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0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "This worksheet contains t he in-class excercises for Week 3 Block 5 and the related homework exe rcises.\nAll of these exercises are part of the homework to be turned \+ in by " }{TEXT 475 31 "5PM, Friday, Spetember 24, 1999" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 360 "to me (JHR) or the box by my office. \nAs this homework is both large and fundamental to Calculus I, it wi ll be work 20 homework points.\nThough we are giving you until Friday \+ to turn the homework in, you are expected to have worked through the e xercises by Thursday's mathematics class so that you will be ready to \+ discuss the product, quotient and chain rules." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Diffe rentiation of \"famous\" functions " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Recall that the derivative of a function " }{XPPEDIT 18 0 "y = \+ f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 13 " is given by\n" } {XPPEDIT 18 0 "df/dx = limit((f(x+h)-f(x))/h,h = 0);" "6#/*&%#dfG\"\" \"%#dxG!\"\"-%&limitG6$*&,&-%\"fG6#,&%\"xGF&%\"hGF&F&-F/6#F2F(F&F3F(/F 3\"\"!" }{TEXT -1 66 " . We need to know the derivative of the followi ng nine functions:" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 1 " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = x^n ;" "6#/-%\"fG6#%\"xG)F'%\"nG" }{TEXT -1 29 " then find the derivative \+ of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6#%\"x G-%$sinG6#F'" }{TEXT -1 30 " then find the derivative of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "functi on 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = cos(x);" "6#/-%\"fG6#%\"xG-%$cosG6#F'" }{TEXT -1 29 " then find the d erivative of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 10 "function 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "I f " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 29 " then find the derivative of " }{XPPEDIT 18 0 "f(x);" "6# -%\"fG6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = ln(x);" "6#/-% \"fG6#%\"xG-%#lnG6#F'" }{TEXT -1 29 " then find the derivative of " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "f(x) = arcsin(x);" "6#/-%\"fG6#%\"xG-%'arcsinG6#F'" } {TEXT -1 30 " then find the derivative of " }{XPPEDIT 18 0 "f(x);" "6 #-%\"fG6#%\"xG" }{TEXT -1 1 "." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "fu nction 7" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f( x) = arctan(x);" "6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT -1 29 " then \+ find the derivative of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x) = cosh(x);" "6#/-%\"fG6#%\" xG-%%coshG6#F'" }{TEXT -1 29 " then find the derivative of " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "function 9" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "f(x) = sinh(x);" "6#/-%\"fG6#%\"xG-%%sinhG6#F'" }{TEXT -1 29 " then find the derivative of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG 6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Diff erentiation of products" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For the exercises 1-9, " }{XPPEDIT 18 0 "h(x) = f(x)*g(x);" "6#/-%\"hG6#%\"xG *&-%\"fG6#F'\"\"\"-%\"gG6#F'F," }{TEXT -1 17 ".\n(A) Write down " } {TEXT 256 1 "g" }{TEXT -1 1 "(" }{TEXT 257 1 "x" }{TEXT -1 3 "), " } {TEXT 258 2 "g'" }{TEXT -1 1 "(" }{TEXT 259 1 "x" }{TEXT -1 3 "), " } {TEXT 262 1 "f" }{TEXT -1 1 "(" }{TEXT 260 1 "x" }{TEXT -1 3 "), " } {TEXT 263 2 "f'" }{TEXT -1 1 "(" }{TEXT 261 1 "x" }{TEXT -1 14 ") and \+ use the " }{TEXT 478 4 "diff" }{TEXT -1 17 " command to find " }{TEXT 264 2 "h'" }{TEXT -1 1 "(" }{TEXT 265 1 "x" }{TEXT -1 12 ").\n(B) Plot " }{TEXT 266 1 "h" }{TEXT -1 1 "(" }{TEXT 267 1 "x" }{TEXT -1 6 ") an d " }{TEXT 268 2 "h'" }{TEXT -1 1 "(" }{TEXT 269 1 "x" }{TEXT -1 44 ") on the same set of axes. Be sure to label " }{TEXT 270 1 "h" }{TEXT -1 1 "(" }{TEXT 271 1 "x" }{TEXT -1 6 ") and " }{TEXT 272 2 "h'" } {TEXT -1 1 "(" }{TEXT 273 1 "x" }{TEXT -1 2 ")." }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 10 "exercise 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "h(x) = x^2*exp(x);" "6#/-%\"hG6#%\"xG*&F'\"\"#-%$exp G6#F'\"\"\"" }{TEXT -1 10 " for -3<= " }{TEXT 274 2 "x " }{TEXT -1 4 " <=1." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = x^2*sin(x);" "6#/-%\"h G6#%\"xG*&F'\"\"#-%$sinG6#F'\"\"\"" }{TEXT -1 12 " for -3 <= " } {TEXT 275 1 "x" }{TEXT -1 5 " <=3." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = x^2*cos(x);" "6#/-%\"hG6#%\"xG*&F'\"\"#-%$cosG6#F'\"\"\" " }{TEXT -1 10 " for -3<= " }{TEXT 276 2 "x " }{TEXT -1 4 "<=3." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = ln(x)*sin(x);" "6#/-%\"hG6# %\"xG*&-%#lnG6#F'\"\"\"-%$sinG6#F'F," }{TEXT -1 12 " for 1/2 <= " } {TEXT 277 1 "x" }{TEXT -1 5 " <=4." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = ln(x)*cos(x);" "6#/-%\"hG6#%\"xG*&-%#lnG6#F'\"\"\"-%$cosG 6#F'F," }{TEXT -1 12 " for 1/2 <= " }{TEXT 278 1 "x" }{TEXT -1 6 " <= \+ 4." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 6" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = ln(x)*exp(x);" "6#/-%\"hG 6#%\"xG*&-%#lnG6#F'\"\"\"-%$expG6#F'F," }{TEXT -1 13 " for 2/10 <= " } {TEXT 279 1 "x" }{TEXT -1 6 " <= 2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 7" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = exp(x)*sin(x);" "6#/-%\"hG6#%\"xG*&-%$expG6#F'\"\"\"-%$si nG6#F'F," }{TEXT -1 11 " for -3 <= " }{TEXT 280 2 "x " }{TEXT -1 7 " < = 3. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exercise 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = exp(x)*cos(x);" "6#/-%\"hG6#%\"x G*&-%$expG6#F'\"\"\"-%$cosG6#F'F," }{TEXT -1 11 " for -3 <= " }{TEXT 281 1 "x" }{TEXT -1 6 " <= 3." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "exe rcise 9" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) \+ = x^2*ln(x);" "6#/-%\"hG6#%\"xG*&F'\"\"#-%#lnG6#F'\"\"\"" }{TEXT -1 13 " for 2/10 <= " }{TEXT 282 1 "x" }{TEXT -1 6 " <= 2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "exercise 10 (summary)" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 19 " When you look at " }{TEXT 283 2 "h'" }{TEXT -1 1 "(" }{TEXT 284 1 "x" }{TEXT -1 61 ") in exercises one through nine, \+ yu should see a pattern. If " }{TEXT 285 2 "f'" }{TEXT -1 1 "(" } {TEXT 286 1 "x" }{TEXT -1 23 ") is the derivative of " }{TEXT 287 1 "f " }{TEXT -1 2 " (" }{TEXT 288 1 "x" }{TEXT -1 3 "), " }{TEXT 289 2 "g' " }{TEXT -1 1 "(" }{TEXT 290 1 "x" }{TEXT -1 23 ") is the derivative o f " }{TEXT 291 1 "g" }{TEXT -1 1 "(" }{TEXT 292 1 "x" }{TEXT -1 7 "), \+ and " }{TEXT 293 1 "h" }{TEXT -1 1 "(" }{TEXT 294 1 "x" }{TEXT -1 2 ") =" }{TEXT 295 1 "f" }{TEXT -1 1 "(" }{TEXT 296 1 "x" }{TEXT -1 2 ")*" }{TEXT 297 1 "g" }{TEXT -1 1 "(" }{TEXT 298 1 "x" }{TEXT -1 11 "), exp ress " }{TEXT 299 2 "h'" }{TEXT -1 1 "(" }{TEXT 300 1 "x" }{TEXT -1 14 ") in terms of " }{TEXT 301 1 "f" }{TEXT -1 1 "(" }{TEXT 302 1 "x" }{TEXT -1 2 ")," }{TEXT 303 1 "g" }{TEXT -1 1 "(" }{TEXT 304 1 "x" } {TEXT -1 2 ")," }{TEXT 305 2 "f'" }{TEXT -1 1 "(" }{TEXT 306 1 "x" } {TEXT -1 7 "), and " }{TEXT 307 2 "g'" }{TEXT -1 1 "(" }{TEXT 308 1 "x " }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "This formula is called the\n " }{TEXT 309 7 "Product" }{TEXT -1 1 " " }{TEXT 310 24 "Rule for Differentiation" }{TEXT -1 112 "\nand should be memorized . \nWrite the product rule as an English sentence without using any ma thematical symbols." }{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "\"home\" portion of the homework " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "problems 1-6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 " For problems 1-5 " }{TEXT 311 1 "f" }{TEXT -1 1 "(" }{TEXT 312 1 "x" }{TEXT -1 2 ")=" }{TEXT 313 1 "g" }{TEXT -1 1 "(" }{TEXT 314 1 "h" } {TEXT -1 1 "(" }{TEXT 315 1 "x" }{TEXT -1 18 "))\n(A) Write down " } {TEXT 316 1 "g" }{TEXT -1 1 "(" }{TEXT 317 1 "x" }{TEXT -1 3 "), " } {TEXT 318 2 "g'" }{TEXT -1 1 "(" }{TEXT 319 1 "x" }{TEXT -1 3 "), " } {TEXT 323 1 "h" }{TEXT -1 1 "(" }{TEXT 322 1 "x" }{TEXT -1 3 "), " } {TEXT 320 2 "h'" }{TEXT -1 1 "(" }{TEXT 321 1 "x" }{TEXT -1 11 ") and \+ have " }{TEXT 324 5 "Maple" }{TEXT -1 11 " calculate " }{TEXT 325 2 "f '" }{TEXT -1 1 "(" }{TEXT 326 1 "x" }{TEXT -1 12 ").\n(B) Plot " } {TEXT 327 1 "f" }{TEXT -1 1 "(" }{TEXT 328 1 "x" }{TEXT -1 6 ") and " }{TEXT 329 2 "f'" }{TEXT -1 1 "(" }{TEXT 330 1 "x" }{TEXT -1 40 ") ove r the stated domain. Clearly label " }{TEXT 331 1 "f" }{TEXT -1 1 "(" }{TEXT 332 1 "x" }{TEXT -1 6 ") and " }{TEXT 333 2 "f'" }{TEXT -1 1 "( " }{TEXT 334 1 "x" }{TEXT -1 2 ")." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "problem 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = ln(sin(x));" "6#/-%\"fG6#%\"xG-%#lnG6#-%$sinG6#F'" }{TEXT -1 9 " for 0.1 " }{TEXT 380 1 "O" }{TEXT -1 1 " " }{TEXT 335 1 "x" } {TEXT -1 1 " " }{TEXT 381 1 "O" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi-.1; " "6#,&%#PiG\"\"\"$\"\"\"!\"\"!\"\"" }{TEXT -1 1 "." }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 9 "problem 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = sin(exp(x));" "6#/-%\"fG6#%\"xG-%$sinG6#-%$expG 6#F'" }{TEXT -1 8 " for -3 " }{TEXT 378 1 "O" }{TEXT -1 1 " " }{TEXT 337 1 "x" }{TEXT -1 1 " " }{TEXT 379 1 "O" }{TEXT -1 3 " 2." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 9 "problem 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x) = cos(ln(x));" "6#/-%\"fG6#%\"xG-%$cosG6#-%#lnG6#F '" }{TEXT -1 9 " for 0.5 " }{TEXT 375 1 "O" }{TEXT -1 1 " " }{TEXT 336 2 "x " }{TEXT 376 1 "O" }{TEXT 377 1 " " }{TEXT -1 4 " 10." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 9 "problem 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x) = exp(cos(x));" "6#/-%\"fG6#%\"xG-%$expG6#-%$cosG6 #F'" }{TEXT -1 5 "for " }{XPPEDIT 18 0 "-2*Pi;" "6#,$*&\"\"#\"\"\"%#P iGF&!\"\"" }{TEXT -1 1 " " }{TEXT 374 1 "O" }{TEXT -1 1 " " }{TEXT 338 1 "x" }{TEXT -1 1 " " }{TEXT 373 1 "O" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " problem 5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x ) = sin(x^3+2*x+1);" "6#/-%\"fG6#%\"xG-%$sinG6#,(*$F'\"\"$\"\"\"*&\"\" #F.F'F.F.\"\"\"F." }{TEXT -1 8 " for -2 " }{TEXT 371 1 "O" }{TEXT -1 1 " " }{TEXT 339 1 "x" }{TEXT -1 1 " " }{TEXT 372 1 "O" }{TEXT -1 3 " \+ 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "problem 6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "C onsidering the results in problems 1-5, suppose " }{XPPEDIT 18 0 "y = \+ g(h(x));" "6#/%\"yG-%\"gG6#-%\"hG6#%\"xG" }{TEXT -1 11 ". \nExpress " }{TEXT 340 2 "y'" }{TEXT -1 13 " in terms of " }{TEXT 341 1 "g" } {TEXT -1 2 " (" }{TEXT 342 1 "x" }{TEXT -1 3 "), " }{TEXT 343 2 "g'" } {TEXT -1 1 "(" }{TEXT 344 1 "x" }{TEXT -1 3 "), " }{TEXT 348 1 "h" } {TEXT -1 1 "(" }{TEXT 347 1 "x" }{TEXT -1 7 "), and " }{TEXT 345 2 "h' " }{TEXT -1 1 "(" }{TEXT 346 1 "x" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 " problems 7-10" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "In problems 7-9, \+ " }{XPPEDIT 18 0 "y = f(g(h(x)));" "6#/%\"yG-%\"fG6#-%\"gG6#-%\"hG6#% \"xG" }{TEXT -1 15 ".\n(A) Identify " }{TEXT 357 1 "f" }{TEXT -1 1 "( " }{TEXT 358 1 "x" }{TEXT -1 1 ")" }{TEXT 349 3 ", g" }{TEXT -1 1 "(" }{TEXT 350 1 "x" }{TEXT -1 3 "), " }{TEXT 354 1 "h" }{TEXT -1 1 "(" } {TEXT 353 1 "x" }{TEXT -1 2 ")," }{TEXT 359 3 " f'" }{TEXT -1 1 "(" } {TEXT 360 1 "x" }{TEXT -1 1 ")" }{TEXT 355 4 ", g'" }{TEXT -1 1 "(" } {TEXT 356 1 "x" }{TEXT -1 1 ")" }{TEXT 351 2 ", " }{TEXT -1 3 "and" } {TEXT 361 3 " h'" }{TEXT -1 1 "(" }{TEXT 352 1 "x" }{TEXT -1 8 "). Hav e " }{TEXT 362 5 "Maple" }{TEXT -1 11 " calculate " }{TEXT 363 2 "y'" }{TEXT -1 11 ".\n(B) Plot " }{TEXT 364 1 "y" }{TEXT -1 1 "(" }{TEXT 365 1 "x" }{TEXT -1 6 ") and " }{TEXT 366 2 "y'" }{TEXT -1 1 "(" } {TEXT 367 1 "x" }{TEXT -1 25 ") over the stated domain." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "problem 7" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sin(ln(x^2+2*x+1));" "6#/%\"yG-%$sinG6#-% #lnG6#,(*$%\"xG\"\"#\"\"\"*&\"\"#F/F-F/F/\"\"\"F/" }{TEXT -1 6 " for 0 " }{TEXT 369 1 "O" }{TEXT -1 1 " " }{TEXT 368 1 "x" }{TEXT -1 1 " " } {TEXT 370 1 "O" }{TEXT -1 3 " 2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " problem 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp(cos(ln(x)));" "6#/%\"yG-%$expG6#-%$cosG6#-%#lnG6#%\"xG" }{TEXT -1 6 " for 1" }{TEXT 382 1 "O" }{TEXT -1 1 " " }{TEXT 383 1 "x" } {TEXT -1 1 " " }{TEXT 384 1 "O" }{TEXT -1 4 " 10." }{MPLTEXT 1 0 2 " \+ \011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "problem 9" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = ln(sin(exp(x)));" "6#/%\"yG-%#lnG6#-%$sinG6#-%$ expG6#%\"xG" }{TEXT -1 8 " for -1 " }{TEXT 387 1 "O" }{TEXT -1 1 " " } {TEXT 385 1 "x" }{TEXT -1 1 " " }{TEXT 386 1 "O" }{TEXT -1 3 " 1." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "problem 10" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = f(g(h(x)));" "6#/%\"yG-% \"fG6#-%\"gG6#-%\"hG6#%\"xG" }{TEXT -1 10 ", express " }{TEXT 388 2 "y '" }{TEXT -1 13 " in terms of " }{TEXT 397 1 "f" }{TEXT -1 2 " (" } {TEXT 398 1 "x" }{TEXT -1 1 ")" }{TEXT 389 3 ", g" }{TEXT -1 1 "(" } {TEXT 390 1 "x" }{TEXT -1 3 "), " }{TEXT 394 1 "h" }{TEXT -1 1 "(" } {TEXT 393 1 "x" }{TEXT -1 2 ")," }{TEXT 399 3 " f'" }{TEXT -1 1 "(" } {TEXT 400 1 "x" }{TEXT -1 1 ")" }{TEXT 395 4 ", g'" }{TEXT -1 1 "(" } {TEXT 396 1 "x" }{TEXT -1 1 ")" }{TEXT 391 2 ", " }{TEXT -1 3 "and" } {TEXT 401 3 " h'" }{TEXT -1 1 "(" }{TEXT 392 1 "x" }{TEXT -1 2 ")." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 10 "problem 11" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "In th is problem let " }{XPPEDIT 18 0 "h(x) = f(x)/g(x);" "6#/-%\"hG6#%\"xG* &-%\"fG6#F'\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 17 ".\n(A) Write down " } {TEXT 402 1 "g" }{TEXT -1 2 " (" }{TEXT 403 1 "x" }{TEXT -1 3 "), " } {TEXT 404 2 "g'" }{TEXT -1 1 "(" }{TEXT 405 1 "x" }{TEXT -1 3 "), " } {TEXT 408 1 "f" }{TEXT -1 1 "(" }{TEXT 406 1 "x" }{TEXT -1 3 "), " } {TEXT 409 2 "f'" }{TEXT -1 1 "(" }{TEXT 407 1 "x" }{TEXT -1 11 "). Use the " }{TEXT 412 4 "diff" }{TEXT -1 17 " command to find " }{TEXT 410 2 "h'" }{TEXT -1 1 "(" }{TEXT 411 1 "x" }{TEXT -1 15 ").\n(B) Use \+ the " }{TEXT 476 8 "simplify" }{TEXT -1 4 " or " }{TEXT 477 6 "normal " }{TEXT -1 38 " command to get a single fraction for " }{TEXT 413 2 " h'" }{TEXT -1 1 "(" }{TEXT 414 1 "x" }{TEXT -1 13 "). \n(C) Plot " } {TEXT 415 1 "h" }{TEXT -1 1 "(" }{TEXT 416 1 "x" }{TEXT -1 6 ") and " }{TEXT 417 2 "h'" }{TEXT -1 1 "(" }{TEXT 418 1 "x" }{TEXT -1 41 ") on \+ the same axes and label your graphs." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "part " }{TEXT 449 1 "i" }{TEXT -1 0 "" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = sin(x)/x;" "6#/-%\"hG6#% \"xG*&-%$sinG6#F'\"\"\"F'!\"\"" }{TEXT -1 8 " for -6 " }{TEXT 420 1 "O " }{TEXT -1 1 " " }{TEXT 419 1 "x" }{TEXT -1 1 " " }{TEXT 421 1 "O" } {TEXT -1 3 " 6." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "part " } {TEXT 448 2 "ii" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "h(x) = cos(x)/x;" "6#/-%\"hG6#%\"xG*&-%$cosG6#F'\"\" \"F'!\"\"" }{TEXT -1 8 " for -6 " }{TEXT 429 1 "O" }{TEXT -1 1 " " } {TEXT 422 2 "x " }{TEXT 430 1 "O" }{TEXT -1 4 " 6. " }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 5 "part " }{TEXT 447 3 "iii" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = exp(x)/sin(x);" "6#/-%\"hG6#%\"xG*&-%$expG6#F'\"\"\"-%$sinG6#F'!\"\"" }{TEXT -1 8 " fo r -3 " }{TEXT 431 1 "O" }{TEXT -1 1 " " }{TEXT 423 1 "x" }{TEXT -1 1 " " }{TEXT 432 1 "O" }{TEXT -1 11 " 3 and -5 " }{TEXT 433 1 "O" } {TEXT -1 1 " " }{TEXT 424 1 "y" }{TEXT -1 1 " " }{TEXT 434 1 "O" } {TEXT -1 4 " 5. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "part " } {TEXT 446 2 "iv" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "h(x) = exp(x)/cos(x);" "6#/-%\"hG6#%\"xG*&-%$expG6#F '\"\"\"-%$cosG6#F'!\"\"" }{TEXT -1 8 " for -3 " }{TEXT 435 1 "O" } {TEXT -1 2 " " }{TEXT 425 1 "x" }{TEXT -1 1 " " }{TEXT 436 1 "O" } {TEXT -1 11 " 3 and -5 " }{TEXT 437 1 "O" }{TEXT -1 1 " " }{TEXT 426 1 "y" }{TEXT -1 1 " " }{TEXT 438 1 "O" }{TEXT -1 3 " 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "part " }{TEXT 445 1 "v" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = ln(x)/exp(x);" "6#/-%\"hG6#%\"xG *&-%#lnG6#F'\"\"\"-%$expG6#F'!\"\"" }{TEXT -1 9 " for 1/2 " }{TEXT 439 1 "O" }{TEXT -1 1 " " }{TEXT 427 1 "x" }{TEXT -1 1 " " }{TEXT 440 1 "O" }{TEXT -1 3 " 2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "part " } {TEXT 444 2 "vi" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x) = x/sin(x);" "6#/-%\"hG6#%\"xG*&F'\"\"\"-%$sinG6#F'!\"\"" } {TEXT -1 8 " for -6 " }{TEXT 441 1 "O" }{TEXT -1 1 " " }{TEXT 428 1 "x " }{TEXT -1 1 " " }{TEXT 442 1 "O" }{TEXT -1 3 " 6." }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "prob lem 12" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Based on problem 11, if \+ " }{XPPEDIT 18 0 "h(x) = f(x)/g(x);" "6#/-%\"hG6#%\"xG*&-%\"fG6#F'\"\" \"-%\"gG6#F'!\"\"" }{TEXT -1 16 ", a pattern for " }{TEXT 450 2 "h'" } {TEXT -1 1 "(" }{TEXT 451 1 "x" }{TEXT -1 42 ") should appear. \nWrite an expression for " }{TEXT 454 2 "h'" }{TEXT -1 1 "(" }{TEXT 455 1 "x " }{TEXT -1 14 ") in terms of " }{TEXT 456 1 "g" }{TEXT -1 3 " (" } {TEXT 457 1 "x" }{TEXT -1 3 "), " }{TEXT 458 2 "g'" }{TEXT -1 1 "(" } {TEXT 459 1 "x" }{TEXT -1 3 "), " }{TEXT 462 1 "f" }{TEXT -1 1 "(" } {TEXT 460 1 "x" }{TEXT -1 8 "), and " }{TEXT 463 2 "f'" }{TEXT -1 1 " (" }{TEXT 461 1 "x" }{TEXT -1 115 ").\nThen, express your observation \+ as an English sentence free of mathematical symbols. You should have d educed the\n" }{TEXT 464 33 "Quotient Rule for Differentiation" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "problem 13" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "A variable line through the point (1,2) intersects t he " }{TEXT 465 1 "x" }{TEXT -1 9 "-axis at " }{TEXT 466 1 "A" }{TEXT -1 1 "(" }{TEXT 467 1 "a" }{TEXT -1 12 ",0) and the " }{TEXT 468 1 "y " }{TEXT -1 9 "-axis at " }{TEXT 469 1 "B" }{TEXT -1 3 "(0," }{TEXT 470 1 "b" }{TEXT -1 33 ").\nMinimize the area of triangle " }{TEXT 471 3 "AOB" }{TEXT -1 15 ", keeping both " }{TEXT 473 1 "a" }{TEXT -1 5 " and " }{TEXT 474 1 "b" }{TEXT -1 37 " positive.\nSubmit a formal w rite-up. " }{TEXT 472 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 "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 }