* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download View PDF
Survey
Document related concepts
Mathematics of radio engineering wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
History of the function concept wikipedia , lookup
History of trigonometry wikipedia , lookup
Vincent's theorem wikipedia , lookup
Series (mathematics) wikipedia , lookup
Dirac delta function wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Transcript
Calculus 1.4 Continuity and One-Sided Limits (Part b) Yesterday we looked at continuity on an open interval so today it is only natural that we look at continuity on a closed interval. To do this we first have to talk about a one-sided limit. One sided limits happen when we look at what is happening to a function, f, on only the right or left side if a member of it’s domain, c. Limit from the left: Limit from the right: lim f ( x) (we look at x’s close to c but less than c) x → c− lim f ( x) (we look at x’s close to c but greater than c) x → c+ Ex: Find the limit of the following lim 2[x] − 1 lim 2[x ] − 1 x → 5+ lim x → 5− x2 − 3 x → 5+ lim x2 − 3 x → 5− THEOREM 1.10 The Existence of a Limit Let f be a function and le t c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim f ( x) x→c − = L = lim f ( x) x → c+ A function is continuous on a closed interval if it is continuous in the interior and possesses one-sided continuity at the endpoints. Ex: Identify an interval for which the following functions are continuous. (a) f ( x) = − x (b) f ( x) = 4 − x (c ) f ( x) = 9 − x 2 THEOREM 1.11 Properties of Continuity If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c. 1. Scalar multiple: bf 2. Sum and difference: f ± g 3. Product: fg 4. Quotient: f , if g g (c) ≠ 0 $$ The following types of functions are continuous at every point in their domains: $$ 1. Polynomial Functions: an x n + an −1 x n −1 + an −2 x n −2 + ... + a2 x 2 + a1 x + a 0 2. Rational Functions: r ( x) = p ( x) , q( x) q (c) ≠ 0 3. Radical Functions: f ( x) = n x 4. Trigonometric Functions: sinx, cosx, tanx, cotx, secx, cscx Ex: Describe the interval(s) on which each function is continuous. x−2 (b) f ( x) = ( x + 1) 2 x − x (a) f ( x) = cot x (c) f ( x) = 4 x −1 THEOREM 1.13 Intermediate Value Theorem: If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b], such that f(c) = k. Ex: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f ( x) = x 2 − 6 x + 8, [0, 3], f (c ) = 0 ES - p. 76: 1 - 10, 15 - 24, 38, 40, 42, 57, 59, 63, 67, 83, 84, 89, 90