Chapter 5: Introduction to Limits

Transcription

1 Chapter 5: Introduction to Limits Lesson Decreases 4. Decreases 5. y = 5-3. a. y = k b. 3 = k! 3 4 = k c. y = 3 4!("3) # y = 3 4! 9 = 7 4 y = 3 4 Review and Preview f () = f () = 5-5. f () = k(! 4) = k(3! 4) = 5k 5 = k f (4) = 5 (4! 4) = 5 (6! 4) = 5 " = 4 5 = y = k = k 3! "! =!0.38 = k y =! y =!0.38 =! =!0.3 CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry

2 5-7. a. sin! " 3 ( ) =! sin " 3 =! 3 ( ) = cos " 3 = ( ) =! tan " 3 =! 3 cos! " 3 tan! " 3 csc (! " 3 ) = sec (! " 3 ) = cot (! " 3 ) = =! 3 sin(! " 3) =! 3 ( ) = =! 3 =! 3 3 = cos! " 3 tan(! " 3) =! =! b. sin!3" ( ) =! sin 3" =!0 = 0 ( ) = cos 3" =! ( ) =! tan 3" =!0 = 0 cos!3" tan!3" csc (!3" ) = sec (!3" ) = cot (!3" ) = sin(!3" ) = 0 # und. cos(!3" ) =! =! tan(!3" ) = 0 # und Length of interval = 3! (!) = 4 3 " h() d = " f () d + 5 # 4 = = 37! 3! 5-9. Volume of Florida grapefruit = 4 3! " r3 Diameter of Teas grapefruit = r where r = diameter of Florida grapefruit. ( ) Volume of Teas grapefruit = 4 3! ( r )3 = 4 3! 8r3 = 8 " 4 3! " r3 # Teas grapefruit is 8 times as big as Florida grapefruit, but only costs 7 times as much, thus making it the better deal ! a. 6,! 6 b. 5! 6,! a. b. c. (+)(+)(+) (+) = ( + )( + ) = ( + )! " (!)(!3) (!)(!) =!3! " (+ y)(! y) (+y) = (! y) "!y 5-. y = k + a!!!!! = length of tube!!!!!a = cross-section area of tube CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry

3 Lesson Numerator and denominate must be polynomials. and are not polynomials a. 0 7 = 3 7 b. 9 7 = = = a. f () = b. f () = This is the graph of f () shifted units to the left and 3 units up g() = + 3! a. b. Graph b. h() =!5!3 = (!3)+!3 = (!3)!3 +!3 = +! It is the graph of y = shifted up units and right 3 units. y =! Multiply the numerator and denominator by y a.! " y +y " y " +y = y y! y + y y " + 3 y = y3 + y + 3 y 3 b. y 3 y 3! y 3 +y "3 y "3 = y y 3 CPM Educational Program 0 Chapter 5: Page 3 Pre-Calculus with Trigonometry

4 5-0. a. u = + y = y y + y = +y y b. c.! " # +y y ' $! % & = " # y +y $ % & = (y ) ( +y ) = y 4 ( +y ) y +y 5-. a. u = + y! = + y = y+ y u! = u = y+ y = y y+ c. u = ab! + a! b = a b + b a = a3 +b a b u! = u = " # $ a 3 +b a b % & ' ( ) = a 4 b = a b a 3 +b (a 3 +b ) b. u = ab! + a! b = a b + b a = a3 +b a b u! = u = a 3 +b a b = a b a 3 +b Review and Preview Eample: Vertical asymptote at =!7 " y = +7 Horizontal asymptote at = 5 " y = (3!) (3!)! (3!) (3!) + 3 (3!) = 6!!3++3 = 6!5+4 (3!) (3!) 5-4. a. Any angle in the 4 th quadrant will satisfy this. Eamples:! 6, "! 3 b. sin =! " = 7# 6, # 6 c. Any angle in the 3 rd quadrant will satisfy this. Eamples: 7! 6, 5! 4 d. Approimately 6! 7 e. Not possible, it does not satisfy the Fundamental Pythagorean Identity. cos! + sin! = (0.9) + (0.8) = =.45 " CPM Educational Program 0 Chapter 5: Page 4 Pre-Calculus with Trigonometry

5 5-5. a. y = k!3 = k!! = k 5-6. y =!!3 b. f () =!!3 f (!3) =! (!3)!3 =! 6 =! 3 f f () =!!3 =! =! ( ) =! f a. A(d) = k d b. A(d) = k (3d) = ( ) =!!3 4! 4 =!!! ( a ) = (! a) =!!3 a!3a a k 9d = 9! k d! 4 =! "! 4 =! "! a It decreases 9 ( ) = 8 ( ) =!a!3a as much.!3a 5-7. a. sec! = " cos! = " = " cos! " = cos!! = # 3, 4# 3 b. cot! = 3 tan! = 3 = 3 tan! 3 = tan!! = " 6, 7" a. b. c ! = ! = " + y! y = + y y! y = + y " 4 y + 3 y! 3 = + y 4 y! 3 3! = ! y y! = y + y 4 4 y! = + y 4 y " 3! = + y 3 y! y CPM Educational Program 0 Chapter 5: Page 5 Pre-Calculus with Trigonometry

6 Lesson The -intercept of the first function is the -value of the vertical asymptote of the second( = 3) a. =, =! b. Yes, y = 0 c. <! or > d.! < < 5-3. f () = sec()! a. (!n,) and (! +!n, " ), n an integer. b.! " + "n, n an integer c. Range: cos! [",]; sec! y # or y $ " 5-3. f () See graph below. Cosecant: Domain:! "n, n an integer Range: y! " or y # CPM Educational Program 0 Chapter 5: Page 6 Pre-Calculus with Trigonometry

7 Review and Preview a. T = 0.075P b. k = a. f (00) = (00) = 03 0 =.0099 f (000) = (000) = =.000 f (0000) = (0000)+3 = =.000 c. Yes, since! ", = " is the vertical asymptote. b. y = d. f () = +3 + = + + This is the graph of, shifted one unit left and two units up Possible answer: g() = ( + )(! ) c + = 5! 0c + c 0 = c! c = (c! 8)(c! 3) c = 3, 8 a() = h(), b() = f () Check 3 + = 5! 3 = 8 + = 3 " 5! 8 =!3 Solution : c = a. d = (5! (!)) + (!! 5) d = = 98 = 7 b. slope = m =!!5 5!(!) =!7 7 =! Point-Slope Form: y! 5 =!(! (!)) y! 5 =!( + ) OR y! (!) =!(! 5) y + =!(! 5) Slope-Intercept Form: y + =!(! 5) y + =! + 5 y =! + 3 CPM Educational Program 0 Chapter 5: Page 7 Pre-Calculus with Trigonometry

8 Lesson f () =! + 3!" f () = a. He thinks that he will get to 3. b. She thinks that she will get to a. She thinks she is getting infinitely far up. b. + means approaching from the right. means approaching from the left. c. f () =,!3 "! f () =!4! a. f () = #!" b. f () = "#!4 " c. f () = +"! a.!" ( ) 3 3 = 0 b.!" ( ) 54 = " c. d.!" sin No it. sin oscillates between and, but does not approach any specific number. e. An eample is f () = cos a. g() = = (+3)! (+3) =! +3 + b. g()!" =,! g()!#3 # = ",! g()!#3 + = #"!" ( ) = 0 must approach! from the left and must approach!" from the right, therefore we do not need etra notation. cannot approach! from the right. # 9 CPM Educational Program 0 Chapter 5: Page 8 Pre-Calculus with Trigonometry

9 Review and Preview ! f () = + 3 = 7 In the first, the it is approaching, and in the second, the it is approaching from the left side. y = k!" = 0 As! ", gets smaller and smaller. a. If u = sin! " u3 #4u u = 3 b. u 3! 4u = 3u c. sin! = " or sin! = ! = 3#,!!!!!!!! sin! $ 4 a. 6 3 b. 3n 0n!5 u 3! 3u! 4u = 0 u(u! 3u! 4) = 0 u(u! 4)(u + ) = a. S(r) = kr b. 6! = k "() 6! = 4k a. b. 5.5 k = 4! S(r) = 4! " r u =!, 0, 4 But u = sin" # 0 So u =!, 4 SA = S(3) = 4! " 3 3 = 36! CPM Educational Program 0 Chapter 5: Page 9 Pre-Calculus with Trigonometry

10 Lesson f. If the procedures are followed accurately, all the last acute angles should be very close to 60º. g. They should say 60º. Once this angle is reached, the obtuse angle will be 0º. This is, of course, interesting since theoretically this angle is never reached. When it is bisected the resulting angle will equal 60º. Do not simply accept as justification that the pattern continues a. They are alternate interior angles. b. 80!! 0! = 60! ; a = 60! = 80! c. n Measure of a n d is obtained by n = 8. Students can repeat this process by using the ANS key on their calculator. Enter 0 and push Í. At the net line use (80 Z)/. Repeatedly hitting Í will generate the sequence. e.!" a n = 60 f. The measure approaches 60º n Angle an Measure of angle a Error: 60 angle measure a 0 40 a a a a a a a CPM Educational Program 0 Chapter 5: Page 0 Pre-Calculus with Trigonometry

11 Review and Preview g() = a 0 < a < a > Eample Sketch g()! " 0! g()! 0! "# f () =! 3!" ( # 3) = " # 3 = " ( # 3) = 0 # 3 = #3! #" The it is 60; it will never actually equal 60 unless the original acute angle was 60º a. S = kp m b. 60 c. The score would be infinite sin 60! = They are the same because 60!! = 3. sin! 3 = a. 4 and 5 should be multiplied, not added. log 4 + log 5 = log 0 b. Base should not change. log log 3 7 = log 3 49 c. Log of a product is the SUM of the logs. log(4! 5) = log 4 + log 5 d. No rule for log(4 + 5) f () =! 3, asymptotes at = and y =!3! CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry

12 5-65. a. Closest that Jade can be is when sin! = ".! d = 5(") + 35 = 0 feet. Farthest away that Jade can be is when sin! =.! d = 5() + 35 = 60 feet ( ) + 35 ( ) ( ) sin (!"t ) b. 35 = 5 sin!"t 5 d. 0 = 5 sin!"t 5 0 = sin!"t 5 sin # 0 = sin #! =!"t 5 30! =! "t ( 5 ) t = 30 seconds c. 60 = 5 sin!"t 5 5 = 5 sin!"t 5 sin # = sin # 5! "! =!"t 5 " 5! ( ) + 35 ( ) sin (!"t ) ( 5 ) t = 5 = 7.5 seconds Second time = 30 sec sec = 37.5 sec a. b. y 3 + y 3 + = 4 y y 3 y 3 = 4 y 3 +! + + = =! + 3 = y 3 + y 3 = 4 y 6 + y y 3 CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry

13 Lesson a. He thinks the height will be y =. b. She thinks she will approach y =. Benny: f () = ;!3 " Bertha:!3 + # c. f () =! + 4 " 3 $ %&! 3 > 3 f () = 3! Right- and left-hand its must be equal for a it to eist #%! for " a. f () = $! b. i.!", ii.!, iii. 4, iv. &% (! ) + 4 for > c. It does not eist since the left hand it and right hand it are not the same # Eample: f () =! + 4 " 3 $ %&! 3 > 3 f() y 5-7. a. f () = 6., f () = 6.! 3 "! 3 + b. The it eists since the its as approaches 3 from both the left and right sides are equal. Review and Preview !3!3 f () = ", f () = " "!3 + f () does not eist as the its from the left and right are not equal f () =!!3 +!"# f () = "!"# "3 + = "0 + = CPM Educational Program 0 Chapter 5: Page 3 Pre-Calculus with Trigonometry

14 !" # 4 = " +!3 % # $! +! & ' = = = ( + = = a, 5 = 4 b6 b 6!b9 5 = 4b 3, 5 4 = b3 b = 3 5 " 0.709, a = ( 4 ) 6 " a. Possible estimate:! g() " b. it a. " 9 = 0!3 b.!3 + "9 = # c.!3 " = "# d. "9!3 Limit does not eist as left and right its are not equal. "9 CPM Educational Program 0 Chapter 5: Page 4 Pre-Calculus with Trigonometry

15 Lesson a. Yes, the it eists, since the it from the left and right are equal. g() = b. h() =!3!3 c. h(3) = 3 d. No, the it is the number f() approaches as gets closer and closer to a a. No, f( 3) and the it do not eist. b. Continuous c. No, it f(). d. No, f(4) does not eist a. b. c. y y y a a a a.! 0 b. c. sin = 0, = 0!0!0 The numerator and denominator of f() are both approaching 0. d.!0 sin = a. n!, n " 0 b. f () = sin oscillates as! " but gets closer to y = 0. c. The line y = 0 IS a horizontal asymptote. d.!0 sin = CPM Educational Program 0 Chapter 5: Page 5 Pre-Calculus with Trigonometry

16 Review and Preview "!, f () = 3!3! Vertical Asymptotes of f () # =!, =!3!" = = sin = sin" " = " = " k 0 = 0!h k(0! h) = 0 k(0!h) 0 = (radius of cross-section) Area of cross-section =! ( k(0"h) 0 ) =! 00 k (0 " h) a. cos 60! = b. sin 3! 4 = c. sin 3! = " d. cos 90! = 0 CPM Educational Program 0 Chapter 5: Page 6 Pre-Calculus with Trigonometry

17 5-90. a. csc + sec! tan! cot = (csc! cot ) + (sec! tan ) = + = = b. sec!!cos = sec sec!!cos ( ) = sec +cos +cos sec +!!cos!cos sec!cos sin!cos cos sin = sec = sec = sec sin cos sin = sec cos = sec 5-9. log A + log B = 4! log (AB) = 4 log 3 A " log 3 B =! log 3 A B 4 = AB 3 = A B AB = 6 9 = 6 B ( ) ( ) = 6 B ( ) = A = 6 B B = 4 3 A = 6 4/3 = 5-9. a. A = kw 5!GPA 4.40 = 0k 5!3.5 = 0k.5 = 0 3 k " k =.6 A =.6w 5!GPA For = : (!) + a = (!)!!4 + a = a = 6 b. 0 =.6(5) 5!GPA 5! GPA =.6(5) 0 =.6 GPA = 5!.6! 3.38 For = 3: 3! =!3 + b 7 =!3 + b 0 = b CPM Educational Program 0 Chapter 5: Page 7 Pre-Calculus with Trigonometry

18 Lesson i.!"# iv.!" f () = 0 ii. f () =!" " iii. f () = 3 f () = no it v.! f () = 3!" + f () = #" vi.!" 5-95.!"#!" f () = 0,!"4 f () =,! f () = #, f () = #, f () does not eist "!"4 +!"4 f () = ", f () =, f () does not eist " +! When is very small, AC and AB! are almost the same length and parallel. Therefore as changes, the lengths of AC and AB! change at similar rates.! a. cos() approaches while approaches 0, so this is little = big. b. approaches 0 while approaches, so this is big a. 0 b. Both approach 0. c. f () = 0 d. Yes, y = 0.!0 Review and Preview " cos!0 = a.! + f () = 3 b.! " f () = c.! f () The it does not eist because the its from the left and the right sides are different. CPM Educational Program 0 Chapter 5: Page 8 Pre-Calculus with Trigonometry

19 5-0. Eample (one of many): 5-0. a. = 4! b. f () = 4!5 Therefore the graph is continuous at = Find the area under the curve. Distance traveled in the first five seconds: 0! 5 = 50 feet Distance traveled in the t 5 seconds: (0 + t)(t! 5) = 00 feet (0 + t)(t! 5) = 400 0t! 50 + t! 0t = 400 t = 450 t = 5 t = a. y = +!3 b. y =, = 3 d. Vertical asymptote because!.5. We know the function will be approaching positive or negative infinity. f (.5) = 4(.5)!5 = 5!!"!! (.5)!3 = +$ #.5 + c. 4/#5/!" /#3/ = 4!" = a. b.! c.! ",, 3 d.!"# h() = g() is the graph of f () with a vertical shift of 3 units. Therefore Bertha thinks that she is getting closer to Benny s height + 3, or + 3 = 5 units CPM Educational Program 0 Chapter 5: Page 9 Pre-Calculus with Trigonometry

20 a. h = kd g b..4 = k(70!50) = 0k = k c. h = 0.036(70!30).5 = 0.036(40).5 =.44.5 = 0.96 BTU CPM Educational Program 0 Chapter 5: Page 0 Pre-Calculus with Trigonometry

21 Closure Problems CL a. f () # = "$!!!!!! " 0.999" f () # 6! = $ Therefore f () does not eist. "! b. f () = 6! = 6 " 3 = c. /+/ f () =!"# /"/ = = CL Yes, for all. CL 5-0. y =!! 8 = (! 4)( + ) This is a parabola opening up with -intercepts at 4 and. Therefore f () will have vertical asymptotes at = 4 and =! CL 5-.!0 CL 5-. f () = "sin " +sin = (+sin )"("sin ) ("sin )(+sin ) = sin "sin = sin cos = sin # cos = # cos (0) = a. b. c. d. 3 e. f. CL 5-3. m = kpq t k(5)(0)!!!!!7 =!!!!!k = = 7 5 = 7 p( p) = 4 5 p 63!5 4 = p 9!5 = p 3!5 = 5 = p CL 5-4. a. f (0.000) = f (!0.000) =!0. b. f () = 3+! 3 = c. "!0 9+3(0) = " 9 3!(3+) 3(3+ ) =! 3(3+ ) =! 9+3 CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry

22 CL 5-5. They will have to agree that they think they are headed to the same place. CL 5-6. a. 3(!)y!! (!) + y! =!3y! + + y! = CL 5-7. a. 3 b. = 3 c.!" #5 + # 3 = #5 y = #5!y! =! y = y = y b. 3 y! + y = 3! y + = y 3 + = y + y 3 + = y( + ) = y CL 5-8. a. See graph at right below. b. Find the area under the curve from t = 0 to t =. c. He is approimating the area under the curve by using right-endpoint rectangles with width = 0.. 0! d. 6(0.k) k= e. 0.5[6(0) + (0.) + (0.) (.9) + 6()] f. Right-endpoint rectangles give 7. mi, left=4.8, trap=6.0 5 ft/sec seconds CPM Educational Program 0 Chapter 5: Page Pre-Calculus with Trigonometry