{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 23 "Trigonometric Functions" }} {PARA 4 "" 0 "" {TEXT 26 19 "Precalculus Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 1 " " }{TEXT 256 10 "Objectives" }}{PARA 0 "" 0 "" {TEXT -1 83 "To study the graphs of y = a + b sin(cx + d) for various values of a, b, c and d. " }}{PARA 0 "" 0 "" {TEXT -1 76 "To study period, \+ amplitude, phase shift and vertical and horizontal shifts." }}{PARA 0 "" 0 "" {TEXT -1 53 "To solve word problems using trigonometric func tions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Use graphical capabilities of " }{TEXT 275 5 "Maple" }{TEXT -1 34 " t o graph trigonometric functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 257 17 "Solved Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 47 "(a) Draw the graphs of sin(x) for two periods." }}{PARA 0 "" 0 "" {TEXT -1 69 "(b) Draw the graphs of sin(x) and sin (2x) on the same set of axes." }}{PARA 0 "" 0 "" {TEXT -1 87 "(c) Draw the graphs of sin(x) a nd -2 + sin (x) for two periods on the same set of axes." }}{PARA 0 " " 0 "" {TEXT -1 44 "(d) Draw the graphs of sin(x) and sin (x + " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 44 "/3) for two periods on the same set of axes." }}{PARA 0 "" 0 "" {TEXT -1 85 "(e) Draw the graph s of sin(x) and - sin (x) for two periods on the same set of axes." }} {PARA 0 "" 0 "" {TEXT -1 85 "(f) Draw the graphs of sin(x) and 4 sin \+ (x) for two periods on the same set of axes." }}{PARA 0 "" 0 "" {TEXT -1 77 "(g) Draw the graphs of sin(x) and 1 + 2sin (3x + 4) on the sam e set of axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 10 " Solution:" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 258 16 "Graph of sin (x) " }}{PARA 0 "" 0 "" {TEXT 259 45 " First define the sine function and \+ graph it." }}{PARA 0 "" 0 "" {TEXT 260 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "f:= x -> sin (x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(f(x),x=-2*Pi..2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The period of the function sin x is 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 ". (2" }{XPPEDIT 18 0 "Pi; " "6#%#PiG" }{TEXT -1 51 " ~ 6.28) That means the function starts rep eating " }}{PARA 0 "" 0 "" {TEXT -1 7 "every 2" }{XPPEDIT 18 0 "Pi;" " 6#%#PiG" }{TEXT -1 37 " interval. In other words sin (x + 2" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 17 ") = sin x, and 2" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 29 " is the smallest such numb er " }}{PARA 0 "" 0 "" {TEXT -1 92 "for which you get one complete cyc le of sin x. Two cycles have been plotted in the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The maximum value of the graph is 1. The maximum value is called the amplitude of sin( x). " }}{PARA 0 "" 0 "" {TEXT -1 75 "The domain of sin(x) is all reals . The range of the function is [-1, 1]. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 261 27 "Graph \+ of sin(x) and sin(2x)" }}{PARA 0 "" 0 "" {TEXT -1 20 "Define the funct ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g := x -> sin(2*x); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{f(x),g(x)\},x=-2*Pi..2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Observe that the pe riod of the function sin 2x is " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 37 ". The above graph shows four cycles " }}{PARA 0 "" 0 "" {TEXT -1 65 "of sin(2x) and only two cycles of sin(x) in the same inte rval [ -" }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 20 " ] . The amplitude " }}{PARA 0 "" 0 "" {TEXT -1 16 "of sin(2x) is 1." } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "To find this manually use the fact that sin(2x + 2" }{XPPEDIT 18 0 "Pi;" "6#% #PiG" }{TEXT -1 15 ") = sin (2x) " }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ sin (2(x + \+ " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 ")) = sin(2x) " }}{PARA 0 "" 0 "" {TEXT -1 17 " In other words " }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT -1 73 " is the smallest interval for which you get a comple cycle of sin(2x). " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 262 31 "Graph of sin(x) and -2 + sin(x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g:= x-> -2 + sin(x);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(\{f(x), g(x)\}, x = -2* Pi..2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 57 " Note that the period of the two functions is the same, 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 36 " . But adding -2, the graph sin(x) " }}{PARA 0 "" 0 "" {TEXT -1 61 "has a vertical shift -2 to get the graph of -2 + sin(x). " }}{PARA 0 "" 0 "" {TEXT -1 19 "The amplitude is 1." }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 263 28 "Graph of sin(x) and sin(x + " }{XPPEDIT 18 0 "2*Pi/3;" "6#*(\" \"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 264 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g:=x->sin(x + Pi/3 );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{f(x),g(x)\}, x=-2*Pi..2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 22 " The period of sin( x +" }{XPPEDIT 18 0 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 7 " \+ ) is 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 57 ". The amplitude \+ is 1. The graph is obtained from sin(x)" }}{PARA 0 "" 0 "" {TEXT -1 27 "by a horizontal shift of - " }{XPPEDIT 18 0 "Pi/3;" "6#*&%#PiG\"\" \"\"\"$!\"\"" }{TEXT -1 24 " . " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 265 27 "Graph of sin(x) and -sin(x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := x -> -sin(x); \n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{f(x),g(x)\},x=-2*Pi.. 2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 86 "Observe that changing the si gn of the function sin x gives a reflection in the x-axis." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " } {TEXT 266 28 "Graphs of sin(x) and 4sin(x)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "g := x -> 4*sin(x); \n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "plot(\{f(x),g(x)\},x=-2*Pi..2*Pi);\n" }}}{PARA 0 " " 0 "" {TEXT -1 15 "The period is 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 38 " and the amplitude of 4sin(x) is 4. " }}{PARA 0 "" 0 " " {TEXT -1 73 " Observe that multiplying sin x by a constant >1 increa ses the amplitude " }}{PARA 0 "" 0 "" {TEXT -1 82 "by the constant. ( Vertical stretch by the constant). Similarly multilying sin x " }} {PARA 0 "" 0 "" {TEXT -1 66 "by a positive constant < 1 will result in shrinking the amplitude." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 267 0 "" }{TEXT 268 39 "Graphs o f sin(x) and 1 + 2*sin(3*x + 4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := x -> 1 + 2*sin(3*x + 4) ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{f(x),g(x)\},x =-2*Pi..2*Pi);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Observe that the graph o f y = sin x is changed into the graph of g by a vertical shift of 1, r educing the period to " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 94 "/3 , taking a horizontal shift of 4, and changing the amplitude to 2. In the same interval [-2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 ", \+ 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 57 "], there are two cycle s of sin x and six cycles of g(x) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 269 17 "Solved Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ Blood Pressure: The function P = 100 - 20 cos (5" }{XPPEDIT 18 0 "Pi " "6#%#PiG" }{TEXT -1 99 " t/3) approximates the blood pressure P in m m of mercury at time t in seconds for a person at rest." }}{PARA 0 "" 0 "" {TEXT -1 39 " (a) Find the period of the function." }}{PARA 0 " " 0 "" {TEXT -1 48 " (b) Find the number of heartbeats per minute." }}{PARA 0 "" 0 "" {TEXT -1 61 " (c) Use a graphing utility to graph \+ the pressure function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 270 11 "(a) Period " }} {PARA 0 "" 0 "" {TEXT -1 67 "The period of this function is found by c onsidering the fact cos (5" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 13 "t/3) = cos (5" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 8 " t/3 + 2 " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {TEXT -1 19 " After taking out 5" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 60 "/3 common from the two terms of the right argument we get " }}{PARA 0 "" 0 "" {TEXT -1 15 " cos(5" }{XPPEDIT 18 0 "P i" "6#%#PiG" }{TEXT -1 14 " t/3) = cos (5" }{XPPEDIT 18 0 "Pi" "6#%#Pi G" }{TEXT -1 13 "/3(t + 6/5))." }}{PARA 0 "" 0 "" {TEXT -1 20 " The p eriod is 6/5." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 271 24 "(b) Number of Heartbeats" }}{PARA 0 "" 0 "" {TEXT -1 78 "Since the period is 6/5 seconds, the number of he artbeats per second is 5/6. " }}{PARA 0 "" 0 "" {TEXT -1 65 " \+ For a minute the number of heartbeats is (5/6)(60) = 50." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 272 36 "(c) Graph of blood pressure function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:= t -> 100 - 20* cos(5*Pi/3* t); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( f(t),t=-2*Pi..2*Pi);\n" }}}}}{PARA 0 "" 0 "" {TEXT -1 67 "____________ _______________________________________________________" }}{PARA 257 " " 0 "" {TEXT 274 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 273 9 "Problems:" }}{PARA 0 "" 0 "" {TEXT -1 104 "Sketch by hand the two functions in each exercise on the same coordinate plane \+ and verify using Maple: " }}{PARA 0 "" 0 "" {TEXT -1 57 " 1. (a) f (x) = sin x (b) g(x) = sin (" }{XPPEDIT 18 0 "x/3;" " 6#*&%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 60 " 2. (a) f(x) = sin x (b) g(x) = cos (x - " } {XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 ")" }} {PARA 0 "" 0 "" {TEXT -1 26 " 3. (a) f(x) = - sin (2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 28 " x) (b) g(x) = 10 cos (" } {XPPEDIT 18 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 4 " x)" } }{PARA 0 "" 0 "" {TEXT -1 100 " 4. Fuel consumption: The daily consu mption C (in gallons) of diesel fuel on a farm is modeled by " }} {PARA 0 "" 0 "" {TEXT -1 55 " C = 30.3 + 21.6 sin ( " }{XPPEDIT 18 0 "2*Pi*t/365" "6#**\"\"#\"\"\"%#PiGF%%\" tGF%\"$l$!\"\"" }{TEXT -1 11 " + 10.9)." }}{PARA 0 "" 0 "" {TEXT -1 71 " where t is the time in days, with t = 1 corresponding to Janua ry 1." }}{PARA 0 "" 0 "" {TEXT -1 72 "(a) What is the period of the mo del? Is it what you expected? Explain." }}{PARA 0 "" 0 "" {TEXT -1 95 "(b) What is the average daily fuel consumption? Which term of the model did you use? Explain." }}{PARA 0 "" 0 "" {TEXT -1 101 "(c) Use Maple to graph the model. Use the graph to approximate the time of th e year when consumption " }}{PARA 0 "" 0 "" {TEXT -1 27 "exceeds 40 ga llons per day." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 73 "_________________________________________________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 82 "MSEIP Grant #P120AA01 0031: \"Four Colleges: Calculus + Enhancements\", 2001-2004 " }}} {MARK "8 0 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }