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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 22 "THE MEAN VALUE THEOREM" }}
{PARA 261 "" 0 "" {TEXT 280 18 "Calculus I Project" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" 
}{TEXT 257 9 "Objective" }}{PARA 0 "" 0 "" {TEXT -1 55 "1.  To underst
and when to apply the Mean Value Theorem." }}{PARA 0 "" 0 "" {TEXT -1 
67 "2.  To develop a graphical interpretation of the Mean Value Theore
m" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Usi
ng Maple allows the student to quickly plot a curve and then visually \+
interpret whether or not the Mean Value Theorem applies to the functio
n in a given interval." }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 42 "  1)  State the Me
an Value Theorem  (MVT)." }}{PARA 0 "" 0 "" {TEXT -1 69 "  2)  Given t
he piecewise defined function   ,   f(x) =     x,  x < 0" }}{PARA 0 "
" 0 "" {TEXT -1 83 "                                                  \+
                                 " }{XPPEDIT 18 0 "x^2-1;" "6#,&*$%\"x
G\"\"#\"\"\"F'!\"\"" }{TEXT -1 5 ",  x " }{TEXT 266 1 ">" }{TEXT -1 3 
" 0," }}{PARA 0 "" 0 "" {TEXT -1 33 "        a) plot the function     \+
" }}{PARA 0 "" 0 "" {TEXT -1 93 "        b) determine if the function \+
satisfies the Mean Value Theorem in (1, 3) and  (-1, 1) " }}{PARA 0 "
" 0 "" {TEXT -1 31 "        c) find all values of  " }{TEXT 270 1 "c" 
}{TEXT -1 56 "  that satisfy the conclusion of the Mean Value Theorem.
" }}{PARA 0 "" 0 "" {TEXT -1 113 "        d) write the equation of the
 secant line for the given interval and the equation of the tangent li
ne at  " }{TEXT 271 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
128 "        e) plot the original function, the secant line, and the t
angent line on one graph.  Are the secant line and tangent line" }}
{PARA 0 "" 0 "" {TEXT -1 21 "            parallel?" }}}{PARA 3 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "
" }{TEXT 260 0 "" }{TEXT 261 0 "" }{TEXT 262 9 "Solution:" }}{PARA 0 "
" 0 "" {TEXT -1 168 "If  y = f(x)  is continuous at every point of the
 closed interval  [a, b]  and differentiable at every point of its int
erior  (a,b), then there is at least one number  " }{TEXT 272 2 "c " }
{TEXT -1 8 "between " }{TEXT 273 1 "a" }{TEXT -1 5 " and " }{TEXT 274 
2 "b " }{TEXT -1 8 "at which" }}{PARA 0 "" 0 "" {TEXT -1 60 "         \+
                                                   " }{XPPEDIT 18 0 "(
f(b)-f(a))/(b-a);" "6#*&,&-%\"fG6#%\"bG\"\"\"-F&6#%\"aG!\"\"F),&F(F)F,
F-F-" }{TEXT -1 13 "  =  f '(c) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "To plot \+
the function, first define it as a piecewise function. Then to plot th
e function so Maple does not connect pixels at points of discontinuity
, use   " }{TEXT 275 14 "discont = true" }{TEXT -1 24 "    in the plot
 command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f:=x -> piecewise(x <
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6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&29$\"\"!F01F1F0,&\"\"
\"!\"\"*$)F0\"\"#F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "plot(f(x), x = -4..4, y = -4..4, title=\"piecewise function\", dis
cont = true,color=black);\n" }}{PARA 8 "" 1 "" {TEXT -1 26 "Plotting e
rror, empty plot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(\{f(x), 4*x -4, 4*x - 5
\}, x = 0..4, y = -2..10); \n" }}{PARA 13 "" 1 "" {GLPLOT2D 413 288 
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lF)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 85.000000 35.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The graphs of the secant on [1,
 3] and the tangent at x = 2, or the point (2, 3) on the graph, are pa
rallel." }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
83 "__________________________________________________________________
_________________" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }
}{PARA 258 "" 0 "" {TEXT 264 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 267 10 "Problem
 1:" }}{PARA 0 "" 0 "" {TEXT -1 31 "Given the function,   f(x) =   " }
{XPPEDIT 18 0 "2*x^3+x+3;" "6#,(*&\"\"#\"\"\"*$%\"xG\"\"$F&F&F(F&F)F&
" }}{PARA 0 "" 0 "" {TEXT -1 33 "        a) plot the function     " }}
{PARA 0 "" 0 "" {TEXT -1 83 "        b) determine if the function sati
sfies the Mean Value Theorem in  (-1, 1)  " }}{PARA 0 "" 0 "" {TEXT 
-1 30 "        c) find all values of " }{TEXT 276 3 " c " }{TEXT -1 
55 " that satisfy the conclusion of the Mean Value Theorem." }}{PARA 
0 "" 0 "" {TEXT -1 114 "        d) write the equation of the secant li
ne for the given interval and the equations of the tangent line at  " 
}{TEXT 277 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 130 "    \+
    e) plot the original function, the secant line, and the tangent li
nes on one graph.  Are the secant line and tangents line" }}{PARA 0 "
" 0 "" {TEXT -1 21 "            parallel?" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 268 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 
54 "Given the piecewise defined function   ,   f(x) =     " }{XPPEDIT 
18 0 "x^3-2;" "6#,&*$%\"xG\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 8 ",  x < 2
" }}{PARA 0 "" 0 "" {TEXT -1 75 "                                     \+
                                      " }{XPPEDIT 18 0 "x^2;" "6#*$%\"
xG\"\"#" }{TEXT -1 5 ",  x " }{TEXT 269 1 ">" }{TEXT -1 3 " 2," }}
{PARA 0 "" 0 "" {TEXT -1 33 "        a) plot the function     " }}
{PARA 0 "" 0 "" {TEXT -1 96 "        b) determine if the function sati
sfies the Mean Value Theorem in  (-1, 1)  and  ( 3, 7) " }}{PARA 0 "" 
0 "" {TEXT -1 31 "        c) find all values of  " }{TEXT 278 1 "c" }
{TEXT -1 56 "  that satisfy the conclusion of the Mean Value Theorem.
" }}{PARA 0 "" 0 "" {TEXT -1 113 "        d) write the equation of the
 secant line for the given interval and the equation of the tangent li
ne at  " }{TEXT 279 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
122 "        e) plot  f(x), the secant line, and the tangent line on o
ne graph.  Are the secant line and tangent line parallel?" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 10 "Problem 3:" }}{PARA 256 "" 0 "
" {TEXT -1 15 " Given f(x) =  " }{XPPEDIT 18 0 "x/(x-4);" "6#*&%\"xG\"
\"\",&F$F%\"\"%!\"\"F(" }{TEXT -1 3 " , " }}{PARA 259 "" 0 "" {TEXT 
-1 58 "a)  determine an interval in which the MVT does not apply." }}
{PARA 0 "" 0 "" {TEXT -1 68 "b)  plot the function and use the graph t
o defend your answer in a)." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 71 "_______________________________________________
________________________" }}{PARA 0 "" 0 "" {TEXT -1 94 "MSIP Grant #P
120A80089-98: \"Three Urban Calculus Reform Programs: Adopting the Bes
t\" 1998-2001" }}}{MARK "0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }