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"" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 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 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 37 "Summary of Calculus I Syntax in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "In thi s worksheet we summarize the syntax that we have used in the previous \+ worksheets and explain a few finer points. In MTH 142 next semester we shall continue working with Maple and use the syntax that we have lea rned this semester." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 " " {TEXT -1 25 "Functions and Expressions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "A function, for example " } {XPPEDIT 18 0 "f(x) = exp(2*x)+x*sin(x);" "6#/-%\"fG6#%\"xG,&-%$expG6# *&\"\"#\"\"\"F'F.F.*&F'F.-%$sinG6#F'F.F." }{TEXT -1 33 " , is defined \+ in Maple as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=x->exp(2*x)+x*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG 6#,$9$\"\"#\"\"\"*&F1F3-%$sinG6#F1F3F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 226 "A function, as far as Maple is concerned, is a rule wh ich to each input, say x, prescribes the output given by the formula f or the function. You can apply a function to anything in place of x an d obtain the corresponding value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(s); f(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$%\"sG\"\"#\"\"\"*&F(F*-%$sinG6#F(F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"#?\"\"\"-%$sinG6#\"#5 F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " If you define" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m:=e xp(2*x)+x*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG,&-%$expG6# ,$%\"xG\"\"#\"\"\"*&F*F,-%$sinG6#F*F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "you have defined not a fu nction but a " }{TEXT 266 7 "formula" }{TEXT -1 7 " or an " }{TEXT 256 10 "expression" }{TEXT -1 262 " m in terms of x. Maple can process expressions, as well as functions, but, as you see below, the appropr iate syntax may look a little different. For example \"m(s)\" is meani ngless to Maple. If you want to substitute s for x in the formula m yo u have to use the \"" }{TEXT 267 4 "subs" }{TEXT -1 9 "\" command" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=s,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$%\" sG\"\"#\"\"\"*&F(F*-%$sinG6#F(F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "There are basically three comma nds to evalute expressions. They are \"" }{TEXT 268 4 "eval" }{TEXT -1 4 "\", \"" }{TEXT 269 5 "evalf" }{TEXT -1 7 "\" and \"" }{TEXT 270 5 "value" }{TEXT -1 17 "\". The command \"" }{TEXT 271 4 "eval" } {TEXT -1 70 "\" will attempt to give you an exact value for your expre ssion, while \"" }{TEXT 272 5 "evalf" }{TEXT -1 97 "\" will evaluate a n expression numerically and return a decimal point approximation. The command \"" }{TEXT 273 5 "value" }{TEXT -1 61 "\" is most often used \+ in cojuction with inert commands, like \"" }{TEXT 274 3 "Int" }{TEXT -1 4 "\", \"" }{TEXT 275 5 "Limit" }{TEXT -1 97 "\" which do not evalu te the input but only print it out. You will see in subsequent section s how \"" }{TEXT 276 5 "value" }{TEXT -1 169 "\" works. The best way t o learn the differences between the evaluation commands is by practice and example. Click on the lines below and see if you can predict an o utput." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval( Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(m,x=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(eval(m,x=3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval(sin(h)/h,h=2); evalf(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Th e symbol \"" }{TEXT 277 1 "%" }{TEXT -1 41 "\" in Release 5 of Maple d enotes the last " }{TEXT 256 9 "executed " }{TEXT -1 15 "output. (It i s " }{TEXT 278 1 "\"" }{TEXT -1 64 " in earlier releases.) In order to avoid confusion, the symbol \"" }{TEXT 279 1 "%" }{TEXT -1 70 "\" sho uld always be used on the same command line as the command that \"" } {TEXT 280 1 "%" }{TEXT -1 12 "\" refers to." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 6 "Note: " }{TEXT -1 34 "Y ou should always use the syntax \"" }{TEXT 281 6 "exp(x)" }{TEXT -1 190 "\" for the natural exponential. You should not use exp(1)^x, or d efine e:=exp(1) and then use e^x. Although all of these options seem e quivalent, Maple works better with the syntax \"exp(x)\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 6 "Limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "To find the limit of a function at a point, use the syntax" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "limit(f(x),x=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We have found the limit at x=0 of the function f(x) defi ned above. If you hadn't defined f(x) you could simply type" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "l imit(exp(2*x)+x*sin(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "You could also find the limit of the expresion m at x=0 by typing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "limit(m,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "You cannot, however, use the syntax \"limit(f,x=0);\" You have t o type \"f(x)\". In most commands involving functions you have to ente r \"f(x)\" and not just the name of the function. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " If you want Mapl e to print out a given limit, use the " }{TEXT 284 5 "inert" }{TEXT -1 25 " version of the command \"" }{TEXT 282 5 "limit" }{TEXT -1 13 " \", that is, \"" }{TEXT 283 5 "Limit" }{TEXT -1 24 "\". Inert commands like \"" }{TEXT 285 5 "Limit" }{TEXT -1 4 "\", \"" }{TEXT 286 3 "Int " }{TEXT -1 211 "\" print out an input without evaluating it. They are useful if you want to check if you entered a correct expression. If \+ you want to evaluate your limit or integral right away, follow an ine rt command by the \"" }{TEXT 287 5 "value" }{TEXT -1 12 "\" command. \+ \"" }{TEXT 288 5 "value" }{TEXT -1 104 "\" command simply changes the \+ upper case inert commands to lower case commands and evaluates. For ex ample" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Limit(f(x),x=0); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$,&-%$expG6#,$%\"xG\"\"#\"\"\"*&F+F--%$sinG6# F+F-F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Maple can find limits at infinity and infinite limits as well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(ln (x)/x,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(exp(-x)/x^2,x=-infinit y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "You can use L'Hospital rules to see that Maple g ave us correct answers. If a given limit does not exist Maple will tel l us so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "limit(1/x,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%* undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "You can find one-si ded limits using Maple. The appropriate syntax is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(1/x,x =0,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(1/x,x=0,right);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 46 "If you want the limits to be printed out, use:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Limit(1/x,x=0,left); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%&LimitG6%*&\"\"\"F'%\"xG!\"\"/F(\"\"!%%leftG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 11 "Derivatives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "To find the derivative of a given function, we can type" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diff(exp(2*x)+x*sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,(-%$expG6#,$%\"xG\"\"#F)-%$sinG6#F(\"\"\"*&F(F--%$cosGF,F-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Since we have defined the function " }{XPPEDIT 18 0 "f(x) = exp(2*x)+x*sin(x);" "6#/-%\"fG6#%\"xG,&-%$ex pG6#*&\"\"#\"\"\"F'F.F.*&F'F.-%$sinG6#F'F.F." }{TEXT -1 34 " above, we can also use the syntax" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"xG\"\"#F)-%$sinG6#F(\"\"\"*&F(F--%$cosG F,F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "The syntax \"diff(f,x);\", however, does not work. You h ave to type \"f(x)\". For expressions the syntax is a little different . To find the derivative of the expression m defined above type " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(m,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"xG \"\"#F)-%$sinG6#F(\"\"\"*&F(F--%$cosGF,F-F-" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The simplest syntax for finding the second order and other higher oreder derivatives is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(f(x),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$ %\"xG\"\"#\"\"%-%$cosG6#F(F)*&F(\"\"\"-%$sinGF-F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "In an ins tant, Maple can calculate for you the tenth derivative of f(x)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(f(x),x$10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#, $%\"xG\"\"#\"%C5-%$cosG6#F(\"#5*&F(\"\"\"-%$sinGF-F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Another w ay of finding derivatives is by using the \"" }{TEXT 289 1 "D" }{TEXT -1 12 "\" operator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f *6#%\"xG6\"6$%)operatorG%&arrowGF&,(-%$expG6#,$9$\"\"#F0-%$sinG6#F/\" \"\"*&F/F4-%$cosGF3F4F4F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "To find the second derivative using t he \"" }{TEXT 290 1 "D" }{TEXT -1 18 "\" syntax, you type" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(D(f ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arro wGF&,(-%$expG6#,$9$\"\"#\"\"%-%$cosG6#F/F0*&F/\"\"\"-%$sinGF4F6!\"\"F& F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 6 "Note: " }{TEXT -1 61 "There is an important diff erence between the outputs of the \"" }{TEXT 291 4 "D(f)" }{TEXT -1 11 "\" and the \"" }{TEXT 292 4 "diff" }{TEXT -1 25 "\" commands. The \+ command \"" }{TEXT 293 4 "D(f)" }{TEXT -1 36 ";\" returns the derivati ve of f as a " }{TEXT 259 10 " function " }{TEXT -1 19 "of x. The comm and \"" }{TEXT 294 12 "diff(f(x),x)" }{TEXT -1 30 ";\" gives the deriv ative as an " }{TEXT 260 11 "expression " }{TEXT -1 127 "in terms of x . Both forms have advantages and disadvantages. For example, if you wa nt to simplify the derivative, the command \"" }{TEXT 295 22 "simplify (diff(f(x),x))" }{TEXT -1 37 ";\" works, in general, better than \" " }{TEXT 296 15 "simplify(D(f));" }{TEXT -1 170 "\". For other purpose s, it may be more convenient to have the derivative D(f) as a function . There is a way of turning an expression into a function, using the s o called \"" }{TEXT 297 7 "unapply" }{TEXT -1 47 "\" command. We shall look at this next semester." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "Maple can find indefinte integrals, as we ll as definite integrals. For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(exp(3*x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$%\"xG\"\"$#\"\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$%\"xG\"\"##\"\"\"F)-%$sinG6#F(F+*&F(F +-%$cosGF.F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "In the latt er example we found an antiderivative of the function f(x) defined in \+ the first section. If an antiderivative cannot be found in a closed fo rm, Maple will simply print out the input." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(sin(cos(x^2)) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$sinG6#-%$cosG6#*$) %\"xG\"\"#\"\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "If you want a given integral printed out, as well as found, use the inert version \"" }{TEXT 298 3 "Int" }{TEXT -1 10 "\" of the \"" }{TEXT 299 3 "int " }{TEXT -1 28 "\" command together with the " }{TEXT 300 7 "\"value\" " }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(x^3,x); value(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$*$)%\"xG\"\"$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"%\"\"\"#F(F'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 45 "Similar syntax applies to definite integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "i nt(f(x),x=0..2); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$exp G6#\"\"%#\"\"\"\"\"#-%$sinG6#F*F)-%$cosGF-!\"##!\"\"F*F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+7m1aG!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "We have just found the exact a nd the decimal value of definite integral of the function f(x) from 0 \+ to 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(cos(3*x),x=-1..3); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#\"\"*#\"\"\"\"\"$-F%6#F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6$GT%=!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "If you want a given integral printed out, as well as evaluated, type" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(cos(3*x),x=-1..3); value (%), evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$cosG6#,$ %\"xG\"\"$/F*;!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&-%$sinG6#\" \"*#\"\"\"\"\"$-F%6#F*F($\"+6$GT%=!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "If Maple prints out a gi ven definite integral instead of evaluating, that means that an antide rivative cannot be found and the Fundamental Theorem cannot be used. F or example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(sin(cos(x^2)),x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-%$sinG6#-%$cosG6#*$)%\"xG\"\"#\"\"\"/F.;\"\"! F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "In such case, we can ask Ma ple to find the integral numerically by typing:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(int(s in(cos(x^2)),x=0..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JG\"*RR! #5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 18 "Solving Equations" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 318 "As you know, solv ing equations is not an easy matter. In many cases there are no reliab le methods of finding exact values of solutions and numerical methods \+ have to be used. Maple knows all the standard techniques and algorithm s, as well as numerical methods. There are essentially two commands fo r solving equations \"" }{TEXT 301 5 "solve" }{TEXT -1 7 "\" and \"" } {TEXT 302 6 "fsolve" }{TEXT -1 12 "\". Command \"" }{TEXT 303 5 "solve " }{TEXT -1 193 "\" attempts to find exact values of as many solutions as possible. It works very well with polynomial equations up to the o rder four and equations which can be reduced to such form. For example :" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2-4*x-8=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"#\"\"\"*$-%%sqrtG6#\"\"$F%F$,&F$F%F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Command \"" }{TEXT 304 5 "solve" }{TEXT -1 64 " \" can be applied to equations containing parameters. For example" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(a*x^2+b*x+c=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&, &%\"bG!\"\"*$-%%sqrtG6#,&*$)F&\"\"#\"\"\"F0*&%\"aGF0%\"cGF0!\"%F0F0F0F 2F'#F0F/,$*&,&F&F'F(F'F0F2F'F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "For more complicated equations \"" } {TEXT 305 5 "solve" }{TEXT -1 42 "\" will return essentially the input itself" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^5-x+1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RootOfG6#,(*$)%#_ZG\"\"&\"\"\"F+F)!\"\"F+F+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 381 "Maple's response \+ indicates that it does not know how to solve this equation exactly. It is not surprising. As you may know there are no formulas for finding \+ roots of polynamials of degree five or higher. In that case, we can as k Maple to solve a given equation numerically. The command \"allvalues \" applied to an output \"RootOf\" will produce approximate solutions, real and complex:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "allvalues(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'$!+yRIn6!\"*,&$!+XWK7=!#5\"\"\"%\"IG$!+,T&R3\"F%,&F'F*F+$\"+,T&R 3\"F%,&$\"+OV%)[wF)F*F+$!+garCNF),&F2F*F+$\"+garCNF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The best way to solve equations numerically is by applying the \"" }{TEXT 306 6 "fsol ve" }{TEXT -1 12 "\" command. \"" }{TEXT 307 6 "fsolve" }{TEXT -1 96 " \" will attempt to find a solution numerically, and in general, it wil l return one real solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g:=t->sin(3*t)+cos(2*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrow GF(,&-%$sinG6#,$9$\"\"$\"\"\"-%$cosG6#,$F1\"\"#F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fsolve(g(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hzxC%*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "A solution that \"" }{TEXT 308 6 "fs olve" }{TEXT -1 111 "\" returns may not be located in a range of inter est to you. The situation can be remedied by adding under the \"" } {TEXT 309 6 "fsolve" }{TEXT -1 364 "\" command a range in which you w ant Maple to find a solution. The latter capabilty of Maple equation s olver combined with Maple's plotting facility provide a very powerful \+ tool of finding solutions relevant to your problem. Suppose that you a re studying a process modeled by the function g(t) and the range of in terest for t is [0,4]. You plot g(t) in that range:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(g(t) ,t=0..4);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "You see from the plot approximately where the zeros of g (t) are loacated. You ask Maple to find them using \"" }{TEXT 310 6 "f solve" }{TEXT -1 41 "\" with the range for t specified. Namely:" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(g(t)=0,t,0.5..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ hzxC%*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolve(g(t)=0, t,2..2.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e[6*>#!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(g(t)=0,t,3..4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>>vbM!\"*" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 8 "Plotting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "We have made and we will continue to make extensive use of Maple's plotting facility. The basic command \"" }{TEXT 320 4 "plot" }{TEXT -1 147 "\" as used in the previous section can be supple mented by many options, and several expressions can be plotted in one \+ coordinate system. For example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot([t*sin(t),t*sin(2*t)], t=0..10,color=[blue,magenta],scaling=constrained,labels=[\"time\",\"po sition\"]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "If you \+ are assigning attributes, for example, colors to several expressions, \+ remember to enter expressions as well as their desired attributes as a list and not as a set. A list is always entered between square bracke ts" }{TEXT 321 6 " [...]" }{TEXT -1 47 ", and a set is entered betwee n curly brackets " }{TEXT 322 5 "\{...\}" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 8 "Packages" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 285 "When you start Maple, it loads into the \+ computer's memory only the so called kernel, which contains the basic commands and functions. More advanced commands are contained in packa ges and libraries which can be loaded with appropriate commands. So fa r we have used two such packages: \"" }{TEXT 311 5 "plots" }{TEXT -1 7 "\" and \"" }{TEXT 312 7 "student" }{TEXT -1 27 "\". To load a packa ge, say \"" }{TEXT 313 5 "plots" }{TEXT -1 24 "\", you use the command \"" }{TEXT 314 12 "with(plots):" }{TEXT -1 121 "\".If you end your co mmand with a colon, the content of the package will not be printed. I f you end it with a semicolon \"" }{TEXT 315 12 "with(plots);" }{TEXT -1 62 "\" Maple will load the package, as well as print its content. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7U%(a nimateG%*animate3dG%-animatecurveG%-changecoordsG%,complexplotG%.compl explot3dG%*conformalG%,contourplotG%.contourplot3dG%*coordplotG%,coord plot3dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*fieldplotG %,fieldplot3dG%)gradplotG%+gradplot3dG%-implicitplotG%/implicitplot3dG %(inequalG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)listplot G%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'paretoG% *pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.polygonplot3dG%4p olyhedra_supportedG%.polyhedraplotG%'replotG%*rootlocusG%,semilogplotG %+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixplotG%+spherepl otG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Among the commands which the package contains we see \"" }{TEXT 316 9 "pointplot" } {TEXT -1 7 "\" and \"" }{TEXT 317 7 "display" }{TEXT -1 50 "\", which \+ we used before. We also see the command \"" }{TEXT 318 12 "implicitplo t" }{TEXT -1 66 "\" which allows plotting curves given by xy-equations . For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "implicitplot(3*x^2+y^2=4,x=-3..3,y=-3..3);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " The package \"" }{TEXT 319 11 "with(plots)" }{TEXT -1 90 "\" contains many important plotting commands and we shall use it extensively in the future." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 11 "Pr oblem 1. " }{TEXT -1 26 "Find the following limits:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " (a) " }{XPPEDIT 18 0 "limit((1+x)^(1/x),x = 0);" "6#-%&limitG6$),&\"\"\"F(%\"xGF(*&F(F (F)!\"\"/F)\"\"!" }{TEXT -1 20 " (b) " }{XPPEDIT 18 0 " limit(x^4/(x^5-1),x = 1,right);" "6#-%&limitG6%*&%\"xG\"\"%,&*$F'\"\"& \"\"\"F,!\"\"F-/F'F,%&rightG" }{TEXT -1 0 "" }{TEXT -1 18 " \+ (c) " }{XPPEDIT 18 0 "limit(exp(-2*x)*x^3,x = infinity);" "6#-%&lim itG6$*&-%$expG6#,$*&\"\"#\"\"\"%\"xGF-!\"\"F-*$F.\"\"$F-/F.%)infinityG " }{TEXT -1 1 "\000" }{TEXT -1 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 12 "Problem 2. " } {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = exp(3*x)*sin(x^2);" "6#/-%\"f G6#%\"xG*&-%$expG6#*&\"\"$\"\"\"F'F.F.-%$sinG6#*$F'\"\"#F." }{TEXT -1 44 ". Find the twenty-fifth derivative of f(x). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 11 "Problem 3 . " }{TEXT -1 40 "Find the following indefinite integrals:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " (a) " } {XPPEDIT 18 0 "int(e^(4*x)*x^3,x);" "6#-%$intG6$*&)%\"eG*&\"\"%\"\"\"% \"xGF+F+*$F,\"\"$F+F," }{TEXT -1 0 "" }{TEXT -1 15 " (b) " } {XPPEDIT 18 0 "int(sin(x)*x^2,x);" "6#-%$intG6$*&-%$sinG6#%\"xG\"\"\"* $F*\"\"#F+F*" }{TEXT -1 0 "" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 11 "Problem 4 . " }{TEXT -1 139 "Plot the following curves. For each of them adjust \+ properly the range for x and y so you can have a good idea about the s hape of the curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " (a) " }{XPPEDIT 18 0 "3*x^2+y^2 = 4;" "6#/,&*&\" \"$\"\"\"*$%\"xG\"\"#F'F'*$%\"yGF*F'\"\"%" }{TEXT -1 20 " \+ (b) " }{XPPEDIT 18 0 "x^2-y^2 = 2;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yG F'!\"\"F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 77 "M TH 141 Maple Worksheets written by B. Kaskosz and L. Pakula, Copyright 1998." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 323 38 "Last mo dified August 1999. Update 2006" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 \+ 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }