1004 Continuity 2 3 AP Calculus Activity TeacherDirected

Activity: Teacher-Directed Instruction C CONVERSATION: Voice level 0. No talking! H HELP: Raise your

Content: SWBAT calculate limits of any functions and apply properties of continuity Language: SW

General Idea: ____________________ Can you draw without picking up your pencil We already know

Continuity on a CLOSED INTERVAL. Theorem: A function is Continuous on a closed interval

Discontinuity No value f(a) DNE a) b) Vertical asymptote hole jump c) Limit does

Discontinuity: cont. Method: Look for f(a) = (a). Test the value = Lim DNE

Examples: Identify the x-values (if any) at which f(x)is not continuous. Identify the reason

Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the

Graph: Determine the continuity at each point. Give the reason and the type of

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

Algebraic Method At x=1 a. Value: f(1) = -1 At x=3 a. Value: b.

Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM f(b) If f(c) is between f(a) and

Consequences: cont. I. V. T - Zero Locator Corollary Intermediate Value Theorem EX: Show

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

Consequences of Continuity: B. EXTREME VALUE THEOREM On every closed interval there exists an

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1004 Continuity (2. 3) AP Calculus

Activity: Teacher-Directed Instruction C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M P S MOVEMENT: Remain in seat during instruction. 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.

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

General Idea: ____________________ Can you draw without picking up your pencil 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) Has a point b) Has a limit c) Limit = value f(a) exists

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 a) b) Vertical asymptote hole jump c) Limit does not equal value Limit ≠ value

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

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: = x≠ 4 Hole discontinuous because f(x) has no value It is 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? x≠ 3 VA discontinuous because no value It is 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? 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 Hole discont. No value removable x = -2 VA discont. Because no value no limit essential x = 0 Hiccup discont. Because limit ≠ value removable x =1 x = 2 x = 3 Continuous limit = value VA discont. No limit essential Jump discont. Because limit DNE essential

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

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

Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM f(b) If f(c) is between f(a) and f(c) f(b) there exists a c f(a) between a and b ** Existence Theorem EX: Verify the I. V. T. for f(c) Then find c. a c b on 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. I. V. T - Zero Locator Corollary Intermediate Value Theorem EX: Show that the function has a ZERO on the interval [0, 1]. 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 CALCULUS AND THE CALCULATOR: The calculator looks for a SIGN CHANGE between Left Bound and Right Bound

Consequences: cont. 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