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3.2 Continuity, IVT and End Behavior

3.2 Continuity, IVT and End Behavior. Study for the Quiz (3.1 – 3.2, Word Problem) Pg. 171 #27, 28 #25 inc : (1.5, ∞); dec : (-∞, 1.5) #26 inc : (-∞, -1.79)U(1.12, ∞), dec : (-1.79, 1.12)

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3.2 Continuity, IVT and End Behavior

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  1. 3.2 Continuity, IVT and End Behavior • Study for the Quiz (3.1 – 3.2, Word Problem)Pg. 171 #27, 28 • #25 inc: (1.5, ∞); dec: (-∞, 1.5) #26 inc: (-∞, -1.79)U(1.12, ∞), dec: (-1.79, 1.12) • #27 inc: (-∞, 0.67)U(2, ∞); dec: (0.67, 2) #28 inc: (-∞, -4.10)U(-0.57, ∞)dec: (-4.10, -0.57) • #29 inc: (1.38, ∞); dec: (-∞, 1.38) #30 inc: (-∞, -1.33)U(0.47, 1.61)dec: (-1.33, 0.47)U(1.61, ∞) • #31 inc: (-∞, 0.33)U(1, ∞); dec: (0.33, 1) #32 inc: (-∞, -0.07)U(0, ∞), dec: (-0.07,0) • #1 Continuous #2 Discontinuous @ 3; cont [-5, -4],[0, ∞); discon [-5, 5],[-4, -2] • #3 Continuous #4 Discontinuous @ 1; cont (1, 2), (-2, 1); discon (0, 2), (-5, 4) • #23 ↙ ↗ #25 ↖ ↘ #27 ↙ ↗ • #29 ↖ ↗ (3x4) #31 ↙ ↗ (4x3) #33 [-30, 30]by[10000,10000] • #35 [0, 1019.62] #36 (0, 676.52) #37 (676.52, 1615.81)

  2. 3.1 Graphs of Polynomial Functions Information from a Function Find all the information for: • End behavior type • End behavior model • Determine domain and range • Determine all zeros • Determine y – intercept • Determine all local min/max values • Determine intervals of increasing/decreasing • Draw a complete graph

  3. 3.2 Continuity, IVT and End Behavior Points of Discontinuity Examples • A function f has a point of discontinuity at x = a if on of the following conditions hold: • The function is not defined at x = a • The graph has a break at x = a • The graph has a hole at x = a

  4. 3.2 Continuity, IVT and End Behavior The cool thing about polynomials… they are continuous EVERYWHERE!!!

  5. 3.2 Continuity, IVT and End Behavior Intermediate Value Theorem (Property) Example For f(x) = 9 – x2on the interval [0, 3], will there be a c such that f(c) = 8? For f(x) = 2x3 – 11x + 12x + 5 on the interval [-1, 3], will there be a c such that f(c) = 6? • If a function is continuous on [a, b], then f assumes every value between f(a) and f(b). • If f(a) < L < f(b), then there is some number c in [a, b] such that f(c) = L.

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