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6 Plot each of these regions in the plane (a) {(x, y) : x 2 + y2 < 2} (b) {(x, y) : y < 3x 2 } (c) {(x, y) : y > x 3 } (d) {(x, y) : x y 2} (e) {(x, y) : y 2x + 1} (f) {(x, y) : x 2y 3} 7 Calculate the slope of each of the following lines: (a) The line through the points ( 4, 3) and (1, 4) (b) The line perpendicular to the line through (1, 1) and (2, 5) (c) The line y 2x = 4 (d) The line x 3y = 5 x+y (e) The line through the points (1, 1) and (8, 3) (f) The line x 2y = 5 8 Write the equation of each of the following lines (a) The line parallel to x + 2y = 1 and passing through the point (2, 3) (b) The line perpendicular to x y = 1 and passing through the point ( 1, 2) (c) The line passing through the point (2, 3) and having slope 4 (d) The line passing through ( 3, 4) and (2, 3) (e) The line passing through the origin and having slope 4 (f) The line perpendicular to y = 3x 2 and passing through ( 2, 1) 9 Graph each of the lines in Exercise 8 on its own set of axes Label your graphs 10 Which of the following is a function and which is not Give a reason in each case (a) f assigns to each person his biological mother (b) g assigns to each man his sister (c) h assigns to each real number its cube root (d) f assigns to each positive integer its square (e) g assigns to each car its passenger (f) h assigns to each toe its foot (g) f assigns to each rational number the least integer that exceeds it (h) g assigns to each integer the previous integer (i) h assigns to each real number its cube plus four.

The conditional expression controlling a for loop can be any valid expression that produces a bool result It does not need to involve the loop control variable For example, in the next program, the for loop is controlled by the value of done

// Loop condition can be any bool expression using System; class forDemo { static void Main() { int i, j; bool done = false; for(i=0, j=100; !done; i++, j--) { if(i*i >= j) done = true; ConsoleWriteLine("i, j: " + i + " " + j); } } }

The output is shown here:

It was decided that team leaders would include Amanda, East Coast, Sam, West Coast, Gloria, Mid Atlantic, and Cal, South It was decided that team leaders would include Amanda, East Coast; Sam, West Coast; Gloria, Mid Atlantic; and Cal, South

Following the way that we usually do things in calculus, we break the problem up into pieces In moving the object from position x to position x + x, the distance moved is x feet and the force applied is about F ( x) = 3x 2 x + 10 See Figure 823 Thus work performed in that little bit of the move is w( x) = ( 3x 2 x + 10) x The aggregate of the work is obtained by summation In this instance, the integral is the appropriate device:

i, i, i, i, i, i, i, i, i, i, i, j: j: j: j: j: j: j: j: j: j: j: 0 100 1 99 2 98 3 97 4 96 5 95 6 94 7 93 8 92 9 91 10 90

In this example, the for loop iterates until the bool variable done is true This variable is set to true inside the loop when i squared is greater than or equal to j

5:

Some interesting for loop variations are created by leaving pieces of the loop definition empty In C#, it is possible for any or all of the initialization, condition, or iteration portions of the for loop to be empty For example, consider the following program:

A man is carrying a 100 lb sack of sand up a 20-foot ladder at the rate of 5 feet per minute The sack has a hole in it and sand leaks out continuously at a rate of 4 lbs per minute How much work does the man do in carrying the sack

// Parts of the for can be empty using System; class Empty { static void Main() { int i; for(i = 0; i < 10; ) { ConsoleWriteLine("Pass #" + i); i++; // increment loop control var } } }

Here, the iteration expression of the for is empty Instead, the loop control variable i is incremented inside the body of the loop This means that each time the loop repeats, i is tested to see whether it equals 10, but no further action takes place Of course, since i is incremented within the body of the loop, the loop runs normally, displaying the following output:

Pass Pass Pass Pass Pass Pass Pass Pass Pass Pass #0 #1 #2 #3 #4 #5 #6 #7 #8 #9

It takes four minutes for the man to climb the ladder At time t, the sack has 100 4t pounds of sand in it From time t to time t + t, the man moves 5 t feet up the ladder He therefore performs about w( t) = ( 100 4t) 5 t foot-pounds of work See Figure 824 The total work is then the integral

In the next example, the initialization portion is also moved out of the for: