AP Calculus 1004 Continuity (2.3)

C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M MOVEMENT: Remain in seat during instruction. P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed. NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK. S Activity: Teacher-Directed Instruction

Content: SWBAT calculate limits of any functions and apply properties of continuity Language: SW complete the sentence “Local linearity means…”

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) Can you draw without picking up your pencil Has a point f(a) exists Has a limit Limit = value

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. From the right From the left

Discontinuity No value f(a) DNE hole Limit does not equal value Limit ≠ value Vertical asymptote a) c) b) jump

Discontinuity: cont. Method: (a). (b). (c). Removable or Essential Discontinuities Test the value =Look for f(a) = Test the limit Holes and hiccups are removable Jumps and Vertical Asymptotes are essential Test f(a) =f(a) = Lim DNE Jump = cont. ≠ hiccup

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? = x≠ 4 Hole discontinuous because f(x) has no value It is removable

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? x≠3 VA discontinuous because no value It is 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? Step 1: Value must look at 4 equation f(1) = 4 Step 2: Limit It is a jump discontinuity(essential) because limit does not exist

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 Hole discont. No value removable VA discont. Because no value no limit essential Hiccup discont. Because limit ≠ value removable Continuous limit = value VA discont. No limit essential Jump discont. Because limit DNE essential

Algebraic Method a. b. c. Value:f(2) = 8 Look at function with equal Limit: Limit = value: 8=8

Algebraic Method At x=1 a. b. c. Value: f(1) = -1 Limit: Jump discontinuity because limit DNE essential At x=3 a. b. c. Value: Hole discontinuity because no value removable

Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM ** Existence Theorem EX: Verify the I.V.T. for f(c) Then find c. on If f(c) is between f(a) and f(b) there exists a c between a and b c ab f(a) f(c) f(b) f(1) =1 f(2) = 4 Since 3 is between 1 and 4. There exists a c between 1 and 2 such that f(c) =3 x 2 =3 x=±1.732

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 f(0) = -1 f(1) = 2 Since 0 is between -1 and 2 there exists a c between 0 and 1 such that f(c) = c Intermediate Value Theorem

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