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"Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Derivatives" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "In this worksheet, we will review the de finition of the derivative of a function, and compute our first exampl es." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "The definition of the Der ivative" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "First, the definition. \+ The derivative of a function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 3 " is " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "D(f)(x[0]) := limi t((f(x[0]+h)-f(x[0]))/h,h = 0);" "6#>--%\"DG6#%\"fG6#&%\"xG6#\"\"!-%&l imitG6$*&,&-F(6#,&&F+6#F-\"\"\"%\"hGF8F8-F(6#&F+6#F-!\"\"F8F9F>/F9F-" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 87 "provided, of course, t hat the limit exists. Since the derivative depends on the point " } {XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 50 " where it is eva luated, it is itself a function. " }{TEXT 256 5 "Maple" }{TEXT -1 19 " uses the notation " }{XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT -1 47 " for this function; the more usual notation is " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 69 " ' . You should by now be familiar wit h several interpretations for " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 47 " ' : it is the instantaneous rate of change of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6# \"\"!" }{TEXT -1 54 "; it is the slope of the tangent line to the grap h of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 "; if " }{XPPEDIT 18 0 "f(x[0]);" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 46 " represents the p osition of an object at time " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"! " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 3 " '( " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 18 ") is its velo city." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 " Computing derivatives" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Traditionally, the first function that one differentiates is the squaring function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "To \+ differentiate " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 14 " at any po int " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 6 ", say " } {XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 149 " = 3, we have t o compute the limit in the previous section. First, of course, we hav e to compute the difference quotient whose limit we are to take:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "With the difference quotient simplified, it is easy to ta ke the limit as " }{XPPEDIT 18 0 "proc (h) options operator, arrow; 0 \+ end;" "6#f*6#%\"hG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Actually, " }{TEXT 270 7 "Maple's " }{TEXT -1 1 " " }{TEXT 271 5 "limit" }{TEXT -1 90 " command is power ful enough that you do not even have to do the simplification explicit ly:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 8 "Warning:" }{TEXT -1 37 " You have just seen what is possibly " }{TEXT 273 7 "Maple's" } {TEXT -1 35 " worst feature: the syntax for its " }{TEXT 274 5 "limit " }{TEXT -1 71 " command. It cannot be stressed too strongly that tak ing the limit as " }{XPPEDIT 18 0 "proc (h) options operator, arrow; 0 end;" "6#f*6#%\"hG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 4 " is " }{TEXT 275 3 "not" }{TEXT -1 27 " the same thing as setting \+ " }{XPPEDIT 18 0 "h = 0;" "6#/%\"hG\"\"!" }{TEXT -1 20 ", but the synt ax of " }{TEXT 276 5 "limit" }{TEXT -1 36 " suggests that that is exac tly what " }{TEXT 277 5 "Maple" }{TEXT -1 94 " is doing. It is not, t hough, as you can see by comparing the previous command with this one: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "We can differentiate the function " }{XPPEDIT 18 0 " f;" "6#%\"fG" }{TEXT -1 123 " at any other point in exactly the same w ay (try it!), but it is no harder to do it once and for all at a gener al point x. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We can now substitute values of " } {XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 43 " to get the derivative at d ifferent points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Finally, we could note that it is often useful to express the derivative of a function as a function, r ather than an expression. " }{TEXT 278 5 "Maple" }{TEXT -1 24 " has t he useful command " }{TEXT 279 7 "unapply" }{TEXT -1 87 " which can be used to convert an expression to a function, so we can define a funct ion " }{TEXT 280 6 "fprime" }{TEXT -1 29 ", the derivative function of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 19 " , with the command" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Let's plot the function " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 40 " and its derivative on the same graph. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 268 7 "Maple's" }{TEXT -1 25 " differentiation commands " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{TEXT 257 1 "D" }{TEXT -1 19 " differentiates a " }{TEXT 258 9 "func tion," }{TEXT -1 74 " and that the derivative of a function is again a function. For example, " }{XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" } {TEXT -1 29 " can be evaluated at a point:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "One drawba ck to this approach to differentiation is that we first have to name t he function, as we did with the function " }{XPPEDIT 18 0 "f;" "6#%\"f G" }{TEXT -1 158 " above. If we are dealing with a lot of functions a t once, we might start to run out of letters, and even with one functi on, naming it is an extra step. In " }{TEXT 259 6 "Maple," }{TEXT -1 86 " you can actually get away without naming the function by defining it directly in the " }{TEXT 260 1 "D" }{TEXT -1 9 " command:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "but " }{TEXT 261 5 "Maple" }{TEXT -1 83 " \+ provides another command which is closer to ordinary mathematical prac tice (it is " }{TEXT 262 7 "Maple's" }{TEXT -1 19 " equivalent of the \+ " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 25 " no tation.) The command " }{TEXT 263 4 "diff" }{TEXT -1 19 " differentia tes an " }{TEXT 269 11 "expression," }{TEXT -1 35 " and gives back ano ther expression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "That's all there is to differentia tion in " }{TEXT 264 5 "Maple" }{TEXT -1 163 ". Just remember that th e derivative of a function is another function, and the derivative of \+ an expression is another expression, and that they are computed with \+ " }{TEXT 265 1 "D" }{TEXT -1 5 " and " }{TEXT 266 4 "diff" }{TEXT -1 42 " respectively. This is a nice example of " }{TEXT 267 5 "Maple" } {TEXT -1 43 " forcing you to think clearly, by the way. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Computing ta ngent lines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "As a first application of derivatives, l et's solve the tangent-line problem: how to write down the equation o f the tangent line to the graph of a function " }{XPPEDIT 18 0 "f;" "6 #%\"fG" }{TEXT -1 13 " at a point (" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6 #\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x[0]);" "6#-%\"fG6#&%\"xG6# \"\"!" }{TEXT -1 91 ") on the graph. We can now compute the slope of \+ the tangent line, using the derivative of " }{XPPEDIT 18 0 "f;" "6#%\" fG" }{TEXT -1 14 " evaluated at " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\" \"!" }{TEXT -1 116 ", and we know one point on the line (which one?), \+ so we can write down its equation in point-slope form. Here is a " } {TEXT 281 5 "Maple" }{TEXT -1 98 " computation to do this, define a ne w function whose graph is the relevant tangent line, and plot " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 61 " and the tangent line on th e same set of axes. We will take " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 94 " again to be the squaring function, and find the tangent \+ line at the point (3,9) on the graph." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "You should experiment with other points, \+ and other functions for " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 18 " " 0 "" {TEXT -1 28 "Higher-order Derivatives in " }{TEXT 282 5 "Maple " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Let's look at " }{TEXT 283 7 "Maple's" }{TEXT -1 313 " notation for derivatives of second and higher order; more precis ely, notations (plural), because there is one notation for derivatives of expressions and another for derivatives of functions. We will sta rt by giving examples of both notations, and then explain later how th ey are consistent with other parts of " }{TEXT 284 7 "Maple's" }{TEXT -1 8 " syntax." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "The Syntax" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We begin with the expression synta x. Recall that the command to differentiate an expression is " } {TEXT 291 4 "diff" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "This command has a simple extension: if you want to differentiate twice, just writ e the independent variable twice, separated with a comma:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The same trick works for higher-order derivatives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 65 "but there is an obvious problem with derivatives o f high order. " }{TEXT 285 5 "Maple" }{TEXT -1 45 " provides a conven ient shorthand: the symbol " }{TEXT 286 1 "$" }{TEXT -1 43 " is used t o produce lists so, for example, " }{TEXT 287 3 "x$2" }{TEXT -1 16 " i s the same as " }{TEXT 288 3 "x,x" }{TEXT -1 5 " and " }{TEXT 289 3 "x $4" }{TEXT -1 18 " is shorthand for " }{TEXT 290 7 "x,x,x,x" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "We now turn to the notation for differentiating fun ctions. Of course, we will first have to define a function that we ca n differentiate." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We already know t he notation for a first derivative:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "(Remember that while the derivative of an expression is an expres sion, the derivative of a function is a function.) Since " }{XPPEDIT 18 0 "D(f);" "6#-%\"DG6#%\"fG" }{TEXT -1 41 " is a function, we should be able to use " }{TEXT 292 1 "D" }{TEXT -1 70 " to differentiate it, so we might guess that the second derivative of " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 26 " can be produced this way:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 " and it works. As with expressions, though, you can imagine that this \+ simple extension would become awkward with high-order derivatives, and so there is a shorthand. It would be nice if the same " }{TEXT 293 1 "$" }{TEXT -1 40 " symbol were used, but the designers of " }{TEXT 294 5 "Maple" }{TEXT -1 31 " chose the intuitively-obvious " }{TEXT 295 2 "@@" }{TEXT -1 9 " instead." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Note that th ese are consistent with the answers we got for the second, fourth and \+ seventh derivatives of the expression " }{XPPEDIT 18 0 "x^7;" "6#*$%\" xG\"\"(" }{TEXT -1 11 " earlier. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 18 "Implicit Fu nctions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "The most direct way to define a function is to describe the expression it produces for a given input:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "It often happens in applications, however, that a functi on is not given in this way, but is given in terms of an equation rela ting two quantities. In this worksheet, we will explore the idea of a function defined implicitly through an equation, and see some of " } {TEXT 296 8 " Maple's" }{TEXT -1 42 " commands for manipulating such f unctions." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Curves and Equation s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Perhaps the best-known exampl e is the circle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 7 " Maple's" }{TEXT -1 44 " command to plot these types of equation is " } {TEXT 298 12 "implicitplot" }{TEXT -1 23 " , and it lives in the " } {TEXT 299 5 "plots" }{TEXT -1 9 " package." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here is an \+ implicit plot of the circle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "(You may have to click on the plot, then click on the 1:1 butt on to get the curve to appear circular.) Note that " }{TEXT 300 12 "i mplicitplot" }{TEXT -1 47 " uses expression syntax, and that the range of " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 57 " values is not optio nal. Here is the folium of Descartes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Well, actually, you should be suspicious about this picture: equation s of the type we are discussing usually describe smooth curves, and th e picture above is certainly not smooth, especially near the origin. \+ To check, lets ask " }{TEXT 301 5 "Maple" }{TEXT -1 35 " to plot the c urve more accurately." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Apparently the default value is more than 100 points! With a lit tle trial and error, you can produce the nice smooth curve below." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 302 4 "That" } {TEXT -1 28 " is the folium of Descartes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 4 "One " }{TEXT 307 8 "warning:" }{TEXT -1 1 " " } {TEXT 303 12 "implicitplot" }{TEXT -1 71 " does a lot more computation and uses a lot more memory than the basic " }{TEXT 304 4 "plot" } {TEXT -1 77 " command, especially when it is asked to use a lot of poi nts. If you use the " }{TEXT 306 9 "numpoints" }{TEXT -1 210 " option \+ with this command, therefore, you should start as we did above with a \+ relatively small number of points and work up slowly. It is also a go od idea to save your worksheet every time just before you use " } {TEXT 305 12 "implicitplot" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Implicit Differentiation" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Equations such as " }{TEXT 308 4 " eqn1" }{TEXT -1 5 " and " }{TEXT 309 4 "eqn2" }{TEXT -1 23 " from the \+ last section " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Consider " }{TEXT 310 4 "eqn1" } {TEXT -1 34 " first. If we treat it as giving " }{XPPEDIT 18 0 "y;" " 6#%\"yG" }{TEXT -1 18 " as a function of " }{XPPEDIT 18 0 "x;" "6#%\"x G" }{TEXT -1 19 " , how can we find " }{XPPEDIT 18 0 "dy/dx;" "6#*&%#d yG\"\"\"%#dxG!\"\"" }{TEXT -1 4 " ? " }{TEXT -1 28 " This procedure i s known as " }{TEXT 311 25 "implicit differentiation," }{TEXT -1 9 " a nd the " }{TEXT 312 5 "Maple" }{TEXT -1 26 " command which does it is \+ " }{TEXT 313 12 "implicitdiff" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The order of the arguments in this command is " }{TEXT 314 4 "very" }{TEXT -1 122 " important. The first argument is the equ ation to be differentiated. In this example, the variables in the equ ation are " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 7 " , and " }{TEXT 315 5 "Maple " }{TEXT -1 144 " needs to be told which one is to be taken as the ind ependent variable (remember that the equation itself is neutral on thi s point). The order " }{TEXT 316 4 "y, x" }{TEXT -1 7 " tells " } {TEXT 317 5 "Maple" }{TEXT -1 27 " to compute the derivative " } {XPPEDIT 18 0 "dy/dx;" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 23 " ; th at is to say that " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 33 " is th e independent variable and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 35 " is to be treated as a function of " }{XPPEDIT 18 0 "x;" "6#%\"xG " }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "For a less f amiliar example, do the same analysis with the folium of Descartes:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }