Also, know how to check if a function is concave up or concave down on a given
interval (i.e. by determining if 2
nd
derivative is positive or negative there), and how to find points of
inflection (these are points where concavity changes).
In addition, know how to use the Second
Derivative Test to check if a point is a local max/min.
Finally, know how to sketch the graph of a
function by finding the following:
domain,
y
-intercept (set
0
x
), regions of increasing/decreasing,
local max/min, regions of concave up/concave down, points of inflection, vertical asymptotes, and
horizontal asymptotes.
It may be helpful to also check if the function is symmetric and/or periodic.
4.7:
Applied Optimization Problems
Don’t panic…once you set these problems up, then all you have to do is find the absolute max or min of
a function (which you know how to do from section 4.3 anyways).
To set the problems up, read the
problem carefully to determine what information is given, and what is required.
Ask yourself:
what are
the variables in the question?
What quantity am I trying to minimize or maximize?
How are the
variables related?
If applicable, draw a diagram to help you. Next, introduce variables and set up
equations based on the given information.
If you set up an equation that you want to optimize and it has
more than 1 variable in it, go back and check how the variables are related in the question…there will be
some relationship that is given between the variables, and you can use this to solve for one of the
variables…now you can write the equation to optimize in terms of 1 variable only!
Once you’ve done
this, the problem has been set up, and it’s just an absolute max/min problem…so find the critical points.
If you’re dealing with a closed interval, compare the values of your function at the critical points and the
endpoints.
If you have an open interval, either use the first derivative test for extreme values, or apply a
variation of the closed interval method by taking limits.
Get lots of practice on these types of questions,
particularly in setting them up.
4.8:
L’Hopital’s Rule
Know what L’Hospital’s rule is, and how to apply it.
You should know that it applies only when a limit
is of the form “
0
0
” or “
”.
If it applies, you basically take the derivative of the numerator and the
denominator (independently), and the point is that the limit remains the same (and should now be easier
to evaluate).
Make sure that the rule applies before actually applying it, and realize that it is possible to
have to apply it more than once.
Technically, there are a few more types of limits that can be converted
into a form where L’Hopital’s rule applies, but we don’t cover that.
Finally, as a word of caution,
watch your notation…infinity is not a number, so do NOT substitute it in (similarly, do NOT divide by
zero).