Sullivan Microsite DigiSample

More documents

Recommendations

Info

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Sullivan 152 Chapter 1 • Limits and Continuity 48. See TSM. 49. See TSM. 50. See TSM. 51. lim fx ( ) = 0; x→0 lim fx ( ) does not exist. x→1 See TSM for discussion and proof. 52. lim fx ( ) = 0. See TSM for discussion x→0 and proof. ⎧ ε ⎫ 53. Given any e > 0, let δ ≤ min⎨1, ⎬. ⎩ 47 See TSM for complete proof. ⎭ 54. See TSM. 55. M = 101. 1 56. Given any e > 0, let δ ≤ 1 + | a | . See TSM for complete proof. 57. See TSM. 58. K = 12. 48. Explain why in the ε-δ definition of a limit, the inequality 0 < |x − c| 0 if n is even x→c x→c (p. 95) [ ] m/n • lim[ f (x)] m/n = lim f (x) , provided [ f (x)] m/n is x→c x→c defined for positive integers m and n (p. 95) [ ] f (x) lim f (x) x→c • lim = , provided lim g(x) = 0 (p. 96) x→c g(x) lim g(x) x→c x→c • If P is a polynomial function, then lim P(x) = P(c). (p. 96) x→c • If R is a rational function and if c is in the domain of R, then lim R(x) = R(c). (p. 96) x→c 1.3 Continuity Definitions • Continuity at a number (p. 103) • Removable discontinuity (p. 105) • One-sided continuity at a number (p. 105) • Continuity on an interval (p. 106) • Continuity on a domain (p. 107) 152 Chapter 1 • Limits and Continuity TE_Sullivan_Chapter01_PART II.indd 35 11/01/17 9:56 am
Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Chapter 1 • Chapter Review 153 Properties of Continuity • A polynomial function is continuous on its domain, all real numbers. (p. 107) • A rational function is continuous on its domain. (p. 107) • If the functions f and g are continuous at a number c, and if k is a real number, then the functions f + g, f − g, f · g, and kf are also continuous at c. If g(c) = 0, the function f g is continuous at c. (p. 108) • If a function g is continuous at c and a function f is continuous at g(c), then the composite function ( f ◦ g)(x) = f (g(x)) is continuous at c. (p. 109) • If f is a one-to-one function that is continuous on its domain, then its inverse function f −1 is also continuous on its domain. (p. 110) The Intermediate Value Theorem Let f be a function that is continuous on a closed interval [a, b] with f (a) = f (b). If N is any number between f (a) and f (b), then there is at least one number c in the open interval (a, b) for which f (c) = N. (p. 110) 1.4 Limits and Continuity of Trigonometric, Exponential, and Logarithmic Functions Basic Limits sin θ • lim = 1 θ→0 θ (p. 119) cos θ − 1 • lim = 0 θ→0 θ (p. 121) • lim sin x = sin c x→c (p. 122) • lim cos x = cos c x→c (p. 122) • lim a x = a c ; x→c a > 0, a = 1 (p. 124) • lim log a x = log a c; x→c a > 0, a = 1, and c > 0 (p. 124) Squeeze Theorem If the functions f , g, and h have the property that for all x in an open interval containing c, except possibly at c, f (x) ≤ g(x) ≤ h(x), and if lim f (x) = lim h(x) = L, x→c x→c then lim g(x) = L. (p. 118) x→c Properties of Continuity • The six trigonometric functions are continuous on their domains. (pp. 122–123) • The six inverse trigonometric functions are continuous on their domains. (p. 123) • An exponential function is continuous on its domain, all real numbers. (p. 124) • A logarithmic function is continuous on its domain, all positive real numbers. (p. 124) 1.5 Infinite Limits; Limits at Infinity; Asymptotes Basic Limits 1 • lim x→0 − x = −∞ lim 1 =∞ x→0 + x (p. 128) 1 • lim =∞ x→0 x2 (p. 128) • lim ln x = −∞ x→0 + (p. 129) 1 • lim x→∞ x = 0 lim 1 = 0 x→−∞ x (p. 132) • lim ln x =∞ x→∞ (p. 136) • lim x→−∞ ex = 0 lim x→∞ ex =∞ (p. 136) Definitions • Vertical asymptote (p. 131) • Horizontal asymptote (p. 138) Properties of Limits at Infinity (p. 132): If k is a real number, n ≥ 2 is an integer, and the functions f and g approach real numbers as x →∞, then: • lim A = A, where A is a constant x→∞ • lim [kf(x)] = k lim f (x) x→∞ x→∞ • lim [ f (x) ± g(x)] = lim f (x) ± lim g(x) x→∞ x→∞ x→∞ [ ][ ] • lim [ f (x)g(x)] = lim f (x) lim g(x) x→∞ x→∞ x→∞ f (x) lim f (x) • lim x→∞ g(x) = x→∞ provided lim g(x) = 0 lim g(x) x→∞ x→∞ [ ] n • lim [ f x→∞ (x)]n = lim f (x) x→∞ √ √ n • lim f (x) = n lim f (x), where f (x) >0 if n is even x→∞ x→∞ 1.6 The ε-δ Definition of a Limit Definitions • Limit of a Function (p. 145) • Limit at Infinity (p. 149) • Infinite Limit (p. 150) • Infinite Limit at Infinity (p. 150) Properties of limits • If lim f (x) >0, then there is an open interval around c, for x→c which f (x) >0 everywhere in the interval, except possibly at c. (p. 149) • If lim f (x)
Page 2 and 3:
Chapter 1 Limits and Continuity Ove
Page 4 and 5:
Chapter 1: Resources TRM Teacher’
Page 6 and 7:
Sullivan AP˙Sullivan˙Chapter01 Oc
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33: Sullivan AP˙Sullivan˙Chapter01 Oc
Page 52 and 53: Teaching Tip Alternate Example Find
show all

Sullivan Microsite DigiSample

Create successful ePaper yourself

Delete template?

Save as template?