axes (a) {(x, y) : y = x 2 + 1} (b) {(x, y) : x 2 + y2 = 3} (c) y = x 2 x (d) x = y2 + y (e) x = y3 2y3 (f) x 4 + y2 = 1

i i i i i and and and and and j: j: j: j: j: 0 1 2 3 4 10 9 8 7 6

Here, commas separate the two initialization statements and the two iteration expressions When the loop begins, both i and j are initialized Each time the loop repeats, i is incremented and j is decremented Multiple loop control variables are often convenient and can simplify certain algorithms You can have any number of initialization and iteration statements, but in practice, more than two make the for loop unwieldy Here is a practical use of multiple loop control variables in a for statement This program uses two loop control variables within a single for loop to find the largest and smallest factor of a number, in this case 100 Pay special attention to the termination condition It relies on both loop control variables

case, that the unpacking activities are separate from the packing and trucking activities If you decide to use the comma in this way, be certain to do so consistently throughout your document It s best to use semicolons to separate units when there are commas within the units In the following example, notice that the sentence s meaning is clearer when it is punctuated with semicolons

// Use commas in a for statement to find the largest and // smallest factor of a number using System; class Comma { static void Main() { int i, j; int smallest, largest; int num; num = 100; smallest = largest = 1; for(i=2, j=num/2; (i <= num/2) & (j >= 2); i++, j--) { if((smallest == 1) & ((num % i) == 0)) smallest = i; if((largest == 1) & ((num % j) == 0)) largest = j; } ConsoleWriteLine("Largest factor: " + largest); ConsoleWriteLine("Smallest factor: " + smallest); } }

.

Part I:

Here is the output from the program:

Largest factor: 50 Smallest factor: 2

Through the use of two loop control variables, a single for loop can find both the smallest and the largest factor of a number The control variable i is used to search for the smallest factor It is initially set to 2 and incremented until its value exceeds one half of num The control variable j is used to search for the largest factor Its value is initially set to one half the num and decremented until it is less than 2 The loop runs until both i and j are at their termination values When the loop ends, both factors will have been found