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From Problem 35.7 we know that ex is positive and increasing, and that it approaches + on the right and approaches 0 2 on the left. Moreover, since Dx ex = ex > 0, the graph will be concave upward for all x. The graph is shown in Fig. 35-1(a). By Theorem 35.1, y = ex is equivalent to x = ln y. So, a point (a, b) is on the graph of y = ex if and only if (b, a) is on the graph of y = ln x. But the points (a, b) and (b, a) are symmetric with respect to the line y = x, by Problem 6.4. In general, the graphs of any pair of inverse functions are mirror images of each other in the line y = x.

(2.48)

In Fig 76, the robot uses a while-wend loop to continue to turn as much as is necessary to stay on the turning line before moving forward This means that the robot will not move forward until it has turned suf ciently to remain on the line Extremely sharp turns, such as the last one in Fig 71b, still present a problem, but the robot performs well in most situations 733 A SECOND STRATEGY The second strategy is implemented in Fig 77 The routine allows the robot to leave the line when it turns sharply and then reacquire it a short distance later Once the robot loses the line it cannot use the line sensors to determine which way to turn.

(2.49)

To solve this problem, we need a way for the robot to remember which way it was turning the last time it saw the line This will normally be the direction the robot should turn if it has lost the line Each time the robot makes a normal turn (the if-decisions in Fig 77) the subroutine stores the turn direction in the variable LastTurn to remember which direction the robot was turning Later in the routine, if none of the sensors are on (indicating we probably have lost the line), the robot will be able to head back toward the line Extremely sharp turns are still a problem even for this algorithm In this algorithm, since it is acceptable to lose the line, we can speed up the robot s progress by moving it forward 2 pixels at a time.

The azimuth depends on which quadrant a is in. With a in degrees the azimuth is as given in Table 2.4.

(a) Let > 0. To prove continuity at x, we must show that there exists > 0 such that |u x| < implies |eu ex | < . Let 1 be the minimum of /2 and ex /2. Since ln is increasing and continuous, the range of ln on ex 1 , ex + 1 is an interval (c, d) containing x. Let > 0 be such that x , x + is included within (c, d). Then, for any u, if |u x| < , it follows that |eu ex | < 1 < . (b) The proof will consist in showing that ex+h ex = ex h h 0 lim ex eh ex ex+h ex eh 1 eh 1 = = ex , it will suf ce to show that lim = 1. h h h h h 0 Let k = eh 1. Then eh = 1 + k and, therefore, h = ln (1 + k), by Theorem 35.1. Since ex is continuous and e = 1, k 0 as h 0. Hence, Because eh 1 k = lim h h 0 k 0 ln 1 + k 1 = lim k 0 ln 1 + k ln 1 k 1 = ln 1 + k ln 1 lim k k 0 lim But lim

Example 2.20 The IJK range vector components for a certain satellite, at GST 240 , are as given below. Calculate the corresponding range and the look angles for an earth station the coordinates for which are latitude 48.42 N, longitude 89.26 W, height above mean sea level 200 m.

While the robot is still over the line it will turn only 1 at a time to stay on it, but if the line is lost, the robot will take a 3 turns to help it get back on course Notice that these choices for how much to move or turn are somewhat arbitrary With a little experimentation, you can determine the optimum values for your situation This reminds us again of the advantage of using a simulation With RobotBASIC you can change the values and see how the robot responds to your changes very quickly 734 VERY SHARP TURNS The routine in Fig 76 stays on the line nicely and even handles 90 turns However, if the line turns much more than 90 , the robot can still lose the line There are many ways to solve this problem Figure 78 shows a potential solution..

240 ;

[since ln 1 = 0]