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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 45 "Derivatives -- Introduct
ion -- Curved Mirrors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 67 "The following cell draws a concave mirror described by
 the function" }}{PARA 257 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 256 30 "mi
rror(x) = 8 - sqrt(64 - x^2)" }}{PARA 0 "" 0 "" {TEXT 257 1 "\n" }
{TEXT -1 34 "\nand the light caustic formed by  " }{TEXT 258 1 "n" }
{TEXT -1 80 "  evenly spaced light rays coming down vertically and bou
ncing off the mirror.  " }{TEXT 259 39 "Evaluate it now.  It may take \+
some time" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1271 "with(plots):\nwith(plottools):\n
\nmirror := x -> 8 - sqrt(64 - x^2):     # Mirror\n\nslope := x -> x/s
qrt(64 - x^2):        # slope at each point on mirror\n\nwidth := 5.0:
                          # x = -slope..slope\n\nheight := 2 * width: \+
                  # y = 0..height\n\nn := 30:                         \+
      # number of light rays\n\nunassign('x'):\nunassign('y'):\n\nblk \+
:= plot(mirror(x), \n            x = -width..width, y = 0..height, \n \+
           axes = NONE, scaling = CONSTRAINED, color = black):\n\nrays
 := \{\}:\n\nfor i from 0 to n do\n    x := -width + 2 * i * width/n:
\n    y := mirror(x):\n    rays := rays union \{line([x, height], [x, \+
y])\}:   # incoming ray\n    dr := slope(x):\n    if (abs(dr) > 0.001)
 then          # if slope = 0 no outgoing ray\n       m := (dr^2 - 1)/
(2 * dr):\n       ix := (height - y)/m:\n       if ix < -width then\n \+
         tx := -width:\n          ty := y + m * (tx - x):\n         el
if\n          ix > width then\n          tx := width:\n          ty :=
 y + m * (tx - x):\n         else\n          ty := height:\n          \+
tx := (ty - y)/m + x:\n        fi:\n    rays := rays union \{line([x,y
], [tx, ty])\}:  # outgoing ray\n    fi:\nod:\n\nrd  := display(rays, \+
color = red, axes = NONE, scaling = CONSTRAINED):\n\nunassign('x'):\nu
nassign('y'):\n\ndisplay(\{blk, rd\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "You can modify the cell \+
above to try several different experiments.  You will need to change b
oth the defintion of the function" }}{PARA 258 "" 0 "" {TEXT -1 2 "\n
\n" }{TEXT 260 25 "mirror(x) = sqrt(64 - x^2" }{TEXT -1 2 ")\n" }}
{PARA 0 "" 0 "" {TEXT -1 76 "\n that represents the shape of the mirro
r and the definition of the function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 27 "slope
(x) = x/sqrt(64 - x^2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "that gives the slop
e of the mirror at each point on the mirror\n\nTry two different kinds
 of mirrors -- " }{TEXT 261 10 "elliptical" }{TEXT -1 43 " mirrors des
cribed by functions of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 36 "mirror(
x) = a * (8 - sqrt(64 - x^2))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }
{TEXT 262 1 "a" }{TEXT -1 20 "  is a constant and " }{TEXT 263 9 "para
bolic" }{TEXT -1 13 " mirrors like" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 18 "mirror(
x) = x^2/25" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "2 20 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }