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}}{PARA 0 "" 0 "" {TEXT -1 91 "In this lab, we wi ll learn the two Maple commands for finding the derivative of a functi on." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 8 "Examples" }{TEXT -1 52 " Differentiate the following functions using Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{TEXT 279 1 "f" }{TEXT -1 1 "(" }{TEXT 280 1 "x " }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x^2+4*x+2;" "6#,(*$%\"xG\"\"#\"\" \"*&\"\"%F'F%F'F'F&F'" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> x^2 + 4*x + 2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " One Maple comma nd for finding the derivative of an " }{TEXT 411 10 "expression" } {TEXT -1 8 " is the " }{TEXT 263 4 "diff" }{TEXT -1 9 " command:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(f(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 5 "NOTE:" }{TEXT -1 25 " The first entry in the " }{TEXT 265 4 "d iff" }{TEXT -1 16 " command is the " }{TEXT 412 10 "expression" } {TEXT -1 159 " that you want to differentiate and the second entry in \+ the command is the variable which you are differentiating with repect \+ to. The answer given will be an " }{TEXT 413 10 "expression" }{TEXT -1 92 ". Later, in our lesson on Rated Rates, we will talk about diff erentiating an expression of " }{TEXT 266 1 "x" }{TEXT -1 5 " and " } {TEXT 267 1 "y" }{TEXT -1 33 " with respect to the variable of " } {TEXT 268 1 "t" }{TEXT -1 116 ", which will represent time. However, \+ until we get to that lesson, if you are asked to differentiate a funct ion of " }{TEXT 269 1 "x" }{TEXT -1 46 ", then you will differentiate \+ with respect to " }{TEXT 270 1 "x" }{TEXT -1 51 ". If you are asked t o differentiate a function of " }{TEXT 271 1 "u" }{TEXT -1 46 ", then \+ you will differentiate with respect to " }{TEXT 272 1 "u" }{TEXT -1 12 ", and so on." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 60 " The other Maple command for finding the deriva tive of a " }{TEXT 414 8 "function" }{TEXT -1 8 " is the " }{TEXT 273 1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 5 "NOTE:" }{TEXT -1 54 " The entry in the first set of parentheses after the " }{TEXT 275 1 "D" }{TEXT -1 28 " \+ command is the name of the " }{TEXT 416 8 "function" }{TEXT -1 159 " t o be differentiated and the entry in the second set of parentheses is \+ the variable which you are differentiating with repect to. The answer given will be a " }{TEXT 415 8 "function" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 5 "NOTE:" }{TEXT -1 21 " The format for the " }{TEXT 276 4 "diff" }{TEXT -1 57 " command \+ may seem to be easier to remember; however, the " }{TEXT 277 1 "D" } {TEXT -1 112 " command is easier to use when we want to evaluate the d erivative at a value as we will see in an example below." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "2. " } {TEXT 281 1 "f" }{TEXT -1 1 "(" }{TEXT 282 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "5+2*x^3-4*x^5;" "6#,(\"\"&\"\"\"*&\"\"#F%*$%\"xG\"\"$F% F%*&\"\"%F%*$F)F$F%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := 5 + 2*x^3 - 4*x ^5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " Using the " }{TEXT 287 4 "diff" }{TEXT -1 9 " command:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "What the (Maple sensors have removed a word here) is all this stuff? The answ er you will get if you did not use correct Maple syntax to store the f unction " }{TEXT 288 1 "f" }{TEXT -1 56 ". Look at what was stored to f. It was the expression " }{XPPEDIT 18 0 "5+2*x^3-4*x^5;" "6#,(\"\" &\"\"\"*&\"\"#F%*$%\"xG\"\"$F%F%*&\"\"%F%*$F)F$F%!\"\"" }{TEXT -1 100 ", which Maple will not recognize as a function unless we put the \"x \+ ->\" in front of this expression." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "However, since the " }{TEXT 289 4 "diff" }{TEXT -1 35 " command only needs to be given an " }{TEXT 417 11 "expression " }{TEXT -1 43 "and not a function and you have stored \+ the " }{TEXT 418 10 "expression" }{TEXT -1 1 " " }{XPPEDIT 18 0 "5+2*x ^3-4*x^5;" "6#,(\"\"&\"\"\"*&\"\"#F%*$%\"xG\"\"$F%F%*&\"\"%F%*$F)F$F%! \"\"" }{TEXT -1 23 " to f, we can use the " }{TEXT 290 4 "diff" } {TEXT -1 69 " command to find the derivative of this expression by the following: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(f,x); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We this work for the " }{TEXT 291 1 "D" }{TEXT -1 26 " command? W ell let's see:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Clearly, the answer is " }{TEXT 292 2 "NO" }{TEXT -1 77 ". The lette r that you put in the first set of parentheses which follows the " } {TEXT 293 1 "D" }{TEXT -1 19 " command must be a " }{TEXT 419 8 "funct ion" }{TEXT -1 92 " stored with the Maple syntax for storing functions . Even though it is not required in the " }{TEXT 294 4 "diff" }{TEXT -1 162 " command, when you use Maple to find a derivative of a functio n, you should store the function as a function using Maple syntax. Th e main difference between the " }{TEXT 349 4 "diff" }{TEXT -1 13 " com mand the " }{TEXT 350 1 "D" }{TEXT -1 57 " command for finding a deriv ative is the following: The " }{TEXT 351 4 "diff" }{TEXT -1 36 " comm and finds the derivative of an " }{TEXT 420 10 "expression" }{TEXT -1 28 " and gives the answer as an " }{TEXT 421 10 "expression" }{TEXT -1 7 ". The " }{TEXT 352 1 "D" }{TEXT -1 35 " command finds the deriv ative of a " }{TEXT 422 8 "function" }{TEXT -1 27 " and gives the answ er as a " }{TEXT 423 8 "function" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Now, let's find th e derivative of the function " }{TEXT 295 1 "f" }{TEXT -1 1 "(" } {TEXT 296 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "5+2*x^3-4*x^5;" "6# ,(\"\"&\"\"\"*&\"\"#F%*$%\"xG\"\"$F%F%*&\"\"%F%*$F)F$F%!\"\"" }{TEXT -1 80 " storing this function to f as a function using Maple syntax an d then using the " }{TEXT 297 4 "diff" }{TEXT -1 5 " and " }{TEXT 298 1 "D" }{TEXT -1 10 " commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := x -> 5 + 2*x^3 - 4*x^5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 " Using the " }{TEXT 283 4 "diff" } {TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "di ff(f(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 " Using the " }{TEXT 284 1 "D" }{TEXT -1 9 " command: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " } {TEXT 285 1 "g" }{TEXT -1 1 "(" }{TEXT 286 1 "t" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "t*(t^5-6*t^3+4*t^2);" "6#*&%\"tG\"\"\",(*$F$\"\"&F%*&\" \"'F%*$F$\"\"$F%!\"\"*&\"\"%F%*$F$\"\"#F%F%F%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "g := t -> t*(t^5 - 6*t^3 + 4*t^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 300 5 "NOTE:" }{TEXT -1 71 " We \+ could ask Maple to simplify this expression. However, to use the " } {TEXT 301 4 "diff" }{TEXT -1 5 " and " }{TEXT 302 1 "D" }{TEXT -1 84 " commands to find the derivative, the given function does not need to \+ be simplified." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " Using the " }{TEXT 299 4 "diff" }{TEXT -1 9 " co mmand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(g(t), t);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 5 " NOTE:" }{TEXT -1 8 " Since " }{TEXT 310 1 "g" }{TEXT -1 18 " is a fun ction of " }{TEXT 311 1 "t" }{TEXT -1 46 " and we want to differentiat e with respect to " }{TEXT 312 1 "t" }{TEXT -1 73 ", we must type the \+ second entry in the parentheses as t. Of course, the " }{TEXT 424 10 "expression" }{TEXT -1 35 " which we want to differentiate is " } {TEXT 313 1 "g" }{TEXT 425 1 "(" }{TEXT 314 1 "t" }{TEXT 426 1 ")" } {TEXT -1 263 " and we type this as our first entry in the parentheses. As you can see, Maple did give us an answer. To obtain this answer, Maple used the Product Rule for differentiation, which we will discus s in a later lesson. Let's have Maple simplify this answer for us." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 8 "COMMENT:" }{TEXT -1 10 " It is a " }{TEXT 304 3 "BAD" }{TEXT -1 263 " idea to do what I did above. That is, to use the % symbol on a command line by itself in order to recall the last command which wa s executed. If you go back and look at ALL the times that we used the % symbol to recall the last executed command, it was on the " }{TEXT 305 4 "SAME" }{TEXT -1 169 " line as the command itself. The reason t hat it is a bad idea is because if we go back up the page to re-execut e a command line and then come back down and execute the " }{TEXT 306 12 "simplify(%);" }{TEXT -1 105 " command line, it's not going to simp lify the expression that we want simplified. On a personal note, I " }{TEXT 307 5 "NEVER" }{TEXT -1 102 " do what I did above because of th e chance for error. Hopefully, you will adopted the same practice. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " Now, let's use the " }{TEXT 308 1 "D" }{TEXT -1 33 " command and \+ simplify the answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D(g )(t); simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 315 5 "NOTE:" }{TEXT -1 18 " The name of our " }{TEXT 427 8 "function" }{TEXT -1 20 " in this example is " } {TEXT 316 1 "g" }{TEXT -1 85 ". This is the letter that we type in th e first set of parentheses which follows the " }{TEXT 317 1 "D" } {TEXT -1 72 " command. The variable which we are differentiating with respect to is " }{TEXT 318 1 "t" }{TEXT -1 86 ". This is the letter \+ that we type in the second set of parentheses which follows the " } {TEXT 319 1 "D" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4. " }{TEXT 320 1 "h" } {TEXT 321 0 "" }{TEXT 322 0 "" }{TEXT -1 1 "(" }{TEXT 323 1 "x" } {TEXT -1 4 ") = " }{XPPEDIT 18 0 "(3*x+2)*(4-x^2);" "6#*&,&*&\"\"$\"\" \"%\"xGF'F'\"\"#F'F',&\"\"%F'*$F(F)!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h := x -> (3*x + 2)*(4 - x^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ Using the " }{TEXT 324 4 "diff" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "diff(h(x), x); simplify(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ Using the " }{TEXT 325 1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D(h)(x); simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 326 0 "" }{TEXT 327 7 "Example" }{TEXT 328 0 "" }{TEXT -1 6 " If " } {TEXT 329 1 "s" }{TEXT -1 1 "(" }{TEXT 330 1 "t" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "1/2*t^4-1/3*t^2+2*t-5/2;" "6#,**(\"\"\"F%\"\"#!\"\"%\"t G\"\"%F%*(F%F%\"\"$F'F(F&F'*&F&F%F(F%F%*&\"\"&F%F&F'F'" }{TEXT -1 14 " , then find " }{TEXT 331 1 "s" }{TEXT -1 5 "'(5)." }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "s := t - > t^4/2 - t^2/3 + 2*t - 5/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "To find " }{TEXT 332 1 "s" }{TEXT -1 15 "'(5) using the " }{TEXT 333 4 "diff" }{TEXT -1 48 " command, you would type the followi ng commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diff(s(t), \+ t); subs(t=5, %);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "To find " }{TEXT 334 1 "s" }{TEXT -1 15 "'(5) using the " }{TEXT 335 1 "D" }{TEXT -1 20 " command, you would " }{TEXT 336 6 "simply" }{TEXT -1 28 " type the following command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(s)(5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The answer we get when we use the " }{TEXT 428 4 "diff" }{TEXT -1 16 " command is the " }{TEXT 429 10 "expression" }{TEXT -1 2 " " }{XPPEDIT 18 0 "2*t^3-2/3*t+2;" " 6#,(*&\"\"#\"\"\"*$%\"tG\"\"$F&F&*(F%F&F)!\"\"F(F&F+F%F&" }{TEXT -1 22 " and we must use the " }{TEXT 430 4 "subs" }{TEXT -1 33 " command in order to replace the " }{TEXT 431 1 "t" }{TEXT -1 9 " in this " } {TEXT 435 10 "expression" }{TEXT -1 35 " by 5 in order to get the valu e of " }{TEXT 432 1 "s" }{TEXT -1 41 "'(5). The answer we get when we use the " }{TEXT 433 1 "D" }{TEXT -1 14 " command is a " }{TEXT 434 8 "function" }{TEXT -1 1 " " }{TEXT 437 1 "s" }{TEXT -1 27 "' and we c an evaluate this " }{TEXT 436 8 "function" }{TEXT -1 56 " at 5 right a way as we did in order to get the value of " }{TEXT 256 1 "s" }{TEXT -1 7 "' at 5." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 337 0 "" }{TEXT 338 7 "Example" }{TEXT 339 0 " " }{TEXT -1 23 " Suppose the function " }{TEXT 340 1 "s" }{TEXT -1 95 " above represents a position function and you want to find the ins tantaneous velocity function " }{TEXT 341 1 "v" }{TEXT -1 74 " and sto re this function to v. You could do it in the following two ways:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " \+ using the " }{TEXT 342 1 "D" }{TEXT -1 47 " command, you would type the following command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " v := D(s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " NOTE: If we ask for " }{TEXT 343 1 "v" }{TEXT -1 14 " (5), which is " }{TEXT 344 1 "s" }{TEXT -1 54 "'(5), we should get the answer we got above, which is " }{XPPEDIT 18 0 "746/3;" "6#*&\"$Y(\" \"\"\"\"$!\"\"" }{TEXT -1 23 " . Let's see if we do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " using the " }{TEXT 345 4 "diff" }{TEXT -1 48 " command, you would type the following comm ands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diff(s(t), t); v : = unapply(%, t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 24 " Let's see if we get " }{XPPEDIT 18 0 "746/3;" "6#*&\"$Y(\"\"\"\"\"$!\"\"" }{TEXT -1 15 " if we ask for " }{TEXT 346 1 "v" }{TEXT -1 4 "(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v (5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 347 7 "RECALL:" }{TEXT -1 6 " The " }{TEXT 348 7 "unapply" }{TEXT -1 209 " command is the command which makes an expression, which is given as the first entry in the command, into a function of the variable, w hich is given as the second entry in the command. NOTE: When we use \+ the " }{TEXT 353 4 "diff" }{TEXT -1 50 " command to find the derivativ e of the expression " }{TEXT 354 1 "s" }{TEXT -1 1 "(" }{TEXT 355 1 "t " }{TEXT -1 50 "), our answer is an expression and not a function." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "L et's see what happens if try to write the function " }{TEXT 356 1 "v" }{TEXT -1 11 " using the " }{TEXT 357 4 "diff" }{TEXT -1 18 " command \+ like the " }{TEXT 358 1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v := t -> diff(s(t), t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Let's ask for \+ " }{TEXT 359 1 "v" }{TEXT -1 1 "(" }{TEXT 360 1 "t" }{TEXT -1 2 "):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(t);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " } {TEXT 361 7 "correct" }{TEXT -1 12 " answer for " }{TEXT 362 1 "v" } {TEXT -1 1 "(" }{TEXT 363 1 "t" }{TEXT -1 28 ")! Now, what if we ask \+ for " }{TEXT 364 1 "v" }{TEXT -1 4 "(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "The error message is indicating that the \+ 5 is the wrong type of parameter to use in the " }{TEXT 365 4 "diff" } {TEXT -1 9 " command." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 397 0 "" }{TEXT 398 7 "Example" } {TEXT 399 0 "" }{TEXT -1 54 " Find the slope of the tangent line to t he graph of " }{TEXT 256 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^3+ 4*x^2-5;" "6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F'*$F%\"\"#F'F'\"\"&!\"\"" } {TEXT -1 25 " at the point for which " }{TEXT 257 1 "x" }{TEXT -1 7 " = - 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y := x -> x^3 + 4*x^2 - 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 400 7 "NOTE: " }{TEXT -1 19 "Since we know that " }{TEXT 256 1 "y" }{TEXT -1 11 "'(- 2) = D(" }{TEXT 257 1 "y" } {TEXT -1 66 ")(- 2) gives the slope of the tangent line at the point f or which " }{TEXT 258 1 "x" }{TEXT -1 93 " = - 2, then we may type the following command in order to get the slope of the tangent line:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(y)(-2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 366 0 "" }{TEXT 367 7 "Example" }{TEXT 368 0 "" }{TEXT -1 57 " Find the equati on of the tangent line to the graph of " }{TEXT 369 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^3+4*x^2-5;" "6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F' *$F%\"\"#F'F'\"\"&!\"\"" }{TEXT -1 25 " at the point for which " } {TEXT 370 1 "x" }{TEXT -1 7 " = - 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y := x -> x^3 + 4*x^2 - 5 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 371 5 " NOTE:" }{TEXT -1 21 " Since we know that " }{TEXT 372 1 "y" }{TEXT -1 11 "'(- 2) = D(" }{TEXT 373 1 "y" }{TEXT -1 66 ")(- 2) gives the sl ope of the tangent line at the point for which " }{TEXT 374 1 "x" } {TEXT -1 44 " = - 2 and that the tangent point is ( - 2, " }{TEXT 375 1 "y" }{TEXT -1 97 "(- 2) ), then we may type the following command in order to get the equation of the tangent line:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "solve(y - y(-2) = D(y)(-2)*(x + 2), y);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 376 2 "OR " }{TEXT -1 38 " you could type the following command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "D(y)(-2)*(x + 2) + y(-2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 377 5 "NOTE:" }{TEXT -1 289 " This command is the result of sol ving the equation y - y(-2) = D(y)(-2)*(x + 2) above for y by adding y (-2) to both sides of the equation. Of course, neither one of these a nswers is an equation. Both answers are expressions, namely the right -side of the equation of the tangent line. " }{TEXT 380 36 "The equat ion of the tangent line is " }{TEXT 378 1 "y" }{TEXT 381 6 " = - 4" } {TEXT 379 1 "x" }{TEXT 382 5 " - 5." }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 402 5 " NOTE:" }{TEXT -1 57 " You can type the following command in order to \+ get the " }{TEXT 403 8 "equation" }{TEXT -1 21 " of the tangent line: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "y = D(y)(-2)*(x + 2) + \+ y(-2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 383 0 "" }{TEXT 384 7 "Example" }{TEXT 385 0 "" } {TEXT -1 22 " Graph the function " }{TEXT 256 1 "y" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "x^3+4*x^2-5;" "6#,(*$%\"xG\"\"$\"\"\"*&\"\"%F'*$F%\" \"#F'F'\"\"&!\"\"" }{TEXT -1 75 " and the tangent line to the graph o f the function at the point for which " }{TEXT 386 1 "x" }{TEXT -1 7 " = - 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "From the example above, \+ we found that the equation of the tangent line to the graph of the fun ction at the point for which " }{TEXT 387 1 "x" }{TEXT -1 23 " = - 2 i s the equation " }{TEXT 388 1 "y" }{TEXT -1 6 " = - 4" }{TEXT 389 1 "x " }{TEXT -1 47 " - 5. So, we could type the following command:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot( [ y(x), -4*x - 5 ], x= -4..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 390 2 "OR" }{TEXT -1 110 " we could type the foll owing command if we don't have the equation of the tangent line at the point for which " }{TEXT 391 1 "x" }{TEXT -1 35 " = - 2 and don't nee d the equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot( [ \+ y(x), D(y)(-2)*(x+2) + y(-2) ], x=-4..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 392 0 "" }{TEXT 393 7 "Example" }{TEXT 394 0 "" }{TEXT -1 56 " Find the equation of t he normal line to the graph of " }{TEXT 256 1 "y" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "2*x^4-x^3-x^2+3*x+9;" "6#,,*&\"\"#\"\"\"*$%\"xG\"\"%F&F &*$F(\"\"$!\"\"*$F(F%F,*&F+F&F(F&F&\"\"*F&" }{TEXT -1 25 " at the poi nt for which " }{TEXT 257 1 "x" }{TEXT -1 5 " = 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "y := x -> 2*x^4 - x^3 - x^2 + 3*x + 9;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 395 5 "NOTE:" }{TEXT -1 19 " We know that 1) " }{TEXT 256 1 "y" }{TEXT -1 9 "'(1) = D(" } {TEXT 257 1 "y" }{TEXT -1 64 ")(1) gives the slope of the tangent line at the point for which " }{TEXT 258 1 "x" }{TEXT -1 294 " = 1, 2) the normal line to the graph of a function at a point is the line perpend icular to the tangent line at that point, and 3) the slopes of perpend icular lines are negative reciprocals of one another. Thus, the slope of the normal line to the graph of the function at the point for whic h " }{TEXT 401 1 "x" }{TEXT -1 93 " = 1 is given by - 1 / D(y)(1). T he normal point, which is also the tangent point, is ( 1, " }{TEXT 396 1 "y" }{TEXT -1 94 "(1) ), then we may type the following command \+ in order to get the equation of the normal line:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y = -1/D(y)(1)*(x-1) + y(1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 404 0 "" }}{PARA 0 "" 0 "" {TEXT 405 7 "Example" }{TEXT 406 0 "" }{TEXT -1 22 " Graph the function " }{TEXT 256 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*x^4-x^3-x^2+3*x+9 ;" "6#,,*&\"\"#\"\"\"*$%\"xG\"\"%F&F&*$F(\"\"$!\"\"*$F(F%F,*&F+F&F(F&F &\"\"*F&" }{TEXT -1 94 " , the tangent line, and the normal line to t he graph of the function at the point for which " }{TEXT 257 1 "x" } {TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 105 "plot( [ y(x), D(y)(1)*(x-1) + y(1), -1/D(y) (1)*(x-1) + y(1) ], x=-3..4, y=4..18, color=[red,green,blue]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Di d we make a mistake? The normal line to graph of the function at the \+ point for which " }{TEXT 407 1 "x" }{TEXT -1 135 " = 1does not look pe rpendicular to the tangent line at this point. Click around the graph in order to make it active and click on the " }{TEXT 408 3 "1:1" } {TEXT -1 47 " icon above in order to make the scales on the " }{TEXT 409 1 "x" }{TEXT -1 5 " and " }{TEXT 410 1 "y" }{TEXT -1 15 " axes the same." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 18 "Practice Problems:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "In Problems 1 -3, find the derivative of \+ the following functions using the " }{TEXT 438 4 "diff" }{TEXT -1 17 " command and the " }{TEXT 439 1 "D" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{TEXT 262 1 "f" }{TEXT -1 1 "(" }{TEXT 261 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "6*x^3-5*x^2-x+9;" "6#,**& \"\"'\"\"\"*$%\"xG\"\"$F&F&*&\"\"&F&*$F(\"\"#F&!\"\"F(F.\"\"*F&" } {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "2. s(" }{TEXT 440 1 "t" } {TEXT -1 4 ") = " }{XPPEDIT 18 0 "15+3*t+4*t^3-4*t^5;" "6#,*\"#:\"\"\" *&\"\"$F%%\"tGF%F%*&\"\"%F%*$F(F'F%F%*&F*F%*$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 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3. " }{TEXT 441 1 "y" } {TEXT -1 4 " = " }{XPPEDIT 18 0 "(u^4-9)*(2*u^2+3);" "6#*&,&*$%\"uG\" \"%\"\"\"\"\"*!\"\"F(,&*&\"\"#F(*$F&F-F(F(\"\"$F(F(" }{TEXT -1 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "4. Find " }{TEXT 442 1 "g" }{TEXT -1 10 "'(- 3) if " }{TEXT 443 1 "g" }{TEXT -1 1 "(" }{TEXT 444 1 "x" } {TEXT -1 4 ") = " }{XPPEDIT 18 0 "5*x^4-8*x^2+x;" "6#,(*&\"\"&\"\"\"*$ %\"xG\"\"%F&F&*&\"\")F&*$F(\"\"#F&!\"\"F(F&" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 59 "5. Find the equation of the tangent line to the graph of " }{TEXT 445 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*x^3+4*x^2-5*x-3;" "6#,**&\"\"#\"\"\"*$%\"xG\"\"$F&F&*&\"\"%F&*$F(F% F&F&*&\"\"&F&F(F&!\"\"F)F/" }{TEXT -1 25 " at the point for which " } {TEXT 446 1 "x" }{TEXT -1 5 " = 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 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "6. Find the equatio n of the normal line to the graph of " }{TEXT 256 1 "y" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x^3+x^2+4*x+5;" "6#,**$%\"xG\"\"$\"\"\"*$F%\"\"#F '*&\"\"%F'F%F'F'\"\"&F'" }{TEXT -1 24 " at the point for which " } {TEXT 447 1 "x" }{TEXT -1 8 " = - 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 "" }}}}{MARK "1 0 0" 8 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }