Solutions to Math 1411- Test 3
1.
Find the derivative of the following functions:
a) 
b) 
c)
d)
e)
f)
g)
h)
i)
2.
At first glance it looks like you can answer
this question using L’Hoptial’s Rule. Alas, it is not so. Because
when you take the derivatives of the numerator and denominator you get 
and this not meet
the requirements of L’Hopital’s Rule. So you need to try something
else. You could graph the function 
and note that as x gets
closer and closer to zero, the function is heading toward infinity. This
means there is no limit. You could also evaluate the function at values
very closer to zero and note the functions are getting very large heading toward
infinity. This means there is no limit.
3.
.
a) Find the local minima and maxima.
Critical points are located at 
This means that the
graph is concave down at 
and there is a local maxima at of 
.
This means that the graph
is concave up at 
and there is a local minima at of 
.
b) Find inflection points.
equals zero when 
. 
means the graph
is concave down to the left of . 
means the graph is concave
up to the right of . Since the concavity changes at , it is the location of an the
inflection point 
.
c) Identify the intervals where the function is increasing. Decreasing.
The function is increasing on the
intervals 
and 
. The function is decreasing on the interval 
.
d) Identify the intervals where the function is concave up. Concave down.
The function is concave up on the
interval 
and concave down on the interval 
4.
Find the equation of the tangent line that approximates the function 
near 0.
