Chapter 12 Exponential and Logarithmic Functions

Transcription

1 Chapter Eponential and Logarithmic Functions. Check Points. f( ).(.6) f ().(.6) The average amount spent after three hours at a mall is $6. This overestimates the amount shown in the figure $.. f( ) f( ) (, ),,,,,,,. f( ) ( ) ( ) ( ) f( ) (, ),,,,,,, Copright Pearson Education, Inc. 6

2 Chapter : Eponential and Logarithmic Functions. f( ) and g ( ) f( ) g( ) 6. is ears after 8.. f( ) 66e.() f() 66e 6 In the gra wolf population of the Western Great Lakes is projected to e aout 6.. a. nt r A P n.8 $, $, 8.. rt.8() A Pe $, e $, 8.. Concept and Vocaular Check. ; (, ); (, ). ; = ; horizontal The graph of g ( ) is the graph of f( ) shifted unit to the right.. f( ) and g ( ) f( ) g( ). e; natural;.. A; P; r; n;. semiannuall; quarterl; continuous. Eercise Set e.6.. e. The graph of g ( ) is the graph of f( ) shifted up units. 66 Copright Pearson Education, Inc.

3 Introductor and Intermediate Algera for College Students E Section. f. f 8. f f This function matches graph ().. f f This function matches graph (c). f 6. f. g g 6 6 This function matches graph (a). Copright Pearson Education, Inc. 6

4 Chapter : Eponential and Logarithmic Functions. h h 6. f and g f( ) g( ) 8 6. f.8 8 The graph of g is the graph of f shifted units to the left. f Copright Pearson Education, Inc.

5 Introductor and Intermediate Algera for College Students E Section. 8. f and g f( ) g( ) 8. f g and f( ) g( ) The graph of g is the graph of f shifted down unit. The graph of g is the graph of f shifted unit to the right.. f g and f( ) g( ) 6. f g and f( ) g( ) The graph of g is the graph of f shifted up units. The graph of g is the graph of f reflected across the ais. Copright Pearson Education, Inc. 6

6 Chapter : Eponential and Logarithmic Functions f and g 6. f( ) g( ) 6 The graph of g is the graph of f shifted unit to the left and units down. 8. f g and f( ) g( ). c..6 A 6. The alance in the account is $6. after ears of monthl compounding..6 Ae. The alance in the account is $. after 6 ears of continuous compounding.. Quarterl Compounding.8 A Semiannual Compounding.8 A Quarterl compounding at 8.% ields the greatest return.. Domain:, Range:, 6. Domain:, Range:, 8. Domain:, Range:,. f( ) g( ) 8 8 The graph of g is the graph of f stretched verticall a factor of.. a..6 A. The alance in the account is $. after ears of semiannual compounding. The point of intersection is,. 6 Copright Pearson Education, Inc.

7 Introductor and Intermediate Algera for College Students E Section.. 6. a. Note t 66. Then nt r. A P n,, 6, With monthl compounding, the investment would e worth $,,6,. rt.. A Pe e,, 66, With continuous compounding, the investment would e worth $,,66,. f 8. e 6.. Approimatel 8.% of -ear-olds have some coronar heart disease. f Chernol will not e safe for human haitation 66. There will still e. kirams of cesium- in Chernol s atmosphere. 6. S,.,., Answers will var. a.. In ears, the house will e worth $,. 8. a. f( ) f () () 88 According to the linear model, there were aout 88 million active Faceook users in Jul...66 g ( ).6e.66() g().6e According to the eponential model, there were aout million active Faceook users in Jul. c. The linear model is the etter model for the data in Jul. c. d. Answers will var. One possiilit follows: The graph of... 6 n! is approaching the graph of e. n. makes sense. makes sense Copright Pearson Education, Inc. 6

8 Chapter : Eponential and Logarithmic Functions 6. false; Changes to make the statement true will var. A sample change is: The graphs do not have the same graph as the do not coincide. 8. true 8. ( )( ) ( ) ( )( ) ( )( ) 8 ( )( ) ( )( ) (8) ( )( ) 8 ( )( ) ( )( ) 8. cosh sinh 8. e e e e e e e e e e e e e e e e e e e e e ee e e ee e ee ee ee e ee e a D a Da a Da D a Da a D Da a D Da a D 8. Appl the zero-product principle. or The solutions are and, and the solution set is,. 8. There is no method for solving for. 8. requires a power of 86. to otain. f Interchange and and solve for. f ( ) 6 Copright Pearson Education, Inc.

9 Introductor and Intermediate Algera for College Students E Section.. Check Points. a... a. c c. e e. a. ecause.. ecause 6 ecause c a. Because, we conclude.. Because, we conclude 8.. a. Because, we conclude Because, we conclude. 6. Set up a tale of coordinates for f( ). f( ) Reverse these coordinates to otain the coordinates of g ( ). g ( ). The domain of h is,. 8. f( ) 8.8( ) f () 8.8( ) A -ear-old o has attained approimatel 8% of his adult height.. I R I, I R I, The magnitude on the Richter Scale is.. a. The domain of f consists of all for which. The domain of f is,.. The domain of g consists of all for which. It follows that the domain is all real numers ecept.,,. The domain of g is. f( ). ln.6 f (). ln.6 The temperature increase after minutes will e. The function models the actual increase shown in the figure etremel well.. Concept and Vocaular Check.. arithmic;.. Copright Pearson Education, Inc. 6

10 Chapter : Eponential and Logarithmic Functions (, ) ; (, ) 8. ; = ; vertical.. common;. natural; ln. Eercise Set Copright Pearson Education, Inc.

11 Introductor and Intermediate Algera for College Students E Section f The domain of f is 6,.. f The domain of f is,. f ln The domain of g is all real numers for which. The onl numer that must e ecluded is. The domain of f is,,.. Since. f g, we have that Since, we conclude that. 6. f g 6. ln e e e e e 6. Because ln e, we conclude that ln e. 6. ln e ln e Because ln e, we conclude that ln e. 66. Because e ln ln, we conclude that e. Copright Pearson Education, Inc. 6

12 Chapter : Eponential and Logarithmic Functions 68. Because ln e, we conclude that ln e.. Because e ln, we conclude that e ln.. Since, we conclude that.. 6. The solution set is {} is 6 ears after 6. f.8 ln. f 6.8 ln 6.. According to the function,.% of first-ear college women will epress antifeminist views in. 6. D The deciel level of a normal conversation is approimatel 6 deciels... Answers will var.. f ln g ln The solution set is {6} ln e 8. (f) The graph is similar to that of ln, ut shifted right units. 8. (a) The graph is similar to that of ln, ut shifted down units. 86. (e) The graph is similar to that of ln, ut reflected across the -ais and then shifted right units. 88. f A -ear-old girl is approimatel 8.% of her adult height.. a. 8 is ears after 6. f.8 ln. f.8 ln..8 According to the function,.8% of first-ear college women epressed antifeminist views in 8. This underestimates the value in the graph.%. The graph of g is the graph of f shifted units to the left.. f g The graph of g is the graph of f reflected across the ais. 66 Copright Pearson Education, Inc.

13 Introductor and Intermediate Algera for College Students E Section. 6. f t ( t ). Rewrite the equations in AB C form. 8. After approimatel months, the average score falls elow 6. Multipl the first equation and the second equation and solve addition. 6 8 Back-sustitute for to find. Use the trace function to compare how quickl the functions increase. In order from slowest to fastest, the functions are: ln,,,, e, and.. does not make sense; Eplanations will var. Sample eplanation: Logarithmic functions do not have horizontal asmptotes.. does not make sense; Eplanations will var. Sample eplanation: An earthquake of magnitude 8 8 on the Richter scale is, times as intense as an earthquake of magnitude.. false; Changes to make the statement true will var. A sample change is: We cannot take the of a negative numer. 6. true 8. 8 The solution is, and the solution set is, or 6. a. The solution set is or or.. a.,,. 8 c. (8 ) 8. 6 c.. a.. c. 8 Copright Pearson Education, Inc. 6

14 Chapter : Eponential and Logarithmic Functions. Check Points. a. 6( ) 6 6. ( ). a a.. e e ln ln ln ln ln ln ln c. ( ) ( ). a... a. (). 6 (6) 68 Copright Pearson Education, Inc.

15 Introductor and Intermediate Algera for College Students E Section. 6. a.. c. ln ln( ) ln ln( ) ln ln ln ( ) ( ) ( ) ln 6 6. ln. Concept and Vocaular Check. M. M. p N ; sum M ; product N ; difference. a a M. Eercise Set ,, 6. Copright Pearson Education, Inc. 6

16 Chapter : Eponential and Logarithmic Functions e e ln ln ln 8 ln 8 8. ln e ln e ln e. ln ln e ln ln M 8 8 M ln ln ln. z z z z z z z z 6. z ln ln ln ln Copright Pearson Education, Inc.

17 Introductor and Intermediate Algera for College Students E Section. 8. ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln ln 8 lnln lnz ln ln lnz ln ln z ln z A C... C 8 6 CA 6. false; Changes to make the statement true will var. A sample change is: ln is undefined. 8. true 8. false; Changes to make the statement true will var. A sample change is: ln( ) ln ln. 8. false; Changes to make the statement true will var. A sample change is: ln ln( ) ln( ). 8. false; Changes to make the statement true will var. A sample change is: ( ) ( ). 86. true 88. false; Changes to make the statement true will var. e A sample change is: e ln e.. a Copright Pearson Education, Inc. 66

18 Chapter : Eponential and Logarithmic Functions. a ( ) 6 ( ) 6 6 ( ) 6 ( ) ( ) 6( ) ( )( ) 6( ) ( ) 6 ln ln c A A ln ln c AN c AN. a. t A AN. A 6 t ln ln c AN. 6 6 ln It will take approimatel weeks for the chimpanzee to master signs. 6.. Answers will var.. ( ) (. ) The graph of is the graph of shifted up unit. The graph of. is the graph of The product rule accounts for this relationship. Consider shifted down unit... Likewise consider (. ) Copright Pearson Education, Inc.

19 Introductor and Intermediate Algera for College Students E Section. 6. Answers will var. One eample follows. To disprove the statement,, let. Graph and. The graphs do not coincide, so the epressions are not equivalent. 8. Answers will var. One eample follows. To ln ln ln, let disprove the statement. Graph ln and ln ln. 6. false; Changes to make the statement true will var. A sample change is:, ut. 8. false; Changes to make the statement true will var. A sample change is:. A B. First, find the intercepts to the equation. The graphs do not coincide, so the epressions are not equivalent.. To verif that ln ln ln ln, let. Graph ln and ln ln. ln The graphs do not coincide, so the epressions are not equivalent.. makes sense Find the intercept setting =. Find the intercept setting =. Net, use the origin as a test point. This is a false statement. This means that the origin will not fall in the shaded half-plane.. does not make sense; Eplanations will var. Sample eplanation: Copright Pearson Education, Inc. 66

20 Chapter : Eponential and Logarithmic Functions 6. The solution set is,. Mid-Chapter Check Points Chapter. f e ln( ) ln Domain:, Range:,. lne () () ( ). f ( )( ) or The solution set is,. Domain:, Range:, 66 Copright Pearson Education, Inc.

21 Introductor and Intermediate Algera for College Students E Section.. f Domain:, Range:,. f. f 6 The argument of the arithm must e positive: 6 6 Domain: 6,. f 6 The argument of the arithm must e positive: Domain:, The argument of the arithm must e positive.. Now 6 is alwas positive, ecept when 6.. Domain: 6 6, f Domain:, Domain:, Range:,. Let Copright Pearson Education, Inc. 66

22 Chapter : Eponential and Logarithmic Functions not possile This epression is impossile to evaluate ln e ln e ln e ln ln ln ln ln ln ln ln z ln ln ln z ln ln z ln z. z. Continuousl: A 8e,.8().8 Monthl: A 8,6,,6 8 Interest returned will e $8 more if compounded continuousl. 666 Copright Pearson Education, Inc.