Section Summary: 4.5

1. Definitions

   slant asymptote: if

   

   or

   

   then the line y=mx+b is called a slant asymptote. Intuitively, the function f approaches the line as .

   In particular, there will be a slant asymptote for any rational function whose numerator polynomial degree exceeds the demoninator polynomial degree by 1 (e.g. quadratic over linear).

2. Theorems

   None.

3. Properties/Tricks/Hints/Etc.

   Guidelines for sketching a curve:

   1. Domain - find it!
   2. Intercepts - crossings of the axes. The y-intercept is the point (0,f(0)) (if 0 is in the domain); the x-crossings are the roots of the equation: solve f(x)=0 for these special values of x.
   3. Symmetry - even and odd functions; functions that have symmetry about some displaced point; periodicity
   4. Asymptotes - vertical, horizontal, slant
   5. Intervals of increase or decrease - use the Increasing/Decreasing Test, based on the sign of the first derivative.
   6. Local maxima and minima - use the first or second derivative tests.
   7. Concavity and points of inflection - use the second derivative and the concavity test.
   8. Sketch the curve - sketch asymptotes as dashed lines; plot any points on the curve desired (e.g. intercepts); finish by connecting the points in accord with the information above.

4. Summary

   We are fortunate to have the calculators we have to help us plot functions. As noted in the book, however, calculators are dumb devices which merely plot many points and then connect them with line segments, leading to a risk of obscuring us rather than enlightening us. We've just encountered some tools which enable us to appreciate much of the qualitative behavior of a function, such as the first and second derivative tests, the concavity tests, the increasing/decreasing test, etc., and we can use these to gain the valuable intuition about a function (even if it is much harder for us to plot 1000 points!). The good news is that we're still good for something!

   In addition, we see in this section that there are other types of asymptotic behavior: in particular, the notion of a slant asymptote is introduced (which is a non-horizontal line which the graph of a function approaches as , or ).

Problems to consider: p. 270, #2, 15, 41, 48, 50