3.1 First Derivative Test
Derivatives can be used to determine if a function is increasing, decreasing, or constant on an interval. f(x) is increasing on an interval of:
1)
2) f(x) is decreasing on an interval if:
1)
2) f(x) is constant on an interval if:
Definition of Critical Numbers:
1)
2)
Let f be defined at c . If
Guidelines for finding intervals on which a function is increasing or decreasing:
(1)
(2)
(3)
Relative Extrema:
Relative extrema are minimum and maximum values of a function. Relative extrema occur at critical numbers. However, a critical number does not insure a relative extremum.
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Example: Find the open intervals on which f ( x )
 x
3 
3
2 x
2
is increasing or decreasing.
Graph to check results.

Definition: A function is ___________________ _______________________ on an interval if it is either increasing on the entire interval or decreasing on the entire interval.
After you have determined the intervals on which a function is increasing or decreasing, it is not difficult to locate the relative extrema.
The First Derivative Test
Let c be a critical number.
(1)
(2)
Example: Find the relative extrema of f ( x )

( x
2 
4) 3
2
. Graph to check.

2

Example: Find the relative extrema of f ( x )
 x
4 x
2

1
.
Graph to check.

Finding Absolute Extrema on a Closed Interval
To find the extrema of a continuous function f on a closed interval [a,b], use the following steps:
1)
2)
3)
4)
Example: Find the absolute extrema of f ( x )

3 x
4 
4 x
3
on the interval [-1,2].

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3.4 Concavity/Points of Inflection/The Second Derivative Test
Definition of Concavity
Let f be differentiable on an open interval. The graph of f is concave upward if and concave downward if
The second derivative of a function f can be used to determine concavity.
1) If f”(x) > 0, then f is
Steps:
2) If f”(x) < 0,the f is
(1)
(2)
(3)
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Example: Determine the open intervals on which the graph of f ( x )

6( x
2 
3)

1
is concave upward or concave downward. Graph to check.

Example: Determine the open intervals in which the graph of f ( x )
 x x
2
2

1

4
is concave upward or downward. Graph to check.

Point of Inflection - points where concavity changes
To locate possible points of inflection, determine the values of x for which f”(x) = 0 or is undefined.
This is similar to the procedures for locating relative extrema, but you are using f”(x) rather than f’(x).
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
The Second Derivative Test – simple test to determine extrema
Let f’(x) = 0, so c is a critical number,
1)
2)
If f”(x) = 0, the Second Derivative Test fails. In this case, use the First
Derivative Test to determine extrema.
Example: Use the Second Derivative Test to find the relative extrema for f ( x )
 
3 x
5 
5 x
3
.
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3.6 Sketching Graphs
It would be difficult to overstate the importance of using graphs in mathematics.
Descartes’ introduction of analytic geometry contributed significantly to the rapid advances in Calculus that began during the mid-seventeenth century. In the words of
Lagrange, “As long as algebra and geometry traveled separate paths, their advance was slow and their applications limited. But when thee two sciences joined company, they drew from each other fresh vitality and thenceforth marched on at a rapid pace toward perfection.”
Up to this point, you have studied the following concepts that are helpful in analyzing and summarizing the graph of a function. x-intercepts and y-intercepts symmetry domain and range continuity vertical asymptotes horizontal asymptotes differentiability relative extrema increasing/decreasing intervals points of inflection concavity
When sketching a graph by hand or with a graphing calculator, you will not b e able to how the entire graph. Therefore, you must be careful to show enough of the graph to represent all of its important characteristics (listed above).
Example: Analyze the graph of f ( x )

2( x
2 x 2

9)

4
.
Domain: x-intercept(s): vertical asymptote(s); y-intercept: horizontal asymptote(s):
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 f’(x) =
Increasing/decreasing intervals:
Relative extrema:
Concavity:
Point(s) of Inflection: f”(x) =
There is one other type of asymptote, called a ________________ asymptote. This kind of asymptote occurs in a rational function (having no common factors) if the degree of the numerator exceeds the degree of the denominator by one.
Example: Analyze f ( x )
 x
2 
2 x

4 x

2
Domain
Range y-intercept
Vertical Asymptote(s)
Horizontal Asymptote(s)
Slant Asmptote:
To find the slant asymptote, use long division: x

2
 x
2 
2 x

4 Equation of Slant Asymptote:
Relative Extrema
Increasing/Decreasing Intervals
Point(s) of Inflection
Concavity
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Example: Analyze and graph
(turning point).
f ( x )

2 x 3
5 4

5 x 3

Example: Analyze and graph f ( x )
 x 4 
12 x 3 
48 x 2 
64 x
In general, a polynomial function of degree n can have at most n-1 relative extrema, and at most n-2 points of inflection . A polynomial
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