Math1011_Week05 - Math1011 Section 2.4 Sketch the graph of...

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Math1011 Section 2.4 9/20/2010 Sketch the graph of an example of a function ƒ that satisfies all of the given conditions. ƒ (0) does not exist 1 ( ) 2 lim x f x = − 2 ( ) 1 lim x f x →− = − 2 ( ) lim x f x DNE →− = 4 ( ) 0 lim x f x = ƒ (-2) = 1 ƒ (1) = 3 ƒ (4) = 4
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1 Math1011 Section 2.4 Sketch the graph of an example of a function ƒ that satisfies all of the given conditions. ƒ (0) does not exist 1 ( ) 2 lim x f x = − 2 ( ) 1 lim x f x →− = − 2 ( ) lim x f x DNE →− = 4 ( ) 0 lim x f x = ƒ (-2) = 1 ƒ (1) = 3 ƒ (4) = 4 Note: This is one of many (infinite) correct answers. Your graph may look quite different from the above graph, and still be absolutely correct. (Doc #1011w.24.03)
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Math1011 Section 2.5 2 Given the graph of ƒ : At what points on the interval [-5, 5] is ƒ discontinuous? Explain by indicating the type of discontinuity. At what points on its domain is ƒ discontinuous?
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Math1011 Section 2.5 2 Given the graph of ƒ : At what points on the interval [-5, 5] is ƒ discontinuous? Explain by indicating the type of discontinuity. -2 (infinite discontinuity) -1 (removable discontinuity) 1 (jump discontinuity) 3 (removable discontinuity) At what points on its domain is ƒ discontinuous? 1 (jump discontinuity) 3 (removable discontinuity) Note: x = -2 and x = -1 are not included as these points are not in the domain of ƒ. (Doc #1011.w.25.01)
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Math1011 Section 2.5 3 Sketch the graph of the function. Show, using the definition of continuity, that f(x) is discontinuous at x = 1. Identify the type of discontinuity ƒ (x) has at x = 1. 2 4 3 1 ( ) 1 3 1 x x if x f x x if x + = = How could we redefine ƒ(x) to make it continuous?
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Math1011 Section 2.5 3 Sketch the graph of the function. Show, using the definition of continuity, that f(x) is discontinuous at x = 1. Identify the type of discontinuity ƒ (x) has at x = 1. 2 4 3 1 ( ) 1 3 1 x x if x f x x if x + = = Definition of Continuity A function is continuous at a number a if lim ( ) ( ) x a f x f a = 2 1 4 3 1 1 ( 1)( 3) 1 1 1 lim ( ) ? (1) lim ? (1) lim ? (1) lim 3 ? (1) 2 3 So, f(x) is not continuous at 1 Removable Discontinuity at x=1 x x x x x x x x x x f x f f f x f + How could we redefine ƒ(x) to make it continuous? 2 4 3 1 ( ) 1 2 1 x x if x f x x if x + = = (Doc #1011.w.25.02)
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Math1011 Section 2.5 4 Prove that the equation s³ = 10 has at least one real root. Prove there is a positive number t such that t³+cos(t π ) = 7.
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Math1011 Section 2.5 4 Prove that the equation s³ = 10 has at least one real root. Let ƒ (s) = s³ - 10 and let us find c such that f(c) = 0. Show f(s) is continuous: f(s) is a polynomial and thus continuous. Pick two numbers a and b such that f (a) < 0 and f ( b) > 0. a = 0: f ( 0 ) = - 10 b = 3: f (3 ) = 27 – 10 = 17 Since ƒ (s) is continuous and ƒ (0) 0 ƒ (3), we know by the Intermediate Value Theorem that there exist some c between 0 and 3 such that ƒ (c) = 0.
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