AP Calculus 1005 Continuity (2.3). General Idea: General Idea: ________________________________________ We already know the continuity of many functions:

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Presentation transcript:

AP Calculus 1005 Continuity (2.3)

General Idea: General Idea: ________________________________________ We already know the continuity of many functions: Polynomial (Power), Rational, Radical, Exponential, Trigonometric, and Logarithmic functions DEFN: A function is continuous on an interval if it is continuous at each point in the interval. DEFN: A function is continuous at a point IFF a) b) c)

Continuity Theorems

Continuity on a CLOSED INTERVAL. Theorem: A function is Continuous on a closed interval if it is continuous at every point in the open interval and continuous from one side at the end points. Example : The graph over the closed interval [-2,4] is given.

Discontinuity Continuity may be disrupted by: (a). c (b). c (c). c

Discontinuity: cont. Method: (a). (b). (c). Removable or Essential Discontinuities

Examples: EX: Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential?

Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential?

Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential?

Graph: Determine the continuity at each point. Give the reason and the type of discontinuity. x = -3 x = -2 x = 0 x =1 x = 2 x = 3

Algebraic Method a. b. c.

Algebraic Method At x=1 a. b. c. At x=3 a. b. c.

Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM ** Existence Theorem EX: Verify the I.V.T. for f(c) Then find c. on

Consequences: cont. EX: Show that the function has a ZERO on the interval [0,1]. I.V.T - Zero Locator Corollary CALCULUS AND THE CALCULATOR: The calculator looks for a SIGN CHANGE between Left Bound and Right Bound

Consequences: cont. EX: I.V.T - Sign on an Interval - Corollary (Number Line Analysis) EX:

Consequences of Continuity: B. EXTREME VALUE THEOREM On every closed interval there exists an absolute maximum value and minimum value.

Updates: 8/22/10