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Let's use Maple to find the derivative in the following two ways: 1) simply ask Maple for the val ue of " }{XPPEDIT 18 0 "Limit((g(x+h)-g(x))/h,h = 0);" "6#-%&LimitG6$ *&,&-%\"gG6#,&%\"xG\"\"\"%\"hGF-F--F)6#F,!\"\"F-F.F1/F.\"\"!" }{TEXT -1 61 " and 2) step by step like we would have to do it ourselves. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "First define the function " }{TEXT 281 1 "g" }{TEXT -1 34 " using Map le syntax for functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g := x -> sqrt(3*x + 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "In order to find the derivative by the first way that is indicated, we would type the following commands:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "Limit( (g(x+h) - g(x))/h, h=0); value(%);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 " In order to find the derivative by the second way that is indicated, w e would do the following:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Find " }{TEXT 285 1 "g" }{TEXT -1 1 "(" } {TEXT 263 3 "x+h" }{TEXT -1 6 ") and " }{TEXT 286 1 "g" }{TEXT -1 1 "( " }{TEXT 264 1 "x" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g(x+h); g(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "Subtract " }{TEXT 287 1 "g" }{TEXT -1 1 "(" }{TEXT 265 3 "x+h" }{TEXT -1 6 ") and " }{TEXT 288 1 "g" }{TEXT -1 1 "(" }{TEXT 266 1 "x" }{TEXT -1 85 "). You can store this result \+ to memory if you want. I chose to store the result to " }{TEXT 267 3 "num" }{TEXT -1 84 " since this is the numerator of the fraction in th e limit which we need to evaluate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g(x+h) - g(x); num := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Divide by " }{TEXT 268 1 "h" }{TEXT -1 35 ". I chose to store this result to " }{TEXT 269 5 "f ract" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "num /h; fract := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "At this point, we need to rationalize the numerator \+ of the fraction. Recall the " }{TEXT 260 12 "user-defined" }{TEXT -1 10 " function " }{TEXT 261 14 "rationalizenum" }{TEXT -1 71 ", defined in Lab 8, which will rationalize the numerator of a fraction:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rationalizenum := q -> 1/rat ionalize(1/q):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Now, use the function (command) " }{TEXT 262 14 "rat ionalizenum" }{TEXT -1 65 " to rationalize the numerator. I am going t o store the result to " }{TEXT 271 5 "fract" }{TEXT -1 155 " again bec ause once we rationalize the fraction above, we obtain a new fraction \+ which we use to evaluate the limit. Hence, we don't need the old frac tion " }{TEXT 272 0 "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rationalizenum(fract); fract := %;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 5 " NOTE:" }{TEXT -1 335 " Maple has already \"simplified\" the answer by combining like terms and cancelling. However, Maple will sometimes g ive strange answers. The 1/3 factor in front of the two square roots \+ in the denominator of the fraction is the equivalent to having a 3 in \+ the numerator, which is what you would have in doing this problem step by step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Now, take the limit as " }{TEXT 273 1 "h" }{TEXT -1 11 " goes to 0 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Limit(fract, h=0); val ue(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 9 "Ex ample 2" }{TEXT -1 90 " In lecture, we used the definition of derivat ive to find the derivative of the function " }{TEXT 284 1 "s" }{TEXT -1 1 "(" }{TEXT 307 1 "t" }{TEXT -1 4 ") = " }{TEXT -1 5 " 3/(2" } {TEXT 308 1 "t" }{TEXT -1 5 " - 9)" }{TEXT -1 107 ". Let's use Maple \+ to find the derivative in the following two ways: 1) simply ask Maple \+ for the value of " }{XPPEDIT 18 0 "Limit((s(t+h)-s(t))/h,h = 0);" "6# -%&LimitG6$*&,&-%\"sG6#,&%\"tG\"\"\"%\"hGF-F--F)6#F,!\"\"F-F.F1/F.\"\" !" }{TEXT -1 61 " and 2) step by step like we would have to do it our selves. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "First define the function " }{TEXT 282 1 "s" }{TEXT -1 34 " using Maple syntax for functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "s := t -> 3/(2*t - 9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 105 "In order to find the derivative by the f irst way that is indicated, we would type the following commands:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Limit( (s(t+h) - s(t))/h, h= 0); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "In order to find the derivative by the second way that is indicated, we would do the following:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{TEXT 289 1 "s" } {TEXT -1 1 "(" }{TEXT 274 3 "t+h" }{TEXT -1 6 ") and " }{TEXT 290 1 "s " }{TEXT -1 1 "(" }{TEXT 291 1 "t" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s(t+h); s(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Subtract " }{TEXT 292 1 "s " }{TEXT -1 1 "(" }{TEXT 275 3 "t+h" }{TEXT -1 6 ") and " }{TEXT 293 1 "s" }{TEXT -1 1 "(" }{TEXT 294 1 "t" }{TEXT -1 85 "). You can store this result to memory if you want. I chose to store the result to " }{TEXT 276 3 "num" }{TEXT -1 84 " since this is the numerator of the f raction in the limit which we need to evaluate:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "s(t+h) - s(t); num := %;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Divide by " } {TEXT 277 1 "h" }{TEXT -1 35 ". I chose to store this result to " } {TEXT 278 5 "fract" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "num/h; simplify(%); fract := %;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 279 5 "NOT E:" }{TEXT -1 247 " Maple has already \"simplified\" the answer by co mbining like terms and cancelling. The factor of - 6 in front of the fraction is the equivalent to having the - 6 in the numerator, which \+ is what you would have in doing this problem step by step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Now, take the lim it as " }{TEXT 280 1 "h" }{TEXT -1 11 " goes to 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Limit(fract, h=0); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 18 "Practice Problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Use the definition of derivative to find the derivative of the following functions in the two ways which we did above." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{TEXT 299 1 "f" }{TEXT -1 1 "(" }{TEXT 298 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x^2-6*x;" "6#,&*$%\"xG\" \"#\"\"\"*&\"\"'F'F%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 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " }{TEXT 300 1 "f" }{TEXT -1 1 "(" } {TEXT 256 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\" \"$" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{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 4 "3. " }{TEXT 301 1 "g" }{TEXT -1 1 "(" }{TEXT 257 1 "x" }{TEXT -1 5 ") = " }{XPPEDIT 18 0 "sqrt(2*x-5);" "6#-%%sqrtG6#,&*&\"\"#\"\"\"% \"xGF)F)\"\"&!\"\"" }{TEXT -1 0 "" }}}{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 4 "4. " }{TEXT 302 1 "s" }{TEXT -1 1 "(" } {TEXT 303 1 "t" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "t^3-t^2+2*t;" "6#,( *$%\"tG\"\"$\"\"\"*$F%\"\"#!\"\"*&F)F'F%F'F'" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "5. " }{TEXT 304 1 "g " }{TEXT -1 1 "(" }{TEXT 258 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 " (4-3*x)/(x+2);" "6#*&,&\"\"%\"\"\"*&\"\"$F&%\"xGF&!\"\"F&,&F)F&\"\"#F& F*" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t:" }}}{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 4 "6. " }{TEXT 305 1 "s" }{TEXT -1 1 "(" }{TEXT 306 1 "t" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "1/(t^2);" "6#*&\"\"\"F$*$%\"tG\"\"#!\"\"" } {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 "" }}}} {MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }