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AP Calculus

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.

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AP Calculus

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  1. AP Calculus 1005 Continuity (2.3)

  2. 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)

  3. Continuity Theorems

  4. 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.

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

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

  7. Examples: 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? EX: removable or essential?

  8. 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?

  9. 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? Examples: cont.

  10. 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

  11. Algebraic Method a. b. c.

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

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

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

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

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

  17. Updates: 8/22/10

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