IncreasingDecreasing

1
Objectives Section 5.1
Upon the completion of this lesson, students will
be able to

• use applications of the first derivative

2
Increasing and Decreasing Functions
A function f is increasing on (a, b) if f (x1) f (x2) whenever x1 A function f is decreasing on (a, b) if f (x1)
f (x2) whenever x1 Increasing
Increasing
Decreasing
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Increasing/Decreasing/ConstantFunctions
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Increasing/DecreasingFunctions
Steps
1. Find all values of x for which
is discontinuous and identify open intervals with
these points.
2. Test a point c in each interval to check the
sign of
f is increasing on that interval.
a. If
f is decreasing on that interval.
b. If
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Example
Determine the intervals where
is increasing and where it is decreasing.
-
0 4
f is decreasing on
f is increasing on
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Objectives Section 5.2
Upon the completion of this lesson, students will
be able to

• use applications of the second derivative

7
Relative Extrema
A function f has a relative maximum at x c if
there exists an open interval (a, b) containing c
such that
for all x in (a, b).
A function f has a relative minimum at x c if
there exists an open interval (a, b) containing c
such that
for all x in (a, b).
Relative Maximums
Relative Minimums
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Critical Points of f
A critical point of a function f is a point in
the domain of f where
(horizontal tangent lines, vertical tangent lines
and sharp corners)
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Procedure for Finding Relative Extrema The
First Derivative Test
1. Determine the critical points of f.

• Determine the sign of the derivative of f to
  the left and right of the critical point.

left
right
f(c) is a relative maximum
f(c) is a relative minimum
No change
No relative extremum
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Example
Find all the relative extrema of
Relative max. f (0) 1
Relative min. f (4) -31
-
0 4
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Example
Find all the relative extrema of
or
Relative max.
Relative min.
- -
-1 0 1
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Objectives Section 5.3
Upon the completion of this lesson, students will
be able to

• use applications of the second derivative

13
Concavity
Let f be a differentiable function on (a, b).
1. f is concave upward on (a, b) if
is increasing on (a, b). That is,
for each value of x in (a, b).
2. f is concave downward on (a, b) if
is decreasing on (a, b). That is,
for each value of x in (a, b).
concave upward
concave downward
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Determining the Intervals of Concavity

• Determine the values for which the second
  derivative of f is zero or undefined. Identify
  the open intervals with these points.

2. Determine the sign of
in each interval from
step 1 by testing it at a point, c, on the
interval.
f is concave up on that interval.
f is concave down on that interval.
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Example
Determine where the function
is concave upward and concave downward.

2
f concave down on
f concave up on
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Inflection Point
A point on the graph of f at which the tangent
line exists and concavity changes is called an
inflection point.
To find inflection points, find any point, c, in
the domain where
is undefined.
changes sign from the left to the right of c,
If
Then (c,f (c)) is an inflection point of f.
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The Second Derivative Test
1. Compute
2. Find all critical points, c, at which
If Then
f has a relative maximum at c.
f has a relative minimum at c.
The test is inconclusive.
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Example
Classify the relative extrema of
using the second derivative test.
Critical points x 0, 1, 2
Relative max.
Relative mins.
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Objectives Section 5.4
Upon the completion of this lesson, students will
be able to

• find horizontal and vertical asymptotes
• use graphing techniques to sketch curves

20
Vertical Asymptote
The line x a is a vertical asymptote of the
graph of a function f if either
is infinite.
Horizontal Asymptote
The line y b is a horizontal asymptote of the
graph of a function f if
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Finding Vertical Asymptotes of Rational Functions
If
is a rational function, then x a is a vertical
asymptote if Q(a) 0 but P(a) ? 0.
Ex.
f has a vertical asymptote at x 5.
22
Finding Horizontal Asymptotes of Rational
Functions
Ex.
0
0
Divide by the highest power of x
0
f has a horizontal asymptote at
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Curve Sketching Guide
1. Determine the domain of f.
2. Find the intercepts of f if possible.
3. Look at end behavior of f.
4. Find all horizontal and vertical asymptotes.
5. Determine intervals where f is inc./dec.
6. Find the relative extrema of f.
7. Determine the concavity of f.
8. Find the inflection points of f.
9. Sketch f, use additional points as needed.
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Example
Sketch
1. Domain (-8, 8).
2. Intercept (0, 1)
3.

• No Asymptotes

5.
f inc. on (-8, 1) U (3, 8), dec. on (1, 3).
6. Relative max. (1, 5) relative min. (3,
1)
7.
f concave down (-8, 2) up on (2, 8).
8. Inflection point (2, 3)
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Sketch
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Example
Sketch
1. Domain x ? -3
2. Intercepts (0, -1) and (3/2, 0)
3.

• Horizontal y 2 Vertical x -3

5.
f is increasing on (-8,-3) U (-3, 8).
6. No relative extrema.
7.
f is concave down on (-3, 8) and concave
up on (-8, -3).
8. No inflection points
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Sketch
y 2
x -3