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In lecture, w e derived the equation of the tangent line for the function f(" } {TEXT 256 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\" \"#" }{TEXT -1 20 " at the point where " }{TEXT 257 1 "x" }{TEXT -1 19 " = 3 (that is, the " }{TEXT 258 1 "x" }{TEXT -1 70 "-coordinate of the point is 3). The equation of this tangent line is " }{TEXT 259 1 "y" }{TEXT -1 4 " = 6" }{TEXT 260 1 "x" }{TEXT -1 77 " - 9. Let's u se Maple to graph the following: (1) the graph of the function " } {TEXT 261 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\" #" }{TEXT -1 69 "; (2) the tangent point (3, f(3)) = (3, 9); and (3) t he tangent line " }{TEXT 262 1 "y" }{TEXT -1 4 " = 6" }{TEXT 263 1 "x " }{TEXT -1 5 " - 9." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f : = x -> x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(f(x), x =-6..6); funct := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plo t( [ [3, f(3)] ], style=point, symbol=box, color=black); tanpt := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(6*x - 9, x=-3..7, co lor=blue); tanline := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(fu nct, tanpt, tanline);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 99 "Let's see how we could use Maple to calcu late the slope of our tangent line at the point for which " }{TEXT 264 0 "" }{TEXT -1 0 "" }{TEXT 265 1 "x" }{TEXT -1 61 " = 3. Of cours e, we could simply type the following command:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Limit( (f(3+h)-f(3))/h, h=0); value(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "L et's see how we could find the slope of the tangent line step by step. This process would be useful for checking each step of our work in c alculating this slope." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "First, let's find f(3+" }{TEXT 269 1 "h" }{TEXT -1 11 ") and f(3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f(3+h); expand(%); f(3); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Now, let's subtract f(3+" }{TEXT 270 1 "h" }{TEXT -1 28 ") and f(3). So, let's type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f(3+h) - f(3); expand(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The next step wou ld be to factor out the " }{TEXT 266 1 "h" }{TEXT -1 65 " that is comm on to both terms. Let's store this answer to \"num\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(6*h+h^2); num := %;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "No w, divide by " }{TEXT 267 1 "h" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "num/h;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "Now, take the limit as " }{TEXT 268 1 "h" }{TEXT -1 11 " goes to 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(6+h, h=0); value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "Maple has a defined function called " } {TEXT 303 11 "rationalize" }{TEXT -1 197 " which will rationalize the \+ denominator of a fraction. However, in our work of evaluating limits \+ involving fractions, we need to be able to rationalize the numerator o f a fraction. The following " }{TEXT 304 12 "user-defined" }{TEXT -1 17 " function called " }{TEXT 305 14 "rationalizenum" }{TEXT -1 46 " w ill rationalize the numerator of a fraction:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rationalizenum := q -> 1/rationalize(1/q):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "F or any problem where you need to rationalize the numerator of a fracti on, copy and execute this command before you do the problem. If you t ype the " }{TEXT 282 7 "restart" }{TEXT -1 110 " command, this functio n will be cleared out of memory and you will need to copy and execute \+ the command again." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "2. In lecture, we derived the equation of the ta ngent line for the function f(" }{TEXT 271 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "sqrt(x);" "6#-%%sqrtG6#%\"xG" }{TEXT -1 21 " at the po int where " }{TEXT 272 1 "x" }{TEXT -1 19 " = 4 (that is, the " } {TEXT 273 1 "x" }{TEXT -1 70 "-coordinate of the point is 4). The equ ation of this tangent line is " }{TEXT 274 1 "y" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "1/4*x+1;" "6#,&*(\"\"\"F%\"\"%!\"\"%\"xGF%F%F%F%" } {TEXT -1 73 ". Let's use Maple to graph the following: (1) the graph \+ of the function " }{TEXT 275 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "s qrt(x);" "6#-%%sqrtG6#%\"xG" }{TEXT -1 70 " ; (2) the tangent point (4 , f(4)) = (4, 2); and (3) the tangent line " }{TEXT 276 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/4*x+1;" "6#,&*(\"\"\"F%\"\"%!\"\"%\"xGF%F %F%F%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f \+ := x -> sqrt(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f( x), x=0..9); funct := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot( [ [4, f(4)] ], style=point, symbol=box, color=black); tanpt := \+ %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(x/4 + 1, x=-2..1 0, color=blue); tanline := %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "displ ay(funct, tanpt, tanline);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 99 "Let's see how we could use Maple to calcu late the slope of our tangent line at the point for which " }{TEXT 277 0 "" }{TEXT -1 0 "" }{TEXT 278 1 "x" }{TEXT -1 61 " = 4. Of cours e, we could simply type the following command:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Limit( (f(4+h)-f(4))/h, h=0); value(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "L et's see how we could find the slope of the tangent line step by step. This process would be useful for checking each step of our work in c alculating this slope." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "First, let's find f(4+" }{TEXT 281 1 "h" }{TEXT -1 11 ") and f(4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f(4+h); expand(%); f(4); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "Notice in this problem, the " }{TEXT 283 6 "expand" }{TEXT -1 67 " command did not do anything because, in \+ this case, the expression " }{XPPEDIT 18 0 "sqrt(4+h);" "6#-%%sqrtG6#, &\"\"%\"\"\"%\"hGF(" }{TEXT -1 47 " can not be expanded. Now, let's s ubtract f(4+" }{TEXT 284 1 "h" }{TEXT -1 59 ") and f(4) and store this answer to \"num\". So, let's type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(4+h) - f(4); num := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The next step would be to divide by " }{TEXT 279 1 "h" }{TEXT -1 56 ". Let's store this fracti on to \"fract\". So let's type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "num/h; fract := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Now, we want to rationalize th e numerator of this fraction using the user-defined function " }{TEXT 285 14 "rationalizenum" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rationalizenum(fract);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Now, take the limit as " }{TEXT 280 1 "h" }{TEXT -1 11 " goes to 0:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "Limit(1/(2+sqrt(4+h)), h=0); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "3. In le cture, we found that the instantaneous velocity function of the positi on function " }{XPPEDIT 18 0 "s(t) = 4*t^2-18*t;" "6#/-%\"sG6#%\"tG,& *&\"\"%\"\"\"*$F'\"\"#F+F+*&\"#=F+F'F+!\"\"" }{TEXT -1 20 " was the f unction " }{XPPEDIT 18 0 "v(t) = 8*t-18;" "6#/-%\"vG6#%\"tG,&*&\"\") \"\"\"F'F+F+\"#=!\"\"" }{TEXT -1 134 " . Let's see how we could use M aple to this instantaneous velocity function. Of course, we could sim ply type the following command: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s := t -> 4*t^2 - 18*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Li mit((s(t+h)-s(t))/h, h=0); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "Let's see how we could find th e instantaneous velocity function step by step. This process would be useful for checking each step of our work in finding this function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "First, let's find s(" }{TEXT 311 1 "t" }{TEXT -1 1 "+" }{TEXT 309 1 " h" }{TEXT -1 8 ") and s(" }{TEXT 312 1 "t" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s(t+h); expand(%); s(t); " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "No w, let's subtract s(" }{TEXT 313 1 "t" }{TEXT -1 1 "+" }{TEXT 310 1 "h " }{TEXT -1 8 ") and s(" }{TEXT 314 1 "t" }{TEXT -1 19 "). So, let's \+ type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s(t+h) - s(t); exp and(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The next step would be to factor out the " }{TEXT 306 1 " h" }{TEXT -1 65 " that is common to both terms. Let's store this answ er to \"num\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "factor(8* t*h+4*h^2-18*h); num := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 319 5 "NOTE:" }{TEXT -1 26 " When Maple execute s the " }{TEXT 316 6 "factor" }{TEXT -1 98 " command, it also factors \+ out the 2 which is common to all terms. We only need to factor out th e " }{TEXT 317 1 "h" }{TEXT -1 52 " in order to do the next step which is to divide by " }{TEXT 318 1 "h" }{TEXT -1 24 ". Now, let's divide by " }{TEXT 315 0 "" }{TEXT 307 1 "h" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "num/h;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 5 "NOTE:" }{TEXT -1 17 " In dividing by " }{TEXT 320 1 "h" }{TEXT -1 240 ", Maple gives the answe r with the factor of 2, which was factored out above, multiplied back \+ through the parentheses. This is the form of the expression that we w ant for doing the last step which is to take the limit of this express ion as " }{TEXT 321 1 "h" }{TEXT -1 36 " goes to 0. Now, take the lim it as " }{TEXT 308 1 "h" }{TEXT -1 11 " goes to 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Limit(4*h+8*t-18, h=0); value(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Pr actice Problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 205 "For Problems 1 - 3, use Maple to graph the function, the tangent point, and the tangent line to the graph of the function at t he given point. These problems are examples that were worked in lectu re notes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "1. Function: f(" }{TEXT 286 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "sqrt(x+4);" "6#-%%sqrtG6#,&%\"xG\"\"\"\"\"%F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 " Tangent Point: (- 3, f(- \+ 3)) = (-3, 1)" }}{PARA 0 "" 0 "" {TEXT -1 19 " Tangent Line: " } {TEXT 287 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2*x+5/2;" "6#,&*( \"\"\"F%\"\"#!\"\"%\"xGF%F%*&\"\"&F%F&F'F%" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "2. Function: f(" }{TEXT 288 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "3*x^2+2*x-20;" "6#,(*&\" \"$\"\"\"*$%\"xG\"\"#F&F&*&F)F&F(F&F&\"#?!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 40 " Tangent Point: (2, f(2)) = (2, - 4)" }}{PARA 0 "" 0 "" {TEXT -1 19 " Tangent Line: " }{TEXT 289 1 "y" }{TEXT -1 5 " = \+ 14" }{TEXT 290 1 "x" }{TEXT -1 5 " - 32" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "3. Function: f(" }{TEXT 291 1 "x" } {TEXT -1 4 ") = " }{XPPEDIT 18 0 "1/(x-2);" "6#*&\"\"\"F$,&%\"xGF$\"\" #!\"\"F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " Tangent P oint: (- 5, f(- 5)) = (- 5, - " }{XPPEDIT 18 0 "1/7;" "6#*&\"\"\"F$\" \"(!\"\"" }{TEXT -1 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 19 " Tangent Line: " }{TEXT 292 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-1/49*x-12 /49;" "6#,&*(\"\"\"F%\"#\\!\"\"%\"xGF%F'*&\"#7F%F&F'F'" }{TEXT -1 1 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "F or Problems 4 - 6, calculate the slope of the tangent line to the grap h of the function at the given point for Problems 1 -3 above by (1) us ing Maple to find the slope using the command " }{TEXT 293 11 "Limit( \+ ( f(" }{TEXT 294 1 "a" }{TEXT 295 3 " + " }{TEXT 296 1 "h" }{TEXT 297 6 ") - f(" }{TEXT 298 1 "a" }{TEXT 299 13 ") ) / h, h=0)" }{TEXT -1 39 " and then by (2) doing it step by step." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "4. Function: f(" } {TEXT 300 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "sqrt(x+4);" "6#-%%s qrtG6#,&%\"xG\"\"\"\"\"%F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 " Tangent Point: (- 3, f(- 3)) = (-3, 1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "5. Function: f(" }{TEXT 301 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "3*x^2+2*x-20;" "6#,(*&\" \"$\"\"\"*$%\"xG\"\"#F&F&*&F)F&F(F&F&\"#?!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 40 " Tangent Point: (2, f(2)) = (2, - 4)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "6. Function: f(" }{TEXT 302 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "1/(x-2);" "6#*&\"\"\"F$,& %\"xGF$\"\"#!\"\"F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " \+ Tangent Point: (- 5, f(- 5)) = (- 5, - " }{XPPEDIT 18 0 "1/7;" "6#*& \"\"\"F$\"\"(!\"\"" }{TEXT -1 2 " )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }