determine where a function is increasing or decreasing when determining if a graph is increasing or...
DESCRIPTION
I. Vocabulary Extrema(Plural of Extreme): This means we are talking about maxima (plural of maximum) and minima (plural of minimum) of a function. Interval: Means we are talking about a part of a function, denoted by interval notation (a, b) or [a, b] Open IntervalClosed IntervalTRANSCRIPT
Determine where a function is increasing or decreasingWhen determining if a graph is increasing or decreasing we
always start from left and use only the x values.
Included and excluded do not apply, we always use ( ).
Increase:
Decrease:
(-∞, -2.4)
(-2.4, 1.6)(1.6, ∞)
Objectives:1. Be able to define various vocabulary terms needed to be
successful in this unit.2. Be able to understand the definition of extrema of a function
on an interval and “The Extreme Value Theorem”.3. Be able to find the relative extrema and critical numbers of a
function.4. Be able to find extrema on a closed interval.
Critical Vocabulary:Extrema, The Extreme Value Theorem, Critical Numbers
I. VocabularyExtrema(Plural of
Extreme): This means we are talking about maxima (plural of maximum) and minima (plural of minimum) of a function.
Interval: Means we are talking about a part of a function, denoted by interval notation (a, b) or [a, b]
Open Interval Closed Interval
II. Extrema of a Function
Let f be defined on an interval I containing c 1. f(c) is the MINIMUM of f on I if f(c) ≤ f(x) for all x in I
2. f(c) is the MAXIMUM of f on I if f(c) ≥ f(x) for all x in I The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval.
Example 1: Function: f(x) = x2 + 3, Interval: [-2, 2], Let c = 0f(c): f(0) = 3
f(-2) = 7
f(-1) = 4
f(1) = 4 f(2) = 7f(c) is less than or equal to all values of f(x) on the interval
making f(c) the MINIMUM of f
II. Extrema of a Function
If f is continuous on a closed interval [a, b], then f has both a minimum and maximum on the interval.
To illustrate this, we will look at the graph of f(x) = x2 + 1 on the following intervals:
[-1, 2]
Min: (0, 1)Max: (2, 5)
(-1, 2)
Min: (0, 1)No Max
[-1, 2]
No MinMax: (2, 5)
Relative Minimum: The smallest point of the graph in a given area. Relative Maximum: The largest point of the graph in a given area.
Relative Max “Hill”
Relative Min
“Valley”
Relative Max “Hill”
Relative Min
“Valley”
Point where a graph changes its behavior (increasing/decreasing) help in determining the maximum and minimum values of a graph.
III. Relative Extrema and Critical Numbers
1. If there is an open interval containing c on which f(c) is a maximum, then f (c) is called a relative maximum of f.
III. Relative Extrema and Critical Numbers
2. If there is an open interval containing c on which f(c) is a minimum, then f (c) is called a relative minimum of f.The plural of relative maximum is relative maxima and
the plural of relative minimum is relative minima.
III. Relative Extrema and Critical NumbersExample 2: Find the value of the derivative at each of the relative extrema shown in the graph
3
2 39)(xxxf
32 279)( xxxf
42 819)(' xxxf
4
2 99)('xxxf
4
2
)3(9)3(9)3('
f
0)3(' f
When the derivative is zero, we call the x-value associated with it a CRITICAL NUMBER.
31 279)( xxxf
42
819)('xx
xf
4
2 819)('xxxf
III. Relative Extrema and Critical NumbersExample 3: Find any critical numbers algebraically: f(x) = x2(x2 - 4) 4)( 22 xxxf
xxxf 84)(' 3
1st: Find the derivative
2nd: Set f’(x) = 0 xx 840 3
240 2 xx04 x 022 x0x 22 x
2x
3rd: Check for any places where the derivative is undefined
24 4)( xxxf
IV. Finding Extrema on a Closed Interval
1. Find the critical numbers of f in (a, b)
To find the extrema of a continuous function f on a closed interval [a, b], use the following steps:
2. Evaluate f at each critical number in (a, b)3. Evaluate f at each end point in [a, b]4. The least of these values is the minimum
and the greatest is the maximum.
IV. Finding Extrema on a Closed IntervalExample 4: Locate the absolute extrema of the function on the
closed interval ]1,1[,3
)( 2
2
xxxf
1st: Find the critical numbers 122 3)(
xxxf2)( xxg xxg 2)('
12 3)(
xxh xxxh 23)(22
12312 3232)('
xxxxxf
2222 332)(' xxxxxf
22 36)('
xxxf 06 x
0x
)0,0(
2nd : Evaluate at the endpointsLeft End point:
41,1 Right End
point:
41,1
Minimum
Maximum
IV. Finding Extrema on a Closed IntervalExample 5: Locate the absolute extrema of the function on the
closed interval ]1,1[,)( 3/1 xxf
1st: Find the critical numbers 32
31)( xxf
3 231)(x
xf
Critical number: x = 0
)0,0(
2nd : Evaluate at the endpoints
Left End point:
1,1 Right End point:
1,1
Minimum
Maximum
Page 319-321 #7-27 odd, 33, 43, 45, 47
1. Find all the relative extrema of the function f(x) = x4 – 8x2
2. Find the absolute extrema of the function:
]5,3[,2923
32)( 23 xxxxf
3. Find the absolute extrema of the function:
]3,3[,1
)( 2
xxxf
Practice Assessment
1. Find all the relative extrema of the function f(x) = x4 – 8x2Practice Assessment
)0,0(:0x)16,2(:2 x
16,2:2 x Relative Minimum
Relative MaximumRelative Minimum
xxxf 164)(' 3
2. Find the absolute extrema of the function:
]5,3[,2923
32)( 23 xxxxf
Practice Assessment
932)(' 2 xxxf
241,3:3 x)879,23(:23 x
25,3:3 x
617,5:5x
Maximum
Minimum
3. Find the absolute extrema of the function:
]3,3[,1
)( 2
xxxf
Practice Assessment
22
2
11)('
xxxf
21,1:1x)21,1(:1 x
103,3:3 x
103,3:3x
MaximumMinimum