Today in Pre-Calculus Go over homework Notes: Finding Extrema –You’ll need a graphing calculator (id’s please) Homework.

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Presentation transcript:

Today in Pre-Calculus Go over homework Notes: Finding Extrema –You’ll need a graphing calculator (id’s please) Homework

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

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.

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.

Example Relative minimum of at x = Relative max of 38.6 at x = Absolute min of at x = 2.21

Example 1 Absolute minimum of at x =

Example 2 Local maximum of at x = Local minimum of 0 at x = 1

Example 3 Absolute minimum of at x = Local maximum of at x = Local minimum of at x = 1.402

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.

Example 5 Absolute minimum of -4 at x = 2 Relative minimum of -1 at x = -3 Relative maximum of 3 at x = 1

Homework Wkst.

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, ∞)