Download Chapter Four Review Some Practice Problems

Chapter Three: Section Four
Concavity
and the Second Derivative Test
Chapter Three: Section Four
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In the last section we talked about using the first
derivative to make distinctions about when a
function is increasing or decreasing. We are working
on our ability to make accurate sketches of complex
functions.
We know that graphs can look different even when
they are both increasing or both decreasing.
Look at the graph on the next slide carefully and try
to describe what looks different about the regions
where the function is increasing.
Chapter Three: Section Four
y  x 4  12 x3  48x 2  64 x
Chapter Three: Section Four
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I want you to complete a sign chart for this function
the way that we have been for the first derivative
test.
After that, repeat the process for the second
derivative and see what happens to the second
derivative over the regions where the function is
increasing.
Remember, the equation we are examining is
y  x 4  12 x3  48x 2  64 x
Chapter Three: Section Four
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Compare the results of the two sign charts and look
again at the graph below;
Chapter Three: Section Four
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The word we will use to describe the different
behaviors of the increasing regions of the function is
the word concavity.
When the second derivative of a function is positive,
we say that the function is concave up. What this
means physically is that the movement of the graph
has positive acceleration.
When the second derivative is negative then the
function is said to be concave down and this means
that the acceleration of the graph is negative.
Chapter Three: Section Four
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To summarize the past two sections;
If both f ‘ and f “ are
positive, then both
velocity and acceleration
are positive. The function
is rising and concave up.
If f ‘ is negative and f “ is
positive, then velocity is
negative and acceleration
is positive. The function
is falling and concave up.
If f ‘ is positive and f “ is
negative, then both
velocity is positive and
acceleration is negative.
The function is rising and
concave down.
If both f ‘ and f “ are
negative, then both
velocity and acceleration
are negative. The
function is falling and
concave down.
Chapter Three: Section Four
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Can you sketch functions (just a sketch, not an
equation) that matches up with these four pieces?
Be prepared to share your ideas in class.