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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Lab 15 Maple Activities fo
r November 7:  (Lesson 12 - Higher-Order Derivatives)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "In this lab, we will lear
n the Maple commands for finding higher-order derivatives of a functio
n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
259 8 "Examples" }{TEXT -1 88 "  Find the second, third, and fourth de
rivatives of the following functions using Maple." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1.  " }{TEXT 460 1 
"f" }{TEXT -1 1 "(" }{TEXT 461 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 
0 "2*x^3-4*x^2+x-5;" "6#,**$)%\"xG\"\"$\"\"\"\"\"#*$)F&F)F(!\"%F&\"\"
\"!\"&F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := x -> 2*x^3 - 4*x^2 + 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 462 4 "diff" }{TEXT -1 75 " command, there are two ways to find \+
the second derivative of the function " }{TEXT 465 1 "f" }{TEXT -1 28 
".  One way is the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "diff(f(x),x, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 35 "    The other way is the following:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(f(x), x$2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
463 5 "NOTE:" }{TEXT -1 110 " The answer given by both commands is an \+
expressional answer and is not a functional answer in terms of Maple.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "    Using the " }{TEXT 464 4 "diff" }{TEXT -1 74 " command, there \+
are two ways to find the third derivative of the function " }{TEXT 
466 1 "f" }{TEXT -1 28 ".  One way is the following:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "diff(f(x), x, x, x);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "    The other way \+
is the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(f
(x), x$3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "    Using the " }{TEXT 467 4 "diff" }{TEXT -1 75 " comman
d, there are two ways to find the fourth derivative of the function " 
}{TEXT 468 1 "f" }{TEXT -1 28 ".  One way is the following:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(f(x), x, x, x, x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "  \+
  The other way is the following:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "diff(f(x), x$4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 469 5 "NOTE:" }{TEXT -1 
50 " The second way is probably easier when using the " }{TEXT 470 4 "
diff" }{TEXT -1 42 " command to find higher-order derivatives." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "  \+
  There is only one way to find the second derivative of the function \+
" }{TEXT 471 1 "f" }{TEXT -1 16 " when using the " }{TEXT 472 1 "D" }
{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(D
@@2)(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 473 5 "NOTE:" }{TEXT -1 53 " The answer is a fu
nctional answer in terms of Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "    To find the third derivativ
e of the function " }{TEXT 474 1 "f" }{TEXT -1 11 " using the " }
{TEXT 475 1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "(D@@3)(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 50 "    To find the fourth derivative of \+
the function " }{TEXT 476 1 "f" }{TEXT -1 11 " using the " }{TEXT 477 
1 "D" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "(D@@4)(f)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 4 "2.  " }{TEXT 478 1 "y" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "x^2-3*x+6;" "6#,(*$)%\"xG\"\"#\"\"\"\"\"\"F&!\"$\"\"'F)
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "y := x -> x^2 - 3*x + 6;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 10 "Using the " }{TEXT 479 4 "diff" }
{TEXT -1 68 " command, the second, third, and fourth derivatives of th
e function " }{TEXT 480 1 "y" }{TEXT -1 40 " are obtained by the follo
wing commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff(y(x)
, x$2); diff(y(x), x$3); diff(y(x), x$4);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3.  " }{TEXT 481 1 "s" }
{TEXT -1 1 "(" }{TEXT 482 1 "t" }{TEXT -1 6 ") = 3/" }{XPPEDIT 18 0 "t
^4;" "6#*$)%\"tG\"\"%\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s := t -> 3*t^(-4
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Using the
 " }{TEXT 484 1 "D" }{TEXT -1 68 " command, the second, third, and fou
rth derivatives of the function " }{TEXT 486 1 "s" }{TEXT -1 40 " are \+
obtained by the following commands:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "(D@@2)(s)(t); (D@@3)(s)(t); (D@@4)(s)(t);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "4.  " }
{TEXT 496 1 "g" }{TEXT -1 1 "(" }{TEXT 498 1 "w" }{TEXT -1 4 ") = " }
{XPPEDIT 18 0 "(5-2*w)^(1/3);" "6#*$),&\"\"&\"\"\"%\"wG!\"##F'\"\"$\"
\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := w -
> (5 - 2*w)^(1/3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(w);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
10 "Using the " }{TEXT 488 4 "diff" }{TEXT -1 68 " command, the second
, third, and fourth derivatives of the function " }{TEXT 501 1 "g" }
{TEXT -1 40 " are obtained by the following commands:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff(g(w), w$2); diff(g(w), w$3); d
iff(g(w), w$4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 4 "5.  " }{TEXT 499 1 "h" }{TEXT -1 1 "(" }{TEXT 492 1 "
x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "tan*x;" "6#*&%$tanG\"\"\"%\"xGF%
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "h := x -> tan(x);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "h(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Using the " }{TEXT 500 1 "D" }{TEXT -1 
68 " command, the second, third, and fourth derivatives of the functio
n " }{TEXT 503 1 "h" }{TEXT -1 40 " are obtained by the following comm
ands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "(D@@2)(h)(x); (D@@
3)(h)(x); (D@@4)(h)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 504 5 "NOTE:" }{TEXT -1 22 "  The
 expression  1 + " }{XPPEDIT 18 0 "tan^2*x;" "6#*&)%$tanG\"\"#\"\"\"%
\"xG\"\"\"" }{TEXT -1 57 "  in all three of the answers is same as the
 expression  " }{XPPEDIT 18 0 "sec^2*x;" "6#*&)%$secG\"\"#\"\"\"%\"xG
\"\"\"" }{TEXT -1 66 ".  We could simplify the answers by replacing th
e expression  1 + " }{XPPEDIT 18 0 "tan^2*x;" "6#*&%$tanG\"\"#%\"xG\"
\"\"" }{TEXT -1 34 "  by the equivalently expression  " }{XPPEDIT 18 
0 "sec^2*x;" "6#*&%$secG\"\"#%\"xG\"\"\"" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "(D@@2)(h)(x); subs(1+tan(x)^2=sec(
x)^2,%); (D@@3)(h)(x); subs(1+tan(x)^2=sec(x)^2,%); (D@@4)(h)(x); subs
(1+tan(x)^2=sec(x)^2,%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 505 0 "" }
{TEXT 506 8 "Examples" }{TEXT 507 0 "" }{TEXT -1 7 "  Find " }{TEXT 
508 1 "y" }{TEXT -1 6 "\" and " }{TEXT 509 1 "y" }{TEXT -1 40 "'\" for
 the following implicit functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1.  " }{XPPEDIT 18 0 "x^2+y^2 = \+
4;" "6#/,&*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF(F)F*\"\"%" }{TEXT -1 1 " \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
8 "To find " }{TEXT 510 1 "y" }{TEXT -1 39 "\", we would type the foll
owing command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "implicitd
iff(x^2 + y^2 = 4, y, x$2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 512 5 "NOTE:" }{TEXT -1 72 " Map
le does not simplify the answer for us by replacing the expression  " 
}{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F'" }
{TEXT -1 7 "  by 4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 8 "To find " }{TEXT 256 1 "y" }{TEXT -1 40 "'\", we
 would type the following command:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "implicitdiff(x^2 + y^2 = 4, y, x$3);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
514 5 "NOTE:" }{TEXT -1 66 " Again this answer can be simplified by re
placing the expression  " }{XPPEDIT 18 0 "x^2+y^2" "6#,&*$%\"xG\"\"#\"
\"\"*$%\"yG\"\"#F'" }{TEXT -1 7 "  by 4:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "implicitdiff(x^2 + y^2 = 4, y, x$3); subs(x^2 + y^2 =
 4, %);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "2.  " }{XPPEDIT 18 0 "x^2-3*x*y+y^4 = 2;" "6#/,(*$)%\"xG\"
\"#\"\"\"\"\"\"*&F'F*%\"yGF*!\"$*$)F,\"\"%F)F*F(" }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "To \+
find " }{TEXT 515 1 "y" }{TEXT -1 2 "\":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "implicitdiff(x^2 - 3*x*y + y^4 = 2, y, x$2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 516 5 "NOTE:" }{TEXT -1 81 " If you check the lecture notes for \+
this problem, the fraction can be simplified." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To find y'\":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "implicitdiff(x^2 - 3*x*y + y
^4 = 2, y, x$3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 517 0 "" }{TEXT 518 7 "Example" }{TEXT 
519 0 "" }{TEXT -1 6 "  If  " }{TEXT 520 1 "s" }{TEXT -1 1 "(" }{TEXT 
521 1 "t" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "t/(t+1);" "6#*&%\"tG\"\"
\",&F$\"\"\"F'F'!\"\"" }{TEXT -1 91 "  is the position function which \+
gives the position (in centimeters) of a particle at time " }{TEXT 
522 1 "t" }{TEXT -1 74 " (in hours), then find the velocity and accele
ration of the particle when " }{TEXT 523 1 "t" }{TEXT -1 12 " = 3 hour
s. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s := t -> t/(t+1);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s(t);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "To find the instan
taneous velocity function " }{TEXT 524 1 "v" }{TEXT -1 13 " using Mapl
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "v := D(s);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(t);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Now, to find the vel
ocity of the particle when " }{TEXT 525 1 "t" }{TEXT -1 11 " = 3 hours
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(3);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 527 7 "
Answer:" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "1/16;" "6##\"\"\"\"#;" }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "cm/hr;" "6#*&%#cmG\"\"\"%#hrG!\"\"" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 48 "To find the instantaneous acceleration function " }
{TEXT 526 1 "a" }{TEXT -1 13 " using Maple:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "a := (D@@2)(s);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "a(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 51 "Now, to find the acceleration of the part
icle when " }{TEXT 528 1 "t" }{TEXT -1 11 " = 3 hours:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a(3);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 529 8 "Answer: " 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "-1/32;" "6##!\"\"\"#K" }{TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "cm/(hr^2);" "6#*&%#cmG\"\"\"*$)%#hrG\"\"#F%!\"\"" }
{TEXT -1 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }