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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 652 "This work
sheet is designed to accompany sections 2.1 and/or 2.2 of the Hughes-H
allett et al Calculus text or sections 2.1 and/or 2.2 of the Smith/Min
ton text.  Its purposes are to illustrate visually how a tangent line \+
to a graph can be approximated by secant lines and how a sequence of s
ecant lines connecting x0 and x0+h converges to the tangent line at x0
 in the limit h -> 0.  The user can specify the function, point, and v
iewing window to be used for the demonstration.  After that, the works
heet is simply executed without any need for editing.  One can skip th
e sections with backward difference and central difference secant line
s if desired." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;  with(plots):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 76 "
Define the function and choose the point of interest and the viewing w
indow." }}{PARA 0 "" 0 "" {TEXT -1 272 "(The user may change these set
tings to get a different demonstration of the same phenomena.  The vie
wing window should be roughly symmetric about the point x0 and the ver
tical range should be sufficient for the graph to illustrate the funct
ion f over the entire interval.)" }}{PARA 257 "" 0 "" {TEXT 262 97 "Th
e rest of the worksheet does not need any editing.  Just hit the enter
 key to run each section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := x^3-5*x^2+8*x-2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x0 := 1;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "window := [0..2.5,-2..3];" }}}{EXCHG {PARA 258 "" 
0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 72 "The program sketch
es the function in black and its tangent line in blue." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y0 := \+
subs(x=x0,f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "mtan := subs(x=x0,
diff(f,x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ytan := y0+mtan*(x-x
0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "p1 := plot(f,x=window[1],win
dow[2],thickness=3,color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 
"p2 := textplot([x0,y0,`o`],font=[HELVETICA,BOLD,12],color=blue):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "p3 := plot(ytan,x=window[1],window[
2],thickness=2,color=blue): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dis
play(\{p1,p2,p3\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 115 "The program constructs a set of secant approx
imations to the tangent, using points to the right of the given point.
" }}{PARA 0 "" 0 "" {TEXT 258 88 "The first of these is displayed in o
range, along with the function and the tangent line." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dx0:=min(
rhs(window[1])-x0,x0-lhs(window[1])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "y1 := dx -> subs(x=x0+dx,f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "msec := dx -> (y1(dx)-y0)/dx:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "ysec := dx -> y0+msec(dx)*(x-x0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 322 "p4 := [textplot([x0+dx0,y1(dx0),`o`],font=[HELVETICA,BOLD,12],c
olor=coral),\n       textplot([x0+.6*dx0,y1(.6*dx0),`o`],font=[HELVETI
CA,BOLD,12],color=coral),\n       textplot([x0+.3*dx0,y1(.3*dx0),`o`],
font=[HELVETICA,BOLD,12],color=coral),\n       textplot([x0+.1*dx0,y1(
.1*dx0),`o`],font=[HELVETICA,BOLD,12],color=coral)]:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 289 "p5 := [plot(ysec(dx0),x=window[1],window[2],thi
ckness=2,color=coral),\n       plot(ysec(.6*dx0),x=window[1],window[2]
,thickness=2,color=coral),\n       plot(ysec(.3*dx0),x=window[1],windo
w[2],thickness=2,color=coral),\n       plot(ysec(.1*dx0),x=window[1],w
indow[2],thickness=2,color=coral)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "display(\{p1,p2,p3,p4[1],p5[1]\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 82 "Choosing comparison poin
ts closer to the given point gives a better approximation." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dis
play(\{p1,p2,p3,p4[2],p5[2]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "display(\{p1,p2,p3,p4[3],p5[3]\});" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "display(\{p1,p2,p3,p4[4],p5[4]\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display(\{p1,p2,p3,op(p4),op(p5)\})
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
259 98 "We can also make secant approximations to the tangent using po
ints to the left of the given point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "p6 := [textplot([x0-dx0,
y1(-dx0),`o`],font=[HELVETICA,BOLD,12],color=coral),\n       textplot(
[x0-.6*dx0,y1(-.6*dx0),`o`],font=[HELVETICA,BOLD,12],color=coral),\n  \+
     textplot([x0-.3*dx0,y1(-.3*dx0),`o`],font=[HELVETICA,BOLD,12],col
or=coral),\n       textplot([x0-.1*dx0,y1(-.1*dx0),`o`],font=[HELVETIC
A,BOLD,12],color=coral)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "p7 :=
 [plot(ysec(-dx0),x=window[1],window[2],thickness=2,color=coral),\n   \+
    plot(ysec(-.6*dx0),x=window[1],window[2],thickness=2,color=coral),
\n       plot(ysec(-.3*dx0),x=window[1],window[2],thickness=2,color=co
ral),\n       plot(ysec(-.1*dx0),x=window[1],window[2],thickness=2,col
or=coral)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "display(\{p1,p2,p3,p
6[1],p7[1]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "display(
\{p1,p2,p3,p6[2],p7[2]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "display(\{p1,p2,p3,p6[3],p7[3]\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "display(\{p1,p2,p3,p6[4],p7[4]\});" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "display(\{p1,p2,p3,op(p6),op(p7)\});" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 117 "
We can also make secant approximations to the tangent using points to \+
both the right and the left of the given point." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dx0:=min(rh
s(window[1])-x0,x0-lhs(window[1])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "y1 := dx -> subs(x=x0+dx,f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"mcd := dx -> (y1(dx)-y1(-dx))/(2*dx):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "ycd := dx -> y1(-dx)+mcd(dx)*(x-x0+dx):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 285 "p8 := [plot(ycd(dx0),x=window[1],window[2],thickness
=2,color=coral),\n       plot(ycd(.6*dx0),x=window[1],window[2],thickn
ess=2,color=coral),\n       plot(ycd(.3*dx0),x=window[1],window[2],thi
ckness=2,color=coral),\n       plot(ycd(.1*dx0),x=window[1],window[2],
thickness=2,color=coral)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "displ
ay(\{p1,p2,p3,p4[1],p6[1],p8[1]\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "display(\{p1,p2,p3,p4[2],p6[2],p8[2]\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(\{p1,p2,p3,p4[3],p6[3],p8[3
]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(\{p1,p2,p3
,p4[4],p6[4],p8[4]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d
isplay(\{p1,p2,p3,op(p4),op(p6),op(p8)\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 116 "The computer now constr
ucts an animation for the case where the additional point is to the ri
ght of the given point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 
109 "To view the animation, click inside the picture to bring up a new
 toolbar.  Then click on the forward button." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "p9:=animate(ysec(
dx0*0.01^t),x=window[1],t=0..1,frames=100,view=window,thickness=2,colo
r=coral):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(\{p1,p2,p3,p9
\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "7 0 0" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }