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# lecture9 - f t =-16 t 2 1600 ±ind the time at which the...

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Lecture 9 1 Lecture 9: Sec. 2.5 Continuity: Applications First, recall that f ( x ) is continuous at x = a if: 1) 2) and 3) Discontinuities at x = a : 1) removable 2) nonremovable

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Lecture 9 2 ex. Let f ( x ) = x 2 - 5 x + 6 x 2 - x - 2 . How can we find the discontinuities of f ? 1) Find lim x 2 + f ( x )
Lecture 9 3 2) Find lim x →- 1 - f ( x ) 3) Find and describe all discontinuities of f .

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Lecture 9 4 4) Can you define f (2) and f ( - 1) to make f ( x ) con- tinuous at x = 2 and x = - 1?
Lecture 9 5 Absolute Value ex. Let f ( x ) = 3 x - 6 | x - 2 | Find: 1. f (0) 2. f (2) 3. f (4)

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Lecture 9 6 4) Find an expression for f ( x ). 5) Evaluate: lim x 2 f ( x )
Lecture 9 7 6) Sketch the graph of y = f ( x ). Find and describe each discontinuity of f ( x ).

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Lecture 9 8 ex. f ( x ) = x 2 if x< - 1 | x | if - 1 <x 1 3 - x if x> 1 1) Find and describe all discontinuities for f ( x ).
Lecture 9 9 2) Sketch the graph of f ( x ). 3) Can you define f ( - 1) to make f continuous at x = - 1? What about f (1)?

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Lecture 9 10 Intermediate Value Theorem Consider the functional model giving the height h of a ball above the ground t seconds after it is dropped off a cliff 1600 feet high:

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Unformatted text preview: f ( t ) =-16 t 2 + 1600 ±ind the time at which the ball hits the ground. Graph h = f ( t ): Lecture 9 11 Note: f ( t ) is continuous over the interval and height h ranges from Now consider h = 1200. Find t so that f ( t ) = 1200. Lecture 9 12 The Intermediate Value Theorem If f is a continuous function on a closed interval [ a, b ], and if M is a number between f ( a ) and f ( b ), then Lecture 9 13 There is an important consequence of the Intermedi-ate Value Theorem: Theorem If f is continuous on [ a, b ], and if f ( a ) and f ( b ) have opposite signs, there is at least one solution of the equation f ( x ) = 0 in the interval ( a, b ). Lecture 9 14 ex. Show that the function f ( x ) = x 2-5 x + 2 must have a zero in the interval (-1 , 3). What about f ( x ) = x 2-5 x + 2 x 2 ?...
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