- I'm sorry: the exams aren't graded yet. Certainly by Monday.
-
Thu 3/25 Section 4.4 & 4.5 Read 4.6 -- optimization is what it's all about! Begin on-line homework for 4.5 (due Thu. 4/1)
Problems to hand in -- see the on-line homework.
- Here's a summary of the concavity stuff in section 4.4:
Four major cases:
- Asymptotic behavior ("horizontal asymptote" as x approaches plus or minus infinity)
Behavior is similar for broader classes of functions (e.g. Example 6, p. 212)
- Vertical Asymptotes (when the function values approach plus or minus infinity)
- We now have everything we need to create a great table to summarize
your study of a function, in preparation for graphing it. It's different from
the table in the book, however (although all of the information is essentially
the same). The following are the elements that we want to include:
- Domain of the function
- function values at endpoints of intervals
- function zeros
- critical points
- monotonicity (increasing/decreasing)
- concavity
- extrema
- inflection points
- asymptotes (vertical and horizontal)
This is similar to the author's steps 0-3, p. 213. My steps are:
- Determine the domain of f.
- Determine symmetry of f (if any).
- Determine the critical values (zeros of f', and places where it DNE)
- Determine the zeros of f'' (looking for inflection)
- Draw a real number line, and indicate the "interesting" parts of the domain on the line.
- Below the line are three lines for f, f', and f''. On each line mark places where the corresponding function is either zero or undefined (with double vertical bars).
- Indicate the sign of each function between zeros and bars.
- Use the four "canonical" function shapes to indicate the shape of the function for each sign combination of f' and f''.
- Describe asymptotic behavior.
- Graph the curve!
Let's see how this works on an example or two.
- Examples:
- #61, p. 218
- #40, p. 217
- #79, p. 218
