Review some applications of derivatives:
Concavity
Concave up/down, depending on whether the derivative funciton is increasing or decreasing.
Note: concavity does not depend on the increasing or decreasing of the funciton itself.
At a point x the convacity of 
changes 
either 
vanishes at x, or 
is undefined at 
.
Note: 
! That is, the above condition (to right of the “
“ sign) is only necessary, not sufficient.
Second derivative test
(a complement to the first derivative test; both of thest tests help to determine whether 
takes an extremum at a point 
)
Recall the example for which the first derivative test fails: 
.
This function also fails the second derivative test >_<
Limits at infinity
We have studied limits of the form
#card=math&code=%5Clim%7Bx%20%5Cto%20a%7D%20f%28x%29&id=PesSN) #card=math&code=%5Clim%7B%20x%5Cto%20a-%7D%20f%28x%29&id=ww7qX) and #card=math&code=%5Clim_%7Bx%20%5Cto%20a%2B%7D%20f%28x%29&id=rAN7W)
where 
What if 
? What are these limits 
#card=math&code=%5Clim%7Bx%20%5Cto%20%2B%5Cinfty%7D%20f%28x%29&id=ASvlW) and #card=math&code=%5Clim_%7Bx%20%5Cto%20-%20%5Cinfty%7D%20f%28x%29&id=gjsyq) ?
These two are both one-sided limits.
means 
. 
this means that 
becomes arbitaryly large.
, this means that 
and the absolute value 
of 
becomes sufficiently large.
Horizontal asymptotes
, where 
#card=math&code=L%20%3D%20%5Clim_%7B%20x%5Cto%20%5Cpm%20%5Cinfty%7D%20f%28x%29&id=PLxdm) .
. 
. So 
is a horizontal asymptote for 
.
Vertical asymptotes
, where $ \lim_{x\to c\pm} f(x) = \pm \infty$
Note: there is still a third possbility of asymptote: oblique asymptote.
%7C%20%5Cleq%201#card=math&code=%7C%5Ccos%20%28x%29%7C%20%5Cleq%201&id=RhKZU). So when 
, 
has an infinite discontinuity at 
.
The squeezing theorem tells us that 
%2Fx%20%5Cto%200#card=math&code=%5Ccos%28x%29%2Fx%20%5Cto%200&id=Lqhnw) as 
. Because
%2Fx%7C%20%5Cleq%201%2F%7Cx%7C#card=math&code=0%5Cleq%20%7C%5Ccos%28x%29%2Fx%7C%20%5Cleq%201%2F%7Cx%7C&id=DsSKW).
L’Hôpital’s rule
L’Hospital
Case 1: Indeterminate form 
-f(a)%7D%7Bx-a%7D#card=math&code=%5Cfrac%7Bf%28x%29-f%28a%29%7D%7Bx-a%7D&id=HEdo3) is an indeterminate form as 
.
#card=math&code=%5Clim%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Be%5E%7B1%2Fx%7D-1%7D%7B1%2Fx%7D%20%3D%20%5Clim%7B%20x%20%5Cto%20%5Cinfty%7D%20x%2A%28e%5E%7B1%2Fx%7D-1%29&id=ADskM) (
) And 
.
Remark: the L’Hôpital’s rule in the cases 0/0 and 
, is based on the assumption that the limit 
%7D%7Bg’(x)%7D#card=math&code=%5Clim%7Bx%20%5Cto%20a%7D%5Cfrac%7Bf%27%28x%29%7D%7Bg%27%28x%29%7D&id=ZmCP5) exists. If this limit does not exist, neither do the original limit %7D%7Bg(x)%7D#card=math&code=%5Clim_%7Bx%20%5Cto%20a%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&id=XuRQL).
Other indeterminant forms: 
. (left for next week)
