10/13/2015 Perkins AP Calculus AB Day 5 Section 1.4.

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Presentation transcript:

10/13/2015 Perkins AP Calculus AB Day 5 Section 1.4

Continuity f(x) will be continuous at x = c unless one of the following occurs: a. f(c) does not exist b. does not exist c. c c c Removable DiscontinuityA graph with a “hole” in it Non-removable Discontinuity Any other type

Discuss the continuity of each. Not continuous at x = 0 (V.A.) Non-removable Continuous function Not continuous at x = 1 Hole in graph at (1,2) Removable

If x < 2, the function is a parabola. (continuous) If x > 2, the function is a line. (continuous) To be continuous, the two sides must also meet when x = 2. D.S.

Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k. The red graph has 1 c-value. Blue has 5 c-values. Orange has 1 c-value. Translation: If you connect two dots with a continuous function, you must hit every y-value between them at least once.

Perkins AP Calculus AB Day 5 Section 1.4

Continuity f(x) will be continuous at x = c unless one of the following occurs: a. f(c) does not exist b. does not exist c. Removable Discontinuity Non-removable Discontinuity

Discuss the continuity of each.

Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k. The red graph has 1 c-value. Blue has 5 c-values. Orange has 1 c-value. Translation: If you connect two dots with a continuous function, you must hit every y-value between them at least once.

Intermediate Value Theorem If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k.