Expression 20: "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "x" Subscript, "s" , Baseline , "y" Subscript, "s" , Baseline , right parenthesis equals left brace, 0 less than "t" less than "l" : "x" Subscript, "s" , Baseline "t" , "l" less than "t" less than StartFraction, "l" plus 1 Over 2 , EndFraction : "l" "x" Subscript, "s" , Baseline plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis minus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis , StartFraction, "l" plus 1 Over 2 , EndFraction less than "t" less than 1 : "l" "x" Subscript, "s" , Baseline plus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis , right bracexarrt,xs,ys=0<t<l:xst,l<t<l+12:lxs+cos5π6xst−l−sin5π6yst−l,l+12<t<1:lxs+sin5π6yst−l+12+cos5π6xst−l+12
20
Expression 21: "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "x" Subscript, "s" , Baseline , "y" Subscript, "s" , Baseline , right parenthesis equals left brace, 0 less than "t" less than "l" : "y" Subscript, "s" , Baseline "t" , "l" less than "t" less than StartFraction, "l" plus 1 Over 2 , EndFraction : "l" "y" Subscript, "s" , Baseline plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis plus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis , StartFraction, "l" plus 1 Over 2 , EndFraction less than "t" less than 1 : "l" "y" Subscript, "s" , Baseline minus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis , right braceyarrt,xs,ys=0<t<l:yst,l<t<l+12:lys+cos5π6yst−l+sin5π6xst−l,l+12<t<1:lys−sin5π6xst−l+12+cos5π6yst−l+12
21
Expression 22: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "x" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "x" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "x" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "x" Subscript, "y" , Baseline , right parenthesis , right parenthesisxarrt,alxx,alxy,yarrt,alxx,alxy
22
Expression 23: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "y" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "y" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "y" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "y" Subscript, "y" , Baseline , right parenthesis , right parenthesisxarrt,alyx,alyy,yarrt,alyx,alyy
23
Expression 24: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "z" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "z" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "z" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "z" Subscript, "y" , Baseline , right parenthesis , right parenthesisxarrt,alzx,alzy,yarrt,alzx,alzy
24
And, finally, we draw the graph in blue and green. Can you figure out what g(t) and h(t) do?
25
Expression 26: "g" left parenthesis, "x" , right parenthesis equals 2 "s" left parenthesis, StartFraction, floor left parenthesis, "x" times left parenthesis, "n" plus 1 , right parenthesis , right parenthesis Over "n" , EndFraction minus .5 , right parenthesisgx=2sfloorx·n+1n−.5
26
Expression 27: "h" left parenthesis, "x" , right parenthesis equals 2 "s" left parenthesis, mod left parenthesis, "x" times left parenthesis, "n" plus 1 , right parenthesis , 1 , right parenthesis minus .5 , right parenthesishx=2smodx·n+1,1−.5
27
Expression 28: left parenthesis, "x" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "f" "z" Subscript, "x" , Baseline , "x" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "f" "z" Subscript, "y" , Baseline left brace, sin left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesisxxgt+yxht+fzx,xygt+yyht+fzysinb<0
28
Expression 29: left parenthesis, "x" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis minus "f" "z" Subscript, "x" , Baseline , "x" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis minus "f" "z" Subscript, "y" , Baseline left brace, sin left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisxxgt+yxht−fzx,xygt+yyht−fzysinb>0
29
Expression 30: left parenthesis, "x" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "f" "z" Subscript, "x" , Baseline , "x" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "f" "z" Subscript, "y" , Baseline left brace, sin left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesisxxht+yxgt+fzx,xyht+yygt+fzysinb<0
30
Expression 31: left parenthesis, "x" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis minus "f" "z" Subscript, "x" , Baseline , "x" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis minus "f" "z" Subscript, "y" , Baseline left brace, sin left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisxxht+yxgt−fzx,xyht+yygt−fzysinb>0
31
Expression 32: left parenthesis, "x" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "f" "y" Subscript, "x" , Baseline plus "z" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis , "x" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "f" "y" Subscript, "y" , Baseline plus "z" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis left brace, cos left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisxxgt+fyx+zxht,xygt+fyy+zyhtcosacosb>0
32
Expression 33: left parenthesis, "x" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis minus "f" "y" Subscript, "x" , Baseline plus "z" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis , "x" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis minus "f" "y" Subscript, "y" , Baseline plus "z" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis left brace, cos left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesisxxgt−fyx+zxht,xygt−fyy+zyhtcosacosb<0
33
Expression 34: left parenthesis, "x" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "f" "y" Subscript, "x" , Baseline plus "z" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis , "x" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "f" "y" Subscript, "y" , Baseline plus "z" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis left brace, cos left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisxxht+fyx+zxgt,xyht+fyy+zygtcosacosb>0
34
Expression 35: left parenthesis, "x" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis minus "f" "y" Subscript, "x" , Baseline plus "z" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis , "x" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis minus "f" "y" Subscript, "y" , Baseline plus "z" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis left brace, cos left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesisxxht−fyx+zxgt,xyht−fyy+zygtcosacosb<0
35
Expression 36: left parenthesis, "f" "x" Subscript, "x" , Baseline plus "y" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis , "f" "x" Subscript, "y" , Baseline plus "y" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis left brace, sin left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisfxx+yxht+zxgt,fxy+yyht+zygtsinacosb>0
36
Expression 37: left parenthesis, negative "f" "x" Subscript, "x" , Baseline plus "y" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis , negative "f" "x" Subscript, "y" , Baseline plus "y" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis left brace, sin left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesis−fxx+yxht+zxgt,−fxy+yyht+zygtsinacosb<0
37
Expression 38: left parenthesis, "f" "x" Subscript, "x" , Baseline plus "y" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis , "f" "x" Subscript, "y" , Baseline plus "y" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis left brace, sin left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis greater than 0 , right brace , right parenthesisfxx+yxgt+zxht,fxy+yygt+zyhtsinacosb>0
38
Expression 39: left parenthesis, negative "f" "x" Subscript, "x" , Baseline plus "y" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis , negative "f" "x" Subscript, "y" , Baseline plus "y" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis left brace, sin left parenthesis, "a" , right parenthesis cos left parenthesis, "b" , right parenthesis less than 0 , right brace , right parenthesis−fxx+yxgt+zxht,−fxy+yygt+zyhtsinacosb<0