It is important to be aware of the disadvantages of using calculators or other automated mathematics systems. This was covered in lesson 1.4, and would be excellent to review.

Example Problem:

Analyze the function: .

	First, restate the function.
	Next, find the first derivative, so that we can find all of the critical points
	Set the first derivative equal to zero, factor and solve. These then are our critical points.
	Now we calculate the second derivative.
	Again, we set the second derivative equal to zero, factor, and solve. We will need to use the quadratic formula for solving the trinomial. These are our inflection points.

Now, we will set up a table with these key points as "dividers" of the domain.

Key Points and Intervals	f(x)	f'(x)	f''(x)	Analysis
interval:	-	+	-	Negative, increasing, and concave down
(0,0)	0	0	0
interval:	-	-	-	Negative, decreasing, and concave down
(.369,-.005)	-.005	-	0
interval:	-	-	+	Negative, decreasing, and concave up
(.571,-.0084)	-.0084	0	+
interval:	-	+	+	Negative, increasing, and concave up
(.773,-.004)	-.004	+	0
interval:	-	+	-	Negative, increasing, and concave down
(1,0)	0	0	0
interval:	+	+	+	Positive, increasing, and concave up

Now let us make use of the information in this table to make a rough sketch of the function.

	First, we set up a scale which covers the ordered pairs in our table above. We include scales which allow us to plot the ordered pairs. Please note that the scale for the y-axis is not proportional to the scale of the x-axis. This is so that we can see what is happening on this small scale.
	We now add points for the zeros of the function.
	Now we add the other key ordered pairs from the above chart.
	Now, using the "Analysis" column from the above table, we sketch a curve which follows the slope and concavity specified there.
	Lastly, we compare this to Maple's version. You can see that our estimate of the shape of the first "hump" was inaccurate. Nevertheless, we have a fairly good idea of the behaviour of the function.