Section 4.2
There are two kinds of
critical points:
1) a point where the derivative is zero, called a stationary
point, and
2) a point where the derivative is undefined, called a point of
nondifferentiability.
If a function has a relative extreme value at x0, then x0 must be a critical point.
However, if f has a critical point at x0, then x0 is not necessarily a relative extreme value.
As an example, remember the graph of y = x3.x = 0 is a critical point (since the derivative is zero there), but the graph has neither a relative minimum nor a relative maximum at x = 0.
Suppose you want to find the relative extrema of a
function. First you find the critical points. Then, to tell whether the critical
points are relative minima, relative maxima, or neither, you can use the First
Derivative Test or the Second Derivative Test. There are two important differences
between these tests to keep in mind.
1)When
using the First Derivative Test, you check the sign of 
on
each side of your critical point. When using the Second Derivative Test, you actually
plug the critical point into 
.
2)The
First Derivative Test can be used on both types of critical points. The Second
Derivative Test can only be used on stationary points.
Homework
hints:
#1-2) There are infinitely many possible answers. You only have to find
one.
#13) Since we haven’t learned how to differentiate the absolute value
function, graph this function and use the “looking at the picture” method to determine
the critical points. Remember, sharp changes of direction (corner points) are
points of nondifferentiability.
#49-50) Same hint as 13.
#63) Same hint
as #13.
I am very finicky about the way graphs are drawn.
1)
Use graph paper and a pencil.
2)
All lines (including the axes) must be drawn using a straightedge.
3)
All graphs must be at least a half page in size. In other words, at most
two graphs on one side of a sheet of graph paper.
4)
After drawing your axes, label the horizontal one x and the vertical
one y.
5)
Go out several squares on the x-axis, and label it one. Continue out the
same number of squares and label it two, etc.
6)
You must use the same number of squares for zero to one on the x-axis as
you use for zero to one on the y-axis. (Exception: If you are forced to graph
points like (2, 31) or (-3, 55), you can use a different scale on the y-axis
to keep from running off your graph paper.)
7)
If a graph continues infinitely in a certain direction, put an arrow on
the end of the graph to indicate this.
To graph a polynomial function:
1)
Use the first derivative to determine the intervals where f is increasing
and decreasing. From the first derivative, you can also find the critical points,
relative minima, and relative maxima.
2)
Use the second derivative to determine the intervals where f is concave
up and concave down. From this you can find the inflection points. (Any change
in the sign of 
will be an inflection point since polynomial functions can’t have discontinuities.)
3)
Plot all relative extrema and inflection points and then use the information
in the table below to connect these points with a smooth curve.
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| Decreasing Concave down | Decreasing Concave up | Increasing Concave down | Increasing Concave up |
4)
Plot one additional point on each end of the graph to determine how fast
the graph shoots up or down.
5)
Look at your picture and see if it agrees with the information you obtained
in steps 1 and 2.
Here are some additional homework problems.
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