Example 3 f(x) = sin(x)+cos(x) , x Î [0,2p] .
Intercepts
Here the intercepts might be harder to find in general, since we are now working with a ``transcendental'' function. However, it works out without too much trouble in this case. To find the y -intercept, f(0) = 1 , so (0,1) is the y -intercept. For the x -intercepts, you have to solve 0 = sin(x)+cos(x), or cos(x) = -sin(x) . If you think about the unit circle, you can see that this happens when the angle is -p/4, or 3p/4 . But, here we need to keep within the domain, so this occurs when x = 3p/4 and x = 7p/4 .
Since this graph has a bounded domain, it also helps to find the endpoints, (0,1) , which we already found since it was the y -intercept, and (2p,1) .
Increasing/decreasing, and critical points
f¢(x) = cos(x)-sin(x) , so critical points are where sin(x) = cos(x) , which occurs, in the domain, at x = p/4 and x = 5p/4 . The critical values are f(p/4) = Ö2 and f(5p/4) = -Ö2 . It's a little harder to see the sign of f¢(x) since the function is not factorable, but remember that the only places that f¢(x) = 0 is at these two points, so if it's positive at some point in between, it's positive at all points in between. I prefer to think about how the various parts of f¢(x) are changing to see the signs. Since f¢(0) = 1 > 0 , then f¢(x) is positive on the whole interval [0,p/4] , and as you pass p/4 , sin(x) increases, while cos(x) decreases, and f¢(p/4) = 0 , so in the next interval, (p/4,5p/4), f¢(x) < 0 . As you cross 5p/4 , sin(x) is decreasing (getting more negative), but cos(x) is increasing (less negative), so since f¢(5p/4) = 0 , f¢(x) > 0 on the interval (5p/4,2p] .
Draw the graph
Use this information to draw a fair representation of the graph.
* The first information you plot are the intercepts and endpoints; [Link to ex3-1.gif]
* Then plot the critical points (and values). I always mark it with a horizontal line to remind myself that it is a critical point. [Link to ex3-2.gif]
* Then fill in the graph, connecting the dots and making sure that the graph you draw is increasing/decreasing where the number line indicates it should be. [Link to ex3-3.gif]
[Each of these items should trigger the appearance of a new drawing with that information added.]
Copyright (c) 2000 by David L. Johnson.