today in pre-calculus go over homework notes: finding extrema –you’ll need a graphing calculator...
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Today in Pre-Calculus
• Go over homework• Notes: Finding Extrema
– You’ll need a graphing calculator (id’s please)
• Homework
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Extrema• Definition: The peaks and valleys where a
graph changes from increasing to decreasing or vice versa.
• Types: Minima and Maxima
Local (relative) and absolute
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Local (or relative) extrema• A local maximum for a function f, is a value f(c) that is greater than or equal to the range values of f on some open interval containing c.
• A local minimum for a function f, is a value f(c) that is less than or equal to the range values of f on some open interval containing c.
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Absolute extrema• An absolute maximum for a function f, is a
value f(c) that is greater than or equal to ALL of the range values of f.
• An absolute minimum for a function f, is a value f(c) that is less than or equal to ALL of the range values of f.
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Example
Relative minimum of
-10.75 at x = -2.56
Relative max of
38.6 at x = -0.40
Absolute min of
-42.93 at x = 2.21
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Example 1
Absolute minimum of -1.688 at x = -1.500
x
yy = x^4+2x^3
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Example 2Local maximum of 9.481 at x = -1.667
Local minimum of 0 at x = 1
x
y
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Example 3Absolute minimum of
-11.2 at x = -1.714
Local maximum of
0.459 at x = 0.312
Local minimum of
-1.758 at x = 1.402
x
y
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Example 4
These are absolute because for the min, there are no values in the range less than -1 and for the max, there are no values in the range greater than 1.
3, 1
2
x
y
3absolute min of -1 at ,
2 2x
-
3absolute max of 1 at ,
2 2x
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Example 5Absolute minimum of -4 at x = 2
Relative minimum of -1 at x = -3
Relative maximum of 3 at x = 1
x
y
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Homework
• Wkst.
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incr: (- ∞, ∞) decr: (- ∞, 0 )incr: (0, ∞)
decr: (- ∞, 0 )incr: (0, ∞)
decr: (- 1, 1)incr: (- ∞, -1 ), ( 1, ∞)
decr: ( 3, 5 )incr: (-∞, 3 )
constant: ( 5, ∞)
decr: ( 3, ∞)incr: (-∞, 0 )
constant: (0, 3)
decr: (- ∞, ∞)
decr: (- ∞, -4)incr: ( 4, ∞)
Inc(0,3)decr: (- ∞, 0)cons: (3, ∞)
incr: (- ∞, 0)decr: (0, ∞)
decr: (2,∞)incr: (-∞,-2)
constant(-2,2)
decr: ( - ∞, 7)υ (7, ∞)