Chapter 8: More on Limits

Transcription

1 Chapter 8: More on Limits Lesson a. 000 lim a() = lim = 0 b. c. lim c() = lim 3 +7 = lim b( ) 3 lim( 0000 ) = # = " 8-. a. lim 0 = " b. lim (#0.5 ) = # lim c. lim 4 = lim 4(/ ) = " d. lim (5 # 3) = " ( ) = a.. c() = 00. a() = h() = 0 4. g() =. 5. b() = f () =!!(tie) 7. d() = 0 8. e() =!988 c.. e() =.7!0 30. b()00 0 =! h() = 0!00 3 = 0, 000, c() = 00!00 =, 000, g() =. 00 = 3, d() = 800 log00 = a() = = f () = 00 b.. b() = 0 0. h() = c() = d() = a() = e() = 4 7. f () = 0 8. g() =.59 d.. e(). g() 3. b() 4. h() 5. c() 6. f () 7. a() 8. d() e. At! a() passes d() a. lim a() = lim 4 = " b. lim b() = # lim = #" CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry

2 8-5. a. lim d() = lim ( 4 # ) = " b. Big positive wins since 4 gets larger a lot quicker than a. The dominant term is!y 5. b. The dominant term is 0 9. c. The dominant term is 6. d. The dominant term is. Review and Preview a. lim 6 = " b. lim ( 6 # # 400) = " c. lim (#3 7 ) = #" d. lim (# , 000) = #" 8-8. All are! a. lim (50 log # ) = #". Dominant term is!. b. lim (.5 # ) = ". Dominant term is.5. c. lim (000() # 3 ) = #". Dominant term is!3. d. lim ( # 30 ) = ". Dominant term is. 8-. a. See graph at right. b. 4! 4 +! 4 +!! + 4! 6 = = 50 c !0 = = a. ( 3 ) = ( ) 3 = 4! 5 = 4 = 4 5 b. 00(.05) = = 5 log = log.05 5 = log 5 log.05 = Solution continues on net page. CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry

3 8-. Solution continued from previous page. c. log 3 ( + 5) = 4 3 log 3( + 5) = = 8 = 76 d. 0m.5 = 000 m.5 = 50 ( m 3 ) 3 = 50 3 m = a. Add 4. b. Divide by. c. Add slope = 3!!!!!! slope = " 3 y " 5 = " 3 ( " 7) y " 5 = " 3 + y " 0 = " y = a. y = 0 ; vertical asymptote at = 0 ; horizontal asymptote at y = 0 b. y = 0.5 ; no vertical asymptote; horizontal asymptote at y = 0 c. y = ; vertical asymptote at =! ; horizontal asymptote at y = 0 + d. y = 3 ; no vertical asymptote; horizontal asymptote at y =! a. 3 f () = c. 5 f () = 5 ( ) = b.! f () =! ( + 0 ) =!! 40 ( ) = ( ) = (sin + cos )(sin! cos ) = d. f sin! cos =! cos! cos =! cos =! cos = 4 cos = ± = ± " 3 + "n =! = CPM Educational Program 0 Chapter 8: Page 3 Pre-Calculus with Trigonometry

4 Lesson The limits for all of the three functions are. a. Answers may vary, but students will probably find iii the easiest to evaluate since they should remember little over big goes to zero leaving. b a. lim b. lim 3! = 3! = To change! 5 3! 7 into # = lim # = lim 3 c. lim # = lim! + 3 =! 3 + =! + 3! 5! 7 3, divide the numerator and denominator by # 4 = lim # 5 = lim # 3 = lim # 3 = 3 6 = # = 4 6 = # 3 = lim 6 = " 8-. a. The limit is zero (0). b. The limit will be! or!". c. The limit will be the ratio of the coefficient of the dominant terms. 8-. a. lim b. lim 8+70# = lim 5 4 # #3 3 = lim # = #" 5 4 # # = 3 c. lim (#7)3 3 3 #+0 = lim (4 #8+49)(#7) 3 3 #+0 = lim 8 3 #8 # # #+0 = 8 3 CPM Educational Program 0 Chapter 8: Page 4 Pre-Calculus with Trigonometry

5 8-3. a. lim +8 3 = lim b. lim +6 0 log # 8 3 Limit = 0 Dominant terms: numerator, denominator 3.. Limit =! Dominant terms: numerator, denominator log. log c. lim 5 +3 = lim log 5 +3 = log 5 Dominant terms: numerator log 5, denominator. d. lim 4 #3+000 (#3) = 4 Dominant terms: numerator 4 =, denominator () = 4. Review and Preview a. lim # 3 = lim # 3 = lim # = 0 b. lim # +5 c. The lim # d. lim!0 " +5 = lim # (cos ) does not eist. = ",! lim!0 + e. lim!0 + log = "# = lim # =, therefore lim!0 ( +5) = lim # + 5 does not eist. = " 8-5. a. It approaches zero since the dominant numerator is and the dominant denominator is. b. We have a horizontal asymptote at y = 0. c. lim f () = # and lim! ""! " + d. There is an error at =!and this is reflected on the graph by a vertical asymptote a. lim f () = b. lim!3 " c. lim!3 + d. lim "6("3)+0! "3 " ("3"3) = e. The function is undefined at = 3. f. The function s value is CPM Educational Program 0 Chapter 8: Page 5 Pre-Calculus with Trigonometry

6 8-7. a. = 4(5)! 0 = 0 3 = 4(0)! 0 = 40! 0 = 30 4 = 4(30)! 0 = 0! 0 = 0 5 = 4(0)! 0 = 440! 0 = 430 c. z = 4 z 3 = /4 =! 4 = 4 z 4 = 4 z 5 = /4 =! 4 = 4 b. y = 0! 4 =!4 y 3 = (!4)! 4 = 6! 4 = y 4 =! 4 = 44! 4 = 40 y 5 = 40! 4 = 9, 596 d. w = 3! 5 = 5 w 3 = 5!5 = 75 w 4 = 5! 75 = 5 w 5 = 75!5 = 84, t = 5 + () = 7,!t 3 = 7 + () =,!t 4 = + (3) = 7,!t 5 = 7 + (4) = The end behavior of the graph a. lim 3 = " b. lim (3 # 6 $ ) = ". 3 is the dominant term in the function. c. lim # 5 = #" d. lim (#5 + 0 $ 4 ) = #".!5 is the dominant term in the function The larger base will determine if the function goes to! or!" csc A = sin A = 3 = 3 CPM Educational Program 0 Chapter 8: Page 6 Pre-Calculus with Trigonometry

7 Lesson a. f () : =!, g() : =!, these values would cause division by zero. b f() und g() 0 und c. f( ) g() d. In the table y = error. The function is not defined. There appears to be a vertical asymptote for one of the graphs but the other looks linear They have similar form; domain is all real numbers ecept one point f () has a vertical asymptote, g() has a hole. Common factor in numerator and denominator creates a hole in the graph of a rational function y =!4 + = (+)(!) Hole at =!. + y =!6 (+3)(+4) =!6 Vertical asymptotes at =!3 and =! Horizontal asymptote at y = 0. y 3 = 4 +9 Horizontal asymptote at y = They simplify to the same function, but have different discontinuities. f () = vertical asymptote at =!. g() =!!4 = (!) (+)(!) = + looks like f() but has a hole at =. + has a CPM Educational Program 0 Chapter 8: Page 7 Pre-Calculus with Trigonometry

8 8-38. Answers will vary. Eamples are given. a.!3 b.!9!3 c. 3!! d. +!4! 4 e.! +!6 f. (+)(!9) + Review and Preview a. lim! "# + 3 " 5 = lim! "# + 3 " 5 = lim + 3! "# " 5 = = b. lim 3 + 7! "# 4 " 4 = lim ! "# 4 3 " 4 3 = lim 5 + 7! "# " 4 = 0 c. "6 lim 3 + 8! "# 5 " = lim " ! "# 5 " = lim 8-4.! "# " " = # a. 5 lim +#3 3 = 0 b. lim 3 # = +4#5 3 c. lim 3 "7!0 3 3 = (0 3 )"7 +4 "5 3(0 3 = 7 )+4(0)" # lim g() = lim #5 = lim 3 # #5 = lim 3# #5 = 3 The limit equals # a. lim h() = + lim #5 b. lim j() = lim $ c. See graphs at right. log 3# % & #5 = + 3 = 5 ' ( ) = log 3 h ( ) j( ) CPM Educational Program 0 Chapter 8: Page 8 Pre-Calculus with Trigonometry

9 8-44. a. m() = log 5 b. lim k() = lim c. lim m() = lim k() ( ) = log( )! log( 5 ) = log! 5 log " 0.30! 5 log ( ) = ". The dominant term is. 5 ( ) = lim log 5 log( ) # lim log( 5 ) = " m () a. Area of a side = L. Surface area of a cube = S = 6L b. Length of the diagonal on the face of the cube: D = L + L Length of the longest diagonal: D = L + ( L) c. S = 6L S 6 = L D = S 6! 3 = 3S 6 = S D = L + L D = 3L D = L 3 D = L D = L a. Initial value = $.87, multiplier = =.0, time is in months. b. Initial value = $,000, multiplier =! 0. = 0.88, time is in years. c. Initial value = $,000, multiplier = =.30, time is in hours. d. Initial value = $5,000, multiplier = =.0, time is in months a. y = 5000!.0 y = $5, c. y = 5000!.00 5 y = $5, b. y = 5000!.0 4 CPM Educational Program 0 Chapter 8: Page 9 Pre-Calculus with Trigonometry. y = $5, 4.6 d. y = 5000! y = $5, 378.6

10 Lesson a. 360! = 30! b. cos5! = h!!!!!h = cos5! c. sin5! = (/)b!!!!! sin5! = b ( )! sin 5!! cos5! = sin 5!! cos5! Area of polygon: A = sin5!! cos5! = 3 d. Area of one triangle: A = a. tan5! = (/)b!!!!!tan5! = b!!!!! tan5! = b b. Area of one triangle: A = ( )! tan5!! = tan 5! Area of polygon: A = tan5!! 3.5 c. 3 and a. n b. ( ) c. cos 360! / n = cos 80 n ( n ) ( ) d. sin 360! / e. n f. n sin 80 n 8-5. a. The smallest number of sides of a polygon is 3. c. The plot shows the areas of the inscribed polygons as n increases ( ) ( ) = sin 80 n ( ) cos ( 80 n ) = n ( ) sin 360 n a. n tan 360 n c.! d. The value of! gets squeezed between the two functions which act as an upper bound and a lower bound lim A c (n) = # n!" CPM Educational Program 0 Chapter 8: Page 0 Pre-Calculus with Trigonometry

11 Review and Preview a. lim n!" # + & $ % n ' ( = lim + lim n!" n!" n a. 050(.075) 0 b ( ) = + 0 = b. lim n!" (.0) n = " ( ) 0 c. P ( + r ) nt 00n a. First four terms: n =!n 3+n =! 3+ = 4 n =!n 3+n =! 3+ = 0 5 = 0 n = 3!n 3+n =!3 3+3 =! 6 n = 4!n 3+n =!4 3+4 =! 7 lim!n n"# 3+n =! n n =! c. First four terms: n = n = n = 3 n = 4 (!) n n = (!) (!) n n = (!) (!) n n = (!)3 =! =! = 4 4 = 3 =! 8 8 =! (!) n n = (!)4 4 = 6 6 = There is no limit for lim n"# (!)n n. b. First four terms: n = n = n = 3 n = 4 (!) lim n n"# 3 n = 0 (!) n 3 n = (!) 3 =! 3 (!) n 3 n = (!) 3 = 4 9 (!) n 3 n = (!)3 3 3 =! 8 7 (!) n 3 n = (!)4 3 4 = See graph at right. a. lim g() = lim g() =!#" c. g() = 3+!4 = (!4)+!4 = +! a. Increasing b. Decreasing c. Decreasing CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry

12 8-59. log b RQ T = log b RQ ( T ) = (log b RQ! log b T ) = (log b R + log b Q! log b T ) = (5.4 +.! (!.4)) = (8) = Center of the circle: (0, 0). Slope from center of circle to point (3, 4) : m = 4!0 3!0 = 4 3 Equation of the line y! 4 =! 3 4 (! 3)!!or!!y =! 3 (! 3) " slope =! a. See graph at right below. b. y = t c. Area under the curve after minute: A = 3 + ( (60)) ( ) (60) = 98 feet Area under the curve after 5 minutes: A = 3+ (3+ 0.0(300)) d. A = Lesson 8.. ( 3 + ( T )) T = 3T T ( ) (300) = 350 feet 8-6. a. goes to zero so the epression in the parentheses goes to and the limit might be, n since n =. b. Infinity, since the number inside the parentheses is larger than. c. A number around.78 It should be surprising that this limit is neither nor infinity a. e 3 b. e a c ( ) 4 b. 000 ( ) ( ) 365 d. 000 ( ) 4!365 4!365 e. 000e 0.06 f. a. $06.36 g. 00e = $93.46 b. $ c. $06.83 d. $06.84 e. more and more CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry velocity (ft/sec) v(t) time (sec) t

13 a. y = log b. y = log e See graph at right below a. log ( ) = log 3 3!4 =!4 ( ) = log 3 7 = log = 3 ( ) = ln e e = ln e e = b. log 3 54! log 3 = log 54 3 c. ln e d. 5 ln(e ) ln( e) 0 ln e 0 = = ln e!!ln e =! a. b. c. CPM Educational Program 0 Chapter 8: Page 3 Pre-Calculus with Trigonometry

14 Review and Preview = 58! e = e e e 3 = e = e ln(0.0489) = " # a.! = 0! +! +! + 3! + 4! = = k! k=0 0 b.! = k! 0! +! +! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 0! =.788 k=0 c. lim n!" 8-7. n k! k=0 # = e a. f (!) = (!)3!7(!)!6 = (!+) 0 b. See division at right. lim 3 "7"6 = lim + c. Hole at =.!! 6 which is undefined. 3!7!6!!"!! ! 7! 6!!!!!!!!!!!!!!!!!!!!!!!!!!!( 3 + ) ( " " 6) = (")!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! " (") " 6 = "4!!!7!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(!!!!)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!6! 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(!6! 6)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! See graph at right. Point of intersection: (, 7) y a. 0 b. 0 c. 3 d. 3 CPM Educational Program 0 Chapter 8: Page 4 Pre-Calculus with Trigonometry

15 8-75. t = t = + () + = 4 t 3 = 4 + () + = 9 t 4 = 9 + (3) + = 6 t 5 = 6 + (4) + = Let a = a 4 + a 3 = 0 a 3 (a + ) = 0 a 3 = 0!!or!!a + = 0 If!!a + = 0 ( + 3) + = = 0 ( + )( + ) = 0 =!,! If!! a 3 = 0 ( + 3) 3 = 0 =!3, intercepts: 0 = 3t! 6t =!6t(! t)!!" t = 0, Verte = avg. of -intercepts or t =. At t =, s = 3()! 6( ) = 6 ft cot! = adjacent opposite = 3 " = " 3 f () = +3! = (!)(+)!4 (!)(+) = (!) (!) Hole at =!!!"!! f (!) = ((!)!) =!5!!!4 = 5 4!!"!!!, 5 4 Asymptotes at = and y =. ( ) Lesson a. A = 00e 0.065! A = $06.7 c. A = 00e 0.065!3 A = $30.5 e. A = 00(e 0.06 ) t b. A = 00e 0.065! A = $3.89 d. A = 00(e 0.06 ) 7 A = $30.9 CPM Educational Program 0 Chapter 8: Page 5 Pre-Calculus with Trigonometry

16 8-8. a. r = G = 500! e!t G = 500! e!( ) G = 500! e 6 G = 938 ZWD t-value is b. A = 500!( + ) A = $, c. A = 500! e! A = $, a. A = 00! e 0.05!3 = 00e 0.5 A = $6.8 c. 600 = 300! e 0.05!t = e 0.05!t ln = ln e 0.05!t ln = 0.05t t = ln = years 0.05 lim P ( + r n!" n ) nt = lim Let k = n r lim k!" P + k ( n r ) nt n!" P + ( ) krt = P lim # ( ) k k!" $ + k % & rt b. = Pe rt 00 = 00! e 0.05!t = e 0.05!t ln = ln e 0.05!t ln = 0.05t t = ln = years a. C = km t C = 0(.07) 0 = $39.34 c. This is a matter of opinion. b. A = Pe rt A = 0e 0.07!0 = $ y = ae k = a(e k ) If a = c and m = e k then y = c! m. CPM Educational Program 0 Chapter 8: Page 6 Pre-Calculus with Trigonometry

17 8-86. a. After another 5730 years there would be 5% remaining. After another 5730 years there would be.5% remaining. b. 0.5 = e k!5730 c = e! ln 5730 " ln = ln e 5730k " ln = 5730k k = " ln 5730 y = a! e " ln 5730 ln 0.78 = ln e! ln 5730 ln 0.78 =! ln 5730 = ln 0.78 "! 5730 ln! 054 years old Review and Preview a. f () =!e b. g() = e + c. h() = e Compounded quarterly: A = 0, 000 ( ) 4 = $0, Continuously compounded: A = 0, 000e 0.05! = $0, 5.7 The quarterly-compounding account is better a. a n = 6! 3 a = 6! 3 a = 6! 3 a 3 = 6! 3 lim 6! n#$ 3 ( ) n" ( ) 0 = 6 ( ) = ( ) = 6!( 9 ) = 3 ( ) n" = 0 b. a n =! "(.) n! a =! "(.) 0 =! a =! "(.)! =!. a 3 =! "(.) 3! =! ". =!.4 lim! "(. n#$ )n! =!$ a = 8 a 3 =! "!4 = a 5 =! "! = 0.5 a =! " 8 =!4 a 4 =! " =! CPM Educational Program 0 Chapter 8: Page 7 Pre-Calculus with Trigonometry

18 8-9. a. 4 b. 4 c. 0 3 log 4 = 0 log 0 43 = 4 3 = 64 d. e 3 ln 4 = e ln e 43 = 4 3 = a. 5 lim +#3 5 3 = lim + #3 5+ # = lim + = 0 b. lim 3 "7!0 3 3 = "7 +4 "5 "5 = tan (! " # ) = (a + b) 4! (a! b) 4 = sin(! "#) sin(! ) cos # "cos(! ) sin # = cos(! "#) cos(! ) cos # +sin(! ) sin # = cos # sin # = cot # a 4 + 4a 3 b + 6a b + 4ab 3 + b 4! (a 4! 4a 3 b + 6a b! 4ab 3 + b 4 ) = a 4 + 4a 3 b + 6a b + 4ab 3 + b 4! a 4 + 4a 3 b! 6a b + 4ab 3! b 4 = 4a 3 b + 4ab 3 = 8ab(a + b ) ab(a + b ) 8 cos sin (cos + sin ) = 8 cos sin = 4( cos sin ) = 4 sin() CPM Educational Program 0 Chapter 8: Page 8 Pre-Calculus with Trigonometry

19 Lesson a. The adjacent terms are proportional b. to each other a. S n = n 3 S n = n + 3 n+ c. S n =! "3 n d. e. lim n!" ( ) = lim # $3 n n!" # lim $3 n = + 0 = S = 3 + ( 3! 3 ) + ( 3! 3! 3 ) + a n = ( 3 ) n b. 3 S n = 3! 3 n+ S 5 =! "3 5 =! 486 # S 7 =! "3 7 =! 4374 # S 9 =! "3 9 =! # The sums seem to be approaching S = = % n & 3 n=0 S = 6 6(#( S n = lim 3)n ) = 6(#0) = 8 # 3 3 ( ) = 6! ( 3 ) 0 + ( 3 ) + ( 3 ) + ( 3 ) 3 + " # $ a. S n = a(!rn )!r b. lim n!" r n = 0 c. lim a(#rn ) = a n!" #r #r 8-0. a. S n = 0 ( ) n S = 0! = 0 = 40 b. S n = 8 (! 3 ) n c. S n = 8 ( 3 ) n Since r >, the sum goes to infinity. The formula gives a sum of! 6. 8 S =! (! 3) = = 43 5 = r is greater than. The terms get bigger. lim a n = ". Since we keep adding larger quantities the sum will go to infinity. n!" CPM Educational Program 0 Chapter 8: Page 9 Pre-Calculus with Trigonometry

20 8-03. a. lim + c. lim z!" ( ) 5 = e 5 b. lim + ( 5 ) ( z ) ( ) z = e 5 ( ( )) h = e # = e h!" + # h Review and Preview a b. 0N = c. 0N = N = N = N = d. 9N = 3 N = a. 0N = N = N = 5 N = N = N = 99N = 36 N = = a. 00N = N = N = 60 N = = 0 33 b. 0N = N = N = 7 N = 7 9 b. 00N =. N = 0. 99N = N = 99 = = = c. 0N = N = N = 9 N = 9 9 = c. 000N = 3.3 N = N = 3 N = = (+ 0.) 4 = 4 + 4! 3! 0.+ 6!! !! = + 4! ! ! = =.464 CPM Educational Program 0 Chapter 8: Page 0 Pre-Calculus with Trigonometry

21 8-09. y = a! b + 0!!" 6 = a!b = a! b7 + 0!!" "4 = a!b 4 # = a! b 7 y =!8 " # 3 3 $ + 0 % y-intercept: y =!8 " # &!!y = 0! 8 0 $ + 0!! % '!0.079!=a"b 7!4=a"b 4 # = b3!!b = 3!4 = a " $ % a =!4 $ & ( % 3 ) ' 3 4 & # ' 4 =!4 3 ( ) 4 3 =! Average rate = 8-. a. P = Pe rt = e rt ln = ln e rt ln = rt total distance total time b. r = R 00 t = ln r d. We use 70 because it makes the mental arithmetic easier. 8-. () + l = l = 00 l = 00! 4 l = 5! c. t = ln r t = = 0.693! 00 R R " 70 R 00 a. Perimeter of the rectangle: P = + + 5! + 5! = ! b. A = () 5! 8-3. a. ln = 3 e ln = e 3 = e 3 c. 5 ln( + ) = 0 ln( + ) = 4 e ln(+) = e 4 ( ) = 4 5! + = e 4 = e 4! 60 = 50 = 50 t +t 5!!!!! = 50 r 60 r 5 r = " 5 50 = = 5 3!!!!!5 5 = r 3!!!!!r = 5(3) 5 = 75 b. e = 00 ln e = ln 00 = ln 00 d. e (+) + 0 = 00 e (+) = 80 ln e (+) = ln 80 + = ln 80 = ln 80! CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry

22 Lesson Infinity ! n= 0! n= a. = + n =.99 b. n = ! c d b. n < +n n since n + n > n = a b d. It is larger than the series that adds ½ an infinite number of times. Since that series has an infinite sum, the harmonic series (which must be larger) must also be infinite. e. Infinity First is the largest, last is the smallest Infinity. If not, the harmonic series would have a finite sum ( ) + ( 3! 4 ) + ( 5! 6 ) + > 0 ( )! ( 4! 5 )!! < d. The eact sum is ln = a.! + 3! 4 + 5! 6 + 7! 8 + 9! 0 b.! c.!! 3 CPM Educational Program 0 Chapter 8: Page Pre-Calculus with Trigonometry

23 Review and Preview Sample graphs: a. b. 8-. n+ = n + = = + = 3 = + = 3 = 3 4 = 3+ = 5 3 = = 3 5+ = 8 5 = a a n = ( ) n Sum =! = = c. a + 0.9a + 0.8a a + r n = a 9 0 Sum = ( ) n a!9 0 = a 0 = 0a b a n = 6 ( 4 ) n Sum = 6! 4 = = 6"4 = d. The sum would be! because r >. 8-4.! = 8-5.! =!! = 0 =!(!)± (!)!4()(!) () = ± 5 +sin! ("sin!)(+sin!) + "sin! (+sin!)("sin!) +sin! +"sin! = "sin ==! cos =! sec! CPM Educational Program 0 Chapter 8: Page 3 Pre-Calculus with Trigonometry

24 8-6. a. f () = 0! = 6 f (4) = 0! 4 =!6 m =!6!6 4! =! =!6 c. f () = 0! = 6 f (.5) = 0!.5 = 3.75 b. f () = 0! = 6 f (3) = 0! 3 = m = 6!!3 =! 5 =!5 d. f () = 0! = 6 f (.) = 0!. = 5.59 m = 6!3.75!.5 =! =!4.5 m = 6!5.59!. =! =!4. e. As the two points get closer and closer, the slope of the line gets closer to a. log( + 3) = 0 log(+3) = = 00 = 97 c. ln 3 + ln = 4 ln(3) = 4 e ln(3) = e 4 3 = e 4 = e a. f () = 8! = 7 f (4) = 8! 4 =!8 m =!8!7 4! =!5 3 =!5 b. ln( + 3) = e ln(+3) = e + 3 = e = e! 3 d. ln( + 4)! ln = ( ) = ( ) = e ln +4 e ln +4 b. f () = = +4 = e + 4 = e f (4) = 4 = 6 4 = e! = 4 e! m = 6! 4! = 4 3 = y = ae k 50 = 00e! y = 00e k 0.5 = e! = 00e 4k ln 0.5 = ln e! = e 4k!0.695 =! ln 0.85 = ln e 4k = 7.06 hours!0.65 = 4k k =! CPM Educational Program 0 Chapter 8: Page 4 Pre-Calculus with Trigonometry

25 8-30. e k = m ln e k = ln m k = ln m k = ln m Lesson Month Baby rabbits Teen rabbits Parent rabbits Total rabbits a = = = 3 + = = = = 44 b. an+ = an + an+ CPM Educational Program 0 Chapter 8: Page 5 Pre-Calculus with Trigonometry

26 a a. a = = a a. a = = a 3 a = = a 4 a3 = 3 =.3 a 5 a4 = 5 3 =.67 a 6 a5 = 8 5 =.6 a 7 a6 = 3 8 =.65 a 8 a7 = 3 =.65 a 9 a8 = 34 =.69 a0 a9 = =.676 a a0 = =.68 a a = 44 a a3 = = 0.5 a 3 a4 = 3 = a 4 a5 = 3 5 = 0.6 a 5 a6 = 5 8 = 0.65 a 6 a7 = 8 3 = a 7 a8 = 3 = a 8 a9 = 34 = a9 a0 = = 0.68 a0 a = = 0.68 a a = = =.68 b. Yes,.68. b. Yes, a 000 = No, it reaches this value by the 5 th term. The limit of the calculator s display is reached by this point so it is forced to round off a. = = (! 5) + 5 =!4 + 5 = 3 = (! 5) + 5 =!4 + 5 = 4 = (! 5) + 5 =!4 + 5 = 5 = (! 5) + 5 =!4 + 5 = b. = = (! 5) + 5 =!6 + 5 =! 3 =!(!! 5) + 5 = = 4 = (! 5) + 5 = = 7 5 = 7(7! 5) + 5 = = = (! 5) + 5 =! =! = (! 5)(! ) =, 5. The only two values are and 5. CPM Educational Program 0 Chapter 8: Page 6 Pre-Calculus with Trigonometry

27 8-38. a. t = c. f = t = ()! = 4! = 3 t 3 = (3)! = 6! = 5 t 4 = (5)! = 0! = 9 t 5 = (9)! = 8! = 7 f = f 3 = + = f 4 = + = 3 f 5 = + 3 = 5 b. a = a = + () = + = 3 a 3 = 3 + () = = 7 a 4 = 7 + (3) = = 3 a 5 = 3 + (4) = = Review and Preview a. a = a = 000 b. a 3 = = 00 c. a 4 = = 00 a 5 = = 300 a 6 = = 5003 a 7 = = 8005 a 8 = = 3008 a 9 = =, 03 a 0 = = 34, 0 a = = 55, 034 a 55, 034 = a 0 34, 0 =.68 a 0 34, 0 = a 55, 034 = a n+ a n = a n+ = a n + n + a = a = + () + = 4 a 3 = 4 + () + = 9 a 4 = 9 + (3) + = 6 a 5 = 6 + (4) + = 5 General term: a n = n CPM Educational Program 0 Chapter 8: Page 7 Pre-Calculus with Trigonometry

28 8-4. a..68 b. L = + L " # $! y 3!5 y 7 3 % " & ' = # $ 5 y 3 y 7 3 % & ' = " # $ 3 y 4 3 % & ' = 89 y L = + L L! L! = 0 L =!(!)± (!)!4()(!) () L = ± f ( + h) = ( + h) + ( + h) + = + h + h + + h + f ( + h)! f () = + h + h + + h +! (!! ) = h + h + h!! a. 00N = c. N = N = 45 N = = 5 000N = N = N = 675 N = = = 5 37 b. 00N = N = N = 8 N = 8 99 = = = 75 CPM Educational Program 0 Chapter 8: Page 8 Pre-Calculus with Trigonometry

29 Lesson a. =! =! = b. Assume that +!+ k! = k!. c. If +!+ n! = n!, then +!+ n! + n = n! + n = " n! = n+! d. Hence, we have established the truth of the statement for n = and proven the induction step, then the statement is true for all positive integers This is Gauss method. Step: For n =, = (+)! =, so the statement is true for n =. Step : Assume that the statement is true for some integer k. That is, that k = k(k+). Step 3: k + (k + ) =. Thus the induction step is true. k(k+) + (k + ) = k(k+) + (k+) = k(k+)+(k+) = (k+)(k+) Step 4: Since we know the statement is true for n = and the induction step is true, we know that the statement is true for all positive integers n a. 3!" + = 3 + = 4 b. Assume that 3 k! + is divisible by 4 for some integer k. c. 3 (k+)! + = 3 k+! + d. 3 n! + = 3 k! " 3 + = 9 " 3 k! + = 9! 3 k" + 9 " 8 = 9( 3 k + ) " 8 = 9( 4A )! 8 = 36A! 8 = 4( 9A ) CPM Educational Program 0 Chapter 8: Page 9 Pre-Calculus with Trigonometry

30 Review and Preview Step : = (+)(i+) 6 =!!3 6 = 6 6 =. Step : Assume k = k(k+)(k+) 6 is true for some integer k. Step 3: k + (k + ) = k(k+)(k+) 6 + (k + ) = k(k+)(k+) 6 + 6(k+) 6 = (k+)[k(k+)+6(k+)] 6 = (k+)(k +k+6k+6) = (k+)(k+)(k+3) 6 6 so the induction step is true. Step 4: Since we know the statement is true for n = and the induction step is true, we know that the statement is true for all positive integers n This is an infinite geometric sum. Therefore S = a!r. Hence 5!r = 0!and!r = Step : 5! = 4, Step : Assume that 5 k! is divisible by 4, therefore 5 k! = 4A. Step 3: 5 k+! = 5 " 5 k! = 5 " 5 k! = 5(5 k! ) + 4 = 0A + 4 = 4(5A + ) so 5 k+! is a multiple of 4. Step 4: Hence, by mathematical induction we have proven that 5 n! is divisible by 4 for all natural numbers ! = 6 /> 3 = 8 a. 4! = 4 > 4 = 6 b. Assume that k! > k for k! 4. c. (k + )! = k! (k + ) (k + )! =!!k!!!(k + ) > k (k + ) k (k + ) > k! = k+ (k+)! > k+ d. Hence, by mathematical induction we have proven that n! > n for n! 4. CPM Educational Program 0 Chapter 8: Page 30 Pre-Calculus with Trigonometry

31 8-53. Answer to Hint: n n + > n n = n = n Step :! k Step : Assume! j= j " k for some integer k. k+ Step 3:! j= j k! j= = j + k+ " k + k+ = k k++ k+ " k+ = k + k+ so the induction step is true. n Step 4: Hence, by mathematical induction we have proven that! positive integer n. j= j " n for any Closure Chapter a. The area is less than a rectangle. b.. A = 3! 6! 9 = 36. A =!! " A =!(! 7.348)! 6 " CPM Educational Program 0 Chapter 8: Page 3 Pre-Calculus with Trigonometry

32 a.! f () d = 8 0 b. See graph at right. c. Too large; graph is downward sloping. d. R 6! Area! L 6 e. For : Left = Right = For 8: Left = Right = 7.36 f. The actual area is between the values of the Left Endpoint and Right Endpoint rectangles. As the number of rectangles approaches infinity, the actual area is squeezed between the two values. CL a. lim f () = b. The lim!0 f () does not eist. c. lim "3"4! = "3"4 "8 #"8 = "6 ("4)(+) (+) = d. lim = lim = 5 "6!4 (" 4)!4 8 CL a. lim!5 g() = lim!5 g() = "3.5 = 7 b. g() lim =!5 lim g() = 3.5 = 6.75!5 c. lim (g() + 0) = lim g() + lim 0 = = 3.5!5!5!5 d. The lim g() is unknown as we don t have enough information.!6 CL a. lim + 5 n!" n ( ( )) n = e 5 b. lim ( ( ) n ) n = e = e n!" + CL a. 5e = 30 e = 6 ln e = ln 6 = ln 6 c. e 3! 6 = e 3 = 8 ln e 3 = ln 8 3 = ln 8 = ln 8 3 b. ln( + 4) = 3 e ln(+4) = e = e 3 = e 3! 4 d. 5 ln! 7 = 8 5 ln = 5 ln = 5 e ln = e 5 = e 5 CPM Educational Program 0 Chapter 8: Page 3 Pre-Calculus with Trigonometry

33 CL a! a + a 4! a ( ) ( ) n a n = a! + 4! 8 + a n = a! S = a!(! ) = a 3 = 3 a 4 = 3 a a = 36 CL 8-6. a. + sin lim = b. No, because b() approaches infinity. c. lim (#a)4 (#b) (#c) (#a) 3 (#b) (#c) $ 7 +! 7 = +! CL 8-6. lim! 4 " 3 + " " 3 (")( = lim 3 +"8) = ()3 +"8 = " 5! 3(") 3 3 CL T = 3 T = T! = (3)! = 5 T 3 = T! = (5)! = 8 T 4 = T 3! 3 = (8)! 3 = 3 T 5 = T 4! 4 = (3)! 4 = CL a. P(t) = 000e 0.t P(7) = 000e 0.!7 P(7) = 4055 CL sec +tan + sec!tan = (sec!tan ) (sec +tan )(sec!tan ) + (sec +tan ) (sec!tan )(sec +tan ) = sec!tan sec!tan + sec +tan sec!tan = sec!tan +sec +tan sec!tan b. 0, 000 = 000e 0.t 0 = e 0.t ln 0 = ln e 0.t ln 0 0. = t t! days from his measurement. CPM Educational Program 0 Chapter 8: Page 33 Pre-Calculus with Trigonometry = sec = sec

34 CL a. See diagram at right. b. V = 30 =!! h 30 = h h = 5 SA =!! +!! h +!! h h SA = 4 + 4h + h = 4 + 6h ( ) SA = 4 + 6h = SA = ( ) = CL m = 7!0!0 = 7 tan" = 3.5 tan! tan" = tan! 3.5 " = 74.! d = (! 0) + (7! 0) = = 53 CL ()+60()+0(.5) 3.5 = 4.86 mph CPM Educational Program 0 Chapter 8: Page 34 Pre-Calculus with Trigonometry