Tuesday, October 25
−
Lecture 18 :
Derivatives and the shape of a graph.
(Refers to
Section 4.3 and 4.4 in your text)
Students who have mastered the content of this lecture know about
:
The First derivative test, concavity,
what defines concave upwards and concave downward, inflection points, the Second derivative test
Students who have practiced the techniques presented in this lecture will be able to
:
Use
t
he First
derivative test to sketch the curve of a function f(x) determining all local max and mins, absolute max and
mins, and inflection points.
Introduction – We have seen how the derivative along with the
Increasedecrease
theorem
,
can help us determine on which intervals a function
f
is increasing and
decreasing.

We will now go a step further and use this information to help us determine when a
critical number points to local maximum or local minimum of the function. This
information will help us sketch the graph of simple functions.

The first of these tools is called the
First derivative test
. It is an immediate
consequence of the
IncreaseDecrease theorem
.
18.1
Theorem
.
First derivative test
– Suppose
f
is continuous function on the interval [
a
,
b
]
and
c
∈
(
a
,
b
) is a critical number of
f
.
a)
If
f
′
changes from positive to negative at
c
then
f
has a local maximum at
c
.
b)
If
f
′
changes from negative to positive at
c
then
f
has a local minimum at
c
.
c)
If
f
′
does not change signs at
c
then
f
has neither a local maximum nor a local
minimum at
c
.
Proof is omitted.
18.1.2
Remember – Numbers in the domain at which the derivative is not defined are
also critical numbers of a function.
18.1.3
Example – Construct a table which can be used to determine the intervals on
which the function
f
(
x
) = 3
x
4
– 4
x
3
– 12
x
2
+5 increases and decreases.

In the previous lecture we constructed this chart from the information provided by
the derivative
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We will use this chart along with the
First derivative test
to determine at which
points the function has a local maximum or a local minimum.
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 Spring '11
 RobertAndre
 Calculus, Derivative, Mathematical analysis, Concave function

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