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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 13 "Math 156 F'05" }}{PARA 
256 "" 0 "" {TEXT -1 5 "Lab 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 20 
"Derivatives in Maple" }{TEXT -1 7 "       " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "If f is a function of x then " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "diff(f
(x),x)" }{TEXT -1 76 "   is the derivative applied to an expressin and
 producing an expression    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 258 4 "D(f)" }{TEXT -1 76 "          is the derivative
 applied to a function and producing  a function." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Example:  Use Maple to f
ind the critical points and inflection points for f(x) = x^3 +12*x^2 -
 3*x +15" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:= x->x^3 + 12
*x^2 -3*x +15;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "plot(f(x),x);" }}{PARA 13 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We can get a better view by s
witching the domain " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plo
t(f(x),x=-20..6);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 109 "There are critical points around 8 and around 0 \+
and an inflection point around 4.  Let's get the exact values" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 12 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "df:=diff(f(x),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(df=0,x);" }}{PARA 11 
"" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Those are
 the critical points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dd
f:=diff(df,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 46 "The second derivative can also be computed by " }
{TEXT 256 15 "diff(f(x),x$2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(f(x),x$2);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fsolve(ddf=0,x);" }}{PARA 
11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "This i
s the inflection point." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 12 "Assig
nment 1" }{TEXT -1 28 "    Let  g(x)=x^4-6x^3+1.   " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 3 "1A)" }{TEXT -1 90 "    \+
 Plot  the graph of g(x) choosing the domain so that one can see all t
he local minima." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 262 4 "1B) " }{TEXT -1 106 "     Find the critical points of  g(
x) by  computing the derivative of g(x) and solve the equation g'(x)=0
" }{TEXT 263 1 "." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 5 "1C)  " }
{TEXT -1 85 "Find the minimum value of g(x)  by checking the values of
 g at the critical points.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 264 3 "1D)" }{TEXT -1 191 "
  Find all inflection points of g(x) by solving the equation g''(x)=0.
   Check on the plotted graph that this coincides with the places that
 the graph  changes from concave upwards to concave" }}{PARA 0 "" 0 "
" {TEXT -1 26 "downwards (or vice versa)." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 12 "Riemann Sums
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Examp
le:  Use Maple to study the Riemann sums for f(x) = x^2 on the interva
l [1,3]" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}
{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 73 
"The relevant Maple commands are leftbox, rightbox, leftsum and righsu
m.  " }}{PARA 0 "" 0 "" {TEXT -1 25 "For each you must specify" }}
{PARA 0 "" 0 "" {TEXT -1 21 "1)    the expression " }}{PARA 0 "" 0 "" 
{TEXT -1 24 "2)    the domain  x=a..b" }}{PARA 0 "" 0 "" {TEXT -1 33 "
t3)    he number of subIntervals " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:= x->x^2;\n" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can see the Riemann boxes using le
ft endpoints and 10 intervals." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "leftbox(f(x),x=1..3,10);" }}
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "
It is clear that the area of the Riemann boxes underestimates the area
 under the curve.   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Next we s
ee the boxes using right hand endpoints." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "rightbox(f(x),x=1..3,10);" }}{PARA 13 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "It is clear that this t
ime the boxes overestimate the area." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 79 "Let's now do the same example with shor
ter subintervals.  This time we use n=50" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "\nleftbox(f(x),x=1..3,50);\n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "rightbox(f(x),x=1..3,50);\n" }}{PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "It is nice to see
 both boxes at one time.  For this we give the pictures names and use \+
the " }{TEXT 268 7 "display" }{TEXT -1 22 " command, which needs " }
{TEXT 269 11 "with(plots)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "left:=leftbox(f(x),x=1..3,10):right:=rightbox(f(x),x=1..3,10):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 7 "" 
1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n
" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "display(left,right);" }}{PARA 13 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Do it again with more subinte
rvals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "left2:=leftbox(f(
x),x=1..3,50):right2:=rightbox(f(x),x=1..3,50):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "display(left2,right2); " }}{PARA 13 "" 1 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "You can see that the boxes
 are now better estimates." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Let
's do the numbers for Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L10:=leftsum(f(x),x=1..
3,10): evalf(L10);\n" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "R10:=rightsum(f(x),x=1..3,10): eval
f(R10);\n" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 155 "Since f(x) is increasing on [1,3] the true answer is bet
ween L10 and R10.  This is not yet a very accurate approximation.  We \+
will try again with larger n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "L1000:=leftsum(f(x),x=1..3,1000): evalf(L1000);\n" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "R100
0:=rightsum(f(x),x=1..3,1000): evalf(R1000);\n" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "To get accuracy \+
of four decimal places we would have to use n= 160,000.  This is a slo
w way to make accurate approximations." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 17 "Assignment P
art 2" }{TEXT -1 36 "   Consider the function f(x) = 1/x." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "2A)  Graph the rig
ht and left hand sums for the interval [1,2] with n=2,4,8,16,32 subint
ervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
61 "2B)  Compute the left and right hand sums at those intervals." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "2C)   How
 accurate are the estimates you get with 32 subintervals?" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 17 "Assignment Part 3
" }{TEXT -1 38 "   Repeat 2 with g(x) = x^4 on [-1,2]." }}}}{MARK "30 \+
0 0" 70 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }