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4.1 Extreme Values for a function
Absolute Extreme Values(a) There is an absolute maximum value at x = c iff
f(c) f(x) for all x in the entire domain.(a) There is an absolute minimum value at x = c iff
f(c) f(x) for all x in the entire domain.
Relative Extreme Values(a) There is a relative maximum value at x = c iff
f(c) f(x) for all x in some open interval containing c.(a) There is a relative minimum value at x = c iff
f(c) f(x) for all x in some open interval containing c.
Maxima and Minima
•Where f (c) =0•Where f (c ) is undefined•At an endpoint of a closed interval
Locations of Extreme values
**Values in the domain of f where f (c) is zero or is undefined are called critical values of the function.***
If f(x) has a maximum or minimum value at x = c it must occur at one of the following locations:
Maxima and Minima on closed interval for continuous function must exist.
Maximum and minimums
A curve with a local maximum value. The slope at c, is zero.
Local Max
4.3 First Derivative test for Increasing and Decreasing functions
If a function is continuous on [a, b] and differentiable on (a,b)
(a) If f > 0 at each point of (a,b) then f increases on [a,b].
(a) If f < 0 at each point of (a,b) then f decreases on [a,b].
Find the critical points and identify intervals on which f is increasing and decreasing.
First derivative test for Local Extrema
At a critical point x = c
1. f has a local minimum if f changes from negative to positive at c.
2. f has a local maximum if f changes from positive to negative at c.
3. There is no extreme value if the sign of f does not change. Could be a horizontal tangent without direction change.
Figure 3.24: The graph of f (x) = x3 is concave down on (–, 0) and concave up on (0, ).Concavity
Second derivative test for Concavity
A graph is concave up on any interval whereThe second derivative is positive.
A graph is concave down on any interval whereThe second derivative is negative.
A point of inflection for a function Occurs where the concavity changes
Second derivative test for extreme values
If there is a critical value at x = c
and
( ) 0f c
( ) 0f c
( ) 0f c
A local max at x = c
conclusion
A local min at x = c
inconclusive
+ ++
- --
Section 1 / Figure 1
Section 4.3 Figures 11Graph of the curve
34 4xxy
© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
A
The graph of f (x) = x4 – 4x3 + 10.
Finding intervals of concave up and concave down and Inflection points
What derivatives tell us about the shape of a graph
Figure 1.42: The blue graph of f (x) = x + e–x looks like the graph of g(x) = x (black) to the right of the y-axis and like the graph of h(x) = e–x (red) to the left of the y-axis. (Example 1)
4. 4 Limits to Infinity (End behavior)
What happens to the value of the function when the value of x increases without bound?
What happens to the value of the function when the value of x decreases without bound?limx
limx
Basic limits to infinity
1lim 0x x
1lim 0 0, 0n
x nn
x
lim *0 0 0x n
kk n
x
3
5: lim 0xExample
x
1lim 0x x
lim *0 0 0x n
kk n
x
Figure 1.27: The function in Example 3.
2
2 2 2
2
2 2
5 8 35 5
lim lim3 33 2
x x
x x
x x xx
x x
Divide each term by highest power of x in the denominatorand calculate limits
2
2
5 8 3lim
3 2x
x x
x
As x gets larger and larger, the function gets closer and closer to 5/3.
Figure 1.27: The function in Example 3.
2
3 3 3
3
3 3
5 8 30
lim lim 033 2
x x
x x
x x xx
x x
Divide each term by highest power of x in the denominator and calculate limits
2
3
5 8 3lim
3 2x
x x
x
As x gets larger and larger, the function gets closer and closer to 0.
Figure 1.27: The function in Example 3.
4 2
22 2 2
3
2 2
5 8 35
lim lim33 2
x x
x xxx x x
x
x x
Divide each term by highest power of x in the denominatorand calculate limits
4 2
3
5 8 3lim
3 2x
x x
x
As x gets larger and larger, the function decreases without bound.
Figure 1.27: The function in Example 3.2
2
5 8 3lim
3 2x
x x
x
As x gets larger and larger, the function gets closer and closer to 5/3.
When the limit to infinity exists, at y = L we can say that the line y = L is a horizontal asymptote.
Figure 1.27: The function in Example 3.Horizontal Asymptote
Limits that are infinite (y increases without bound)
41
lim4x x
21
lim2x x
31
lim3x x
An infinite limit will exist as x approaches a finite value when direct substituion produces
0
not zero
If an infinite limit occurs at x = c we have a vertical asymptote with the equation x = c.
Figure 1.29: The function in Example 5(a).Slant Asymptote
As x gets larger and larger, the graph gets closer and closer to the line 2 8
7 49y x
As x gets smaller and smaller, the graphgets closer to the same line. You can use long division to rewrite the given function.
There is a vertical asymptote at x = - 4 / 7
Section 1 / Figure 1
Graphs of the polynomial 235 2332)( xxxxxf 6
© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
4.6Different scales
Section 1 / Figure 1
Derivatives of the polynomial 235 2332)( xxxxxf 6
© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
Graphs of derivatives
Section 1 / Figure 1
2 3
2 4
( 1)( )
( 2) ( 4)
x xf x
x x
Section 1 / Figure 1
Section 1 / Figure 1
Section 4.6 Figures 19, 20The family of functions
221
2
xxy
221
2
xxy
)2/(1)( 2 cxxxf
The family of functions )2/(1)( 2 cxxxf
1c 0c 1c
2c 3c