chapter 2 73

# chapter 2 73 - 4 5 10 11 12 13 CHAPTER 2 REVIEW 137 See...

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Unformatted text preview: 4. 5. 10. 11. 12. 13. CHAPTER 2 REVIEW 137 See Theorem 3 in Section 2.3. (a) See Deﬁnition 2.2.6 and Figures 12—14 in Section 2.2. (b) See Deﬁnition 2.6.3 and Figures 3 and 4 in Section 2.6. (a) y = 1:42 No asymptote (b) y = sin x: No asymptote (c) y : tan cc: Vertical asymptotes a: = g + 7m. (d) y = tan‘1 3:: Horizontal asymptotes y 2 :lzg 72 an integer (e) y 2 er: Horizontal asymptote y : 0 (f) y : 111er Vertical asymptote :c = 0 ( lirn em = 0) ( lim lnm : ~00) za—oo {1390+ (g) y : 1/31: Vertical asymptote ac : 0. (h) y 2 ﬂ; No asymptote horizontal asymptote y = 0 (a) A function f is continuous at a number a if f (:c) approaches f (a) as ac approaches a; that is, lim f(\$) = 1‘01)- I—Hl (b) A function f is continuous on the interval (~007 00) if f is continuous at every real number a. The graph of such a function has no breaks and every vertical line crosses it. . See Theorem 2.5.10. See Deﬁnition 2.7.]. See the paragraph containing Formula 3 in Section 2.7. (a) The average rate of change of y with respect to w over the interval [\$1. 2:2] is W. 2 — 1 _ . . . . \$2 * IE1 (b) The instantaneous rate of change of y With respect to at at a: = \$1 IS hm “23%;“. See Deﬁnition 2.8.2. The pages following the deﬁnition discuss interpretations of f’(a) as the slope of a tangent line to the graph off at m 2 a and as an instantaneous rate of change of f(3:) with respect to x when ac = a. (a) A function f is differentiable at a number a if its derivative f’ exists at m : a; that is. if f’(a) exists. (b) See Theorem 2.9.4. This theorem also tells us that if f is not continuous at a. then f is not differentiable at a. (C) 14. See the discussion and Figure 8 on page 172. a TRUE-FALSE QUIZ \ 1. False. Limit Law 2 applies only if the individual limits exist (these don't). 2. False. Limit Law 5 cannot be applied if the limit of the denominator is 0 (it is). 3. True. Limit Law 5 applies. 4. True. The limit doesn’t exist since f(m)/g(a:) doesn’t approach any real number as m approaches 5. (The denominator approaches 0 and the numerator doesn‘t.) ...
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