{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 18 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
278 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 18 0 
0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 282 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
283 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 1 18 0 0 0 0 1 1 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 286 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 287 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
288 "" 1 18 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 18 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 18 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 }
{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Warning" -1 261 1 {CSTYLE "" -1 -1 "Courier" 1 18 0 0 255 1 
1 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu
t" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 1 2 2 2 2 2 1 1 1 1 
}3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 263 1 {CSTYLE "
" -1 -1 "Times" 1 18 0 0 0 1 1 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 260 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 44 "PARTIAL DERIVATIVES. DIRECTIONAL DERIVATI
VES" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 279 48 "DIFFERENTIABILIT
Y. TANGENT PLANE AND NORMAL LINE" }}{PARA 259 "" 0 "" {TEXT 267 43 "  \+
COMPUTATION AND GEOMETRIC  INTERPRETATION" }}{PARA 260 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT 280 69 "We will consider the real-va
lued function of two variables defined by" }}{PARA 256 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 256 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 
256 "> " 0 "" {MPLTEXT 1 0 32 "f:=(x,y)->exp(-x^2)+exp(-4*y^2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 85 "T
he graph of this function of two variables is represented by the follo
wing surface S" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 ">
 " 0 "" {MPLTEXT 1 0 106 "plot3d(f(x,y),x=-3..3,y=-3..3,grid=[49,49],a
xes=framed,shading=ZHUE,orientation=[142,65],labels=[x,y,z]); " }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 284 20 "PARTIAL DERIVATIVES:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 105 "We want to \+
obtain the partial derivatives at the point (x,y)=(-1,0) and their geo
metrical interpretation." }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
259 106 "First, we will intersect the surface S, corresponding to the \+
function f(x,y) with the verical plane x=-1. " }}{PARA 256 "" 0 "" 
{TEXT 289 87 "It will result in a curve lying in S and in this plane, \+
given by the function f(-1,y). " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 111 "Z3:=plot3d(f(x,y),x=-3..3
,y=-3..3,grid=[49,49],axes=framed,shading=ZHUE,orientation=[-145,45],l
abels=[x,y,z]): " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT 271 10 "Plane x=-1" }}{PARA 256 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 17 "z22:=(x,y,z)-> x;" }}
}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 70 "Z4:=implicitplot3d(z22(x,
y,z)=-1,x=-2..2,y=-3..3, z=0..1.5,color=red):" }}}{EXCHG {PARA 256 "> \+
" 0 "" {MPLTEXT 1 0 67 "display3d([Z3,Z4],axes=framed,labels=[x,y,z],o
rientation=[142,65]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 272 22 "Curve of Intersection:
" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 8 "f(-1,y);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
22 "plot(f(-1,y),y=-3..3);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT 260 136 "The slope of the tangent to the gra
ph of this curve at the given point is the derivative of f(-1,y) with \+
respect to y evaluated at y=0. " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 32 "PDY:=unapply(diff(f(-1,y),
y),y);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 7 "PDY(0);" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 262 
205 "This value is also called the partial derivative of f with respec
t to y, f_y (x,y), evaluated at (x,y)=(-1,0). In the graph below, a di
rection vector of the tangent line at the given point has been drawn. \+
" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
256 "> " 0 "" {MPLTEXT 1 0 85 "tv:=arrow([-1,0,exp(-1)+1],[-1,-5,exp(-
1)+1],width=.035,color=black,difference=true):" }}}{EXCHG {PARA 256 ">
 " 0 "" {MPLTEXT 1 0 70 "display3d([Z3,Z4,tv],axes=framed,labels=[x,y,
z],orientation=[142,65]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }
}{PARA 256 "" 0 "" {TEXT 258 165 "Now, we will obtain an intersection \+
of the surface S, with the vertical plane y=0. It will result in a dif
ferent curve lying in S and also located on the new plane. " }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
17 "z33:=(x,y,z)-> y;" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 69 "
Z5:=implicitplot3d(z33(x,y,z)=0,x=-3..3,y=-3..3, z=0..2.2,color=red):
" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 67 "display3d([Z3,Z5],axe
s=framed,labels=[x,y,z],orientation=[142,65]);" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 273 22 "
Curve of Intersection:" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 256 "> " 0 "" {MPLTEXT 1 0 7 "f(x,0);" }}}{EXCHG {PARA 256 "> " 
0 "" {MPLTEXT 1 0 22 "plot(f(x,0), x=-3..3);" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 261 136 "The slope of the
 tangent to the graph of this curve at the given point is the derivati
ve of f(x,0) with respect to x evaluated at x=-1. " }}{PARA 256 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 31 "PDX:=un
apply(diff(f(x,0),x),x);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
8 "PDX(-1);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 265 110 "This value is \+
also called the partial derivative of f with respect to x, f_x (x,y), \+
evaluated at (x,y)=(-1,0)." }{TEXT -1 0 "" }{TEXT 281 0 "" }{TEXT 282 
0 "" }{TEXT 283 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 
"" {TEXT 264 82 "We can also obtain the direction vector of the tangen
t line using vector calculus." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 58 "i:=vector([1,0,0]); j:=vec
tor([0,1,0]);k:=vector([0,0,1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 90 "Parametric representation of the
 curve of intersection with the plane y=0. Parameter: \"x\"." }}{PARA 
256 "" 0 "" {TEXT -1 28 "Tangent vector to the curve." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 31 "r1:=x-
>x*i+0*j+(exp(-x^2)+1)*k;" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
6 "D(r1);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 16 "aux1:=D(r1)(
-1);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 16 "hv:=evalm(aux1);
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
263 95 " In the graph below, a direction vector of the tangent line at
 the given point has been drawn. " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 63 "tv2:=arrow([-1,0,exp(-1)
+1],hv,width=.09,color=black,length=2):" }}}{EXCHG {PARA 256 "> " 0 "
" {MPLTEXT 1 0 71 "display3d([Z3,Z5,tv2],axes=framed,labels=[x,y,z],or
ientation=[142,65]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT 266 52 "The surface with the two tangent vecto
r is shown now" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 
"> " 0 "" {MPLTEXT 1 0 71 "display3d([Z3,tv,tv2],axes=framed,labels=[x
,y,z],orientation=[142,65]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 256 "" 0 "" {TEXT -1 133 "We will see later that these two vec
tors are located in a special plane called the tangent plane to the su
rface S at the given point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 285 24 "DIRECTIONAL DERIVATIVES:" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 524 "We can choose a different plane to intersect the given s
urface and obtain a new tangent line at the given point, for the curve
 of intersection of this plane with the surface. In fact, if we choose
 the plane y = 2x+2 in the xyz-space, the above procedure will lead us
 to find a derivative of f(x,y) in the direction of the vector (1,2) (
which is the direction vector of the line y=2x+2 in the xy-plane) at t
he point (-1,0). This will be called  directional derivative of  f(x,y
) in the direction of (1,2) at the point (-1,0)." }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 23 "z66:=(x,y
,z)-> y-2*x-2;" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 71 "Z6:=imp
licitplot3d(z66(x,y,z)=0,x=-2.5..1,y=-3..3, z=0..1.5,color=red):" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 67 "display3d([Z6,Z3],axes=fra
med,labels=[x,y,z],orientation=[142,65]);" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT 268 90 "Parametric representation of the curve at the intersec
tion in terms of  the parameter \"x\"." }}{PARA 256 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 17 "aux3:=f(x,2*x+2);
" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 36 "r3:=unapply(x*i+(2*x+
2)*j+aux3*k,x);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT 270 57 "Direction vector of the curve at the interse
ction (x=-1)." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "
> " 0 "" {MPLTEXT 1 0 6 "D(r3);" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 16 "aux2:=D(r3)(-1);" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 17 "hv2:=evalm(aux2);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT 269 1 " " }{TEXT 291 95 "In the grap
h below, a direction vector of the tangent line at the given point wil
l been drawn. " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 
"> " 0 "" {MPLTEXT 1 0 66 "tv3:=arrow([-1,0,exp(-1)+1],hv2,width=.04,c
olor=black,length=2.5):" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
71 "display3d([Z3,Z6,tv3],axes=framed,labels=[x,y,z],orientation=[142,
65]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 208 "The angle theta betwe
en the vector hv2=(1,2,2*exp(-1)) and its projection in the xy-plane, \+
the vector (1,2,0) can be used to compute the slope of the intersectin
g curve at the point (-1,0,2*exp(-1)). In fact " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 23 "aux11:=ve
ctor([1,2,0]);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 26 "aux12:=
dotprod(hv2,aux11);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 19 "au
x13:=norm(hv2,2);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 21 "aux1
4:=norm(aux11,2);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 35 "thet
a:=arccos(aux12/(aux13*aux14));" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 33 "slopeCurve:=simplify(tan(theta));" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 164 "This is th
e same value obtained from the more common formula consisting of the d
ot product of the gradient at (-1,0,exp(-1)+1) with the unit vector (1
,2,0)/sqrt(5)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "
" 0 "" {TEXT -1 160 "We will now find the tangent vector for the curve
 of intersection of the surface S with another oblique plane, the plan
e y = -2x-2, the above procedure leads to" }}{PARA 256 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 23 "z77:=(x,y,z)-> \+
y+2*x+2;" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 71 "Z7:=implicitp
lot3d(z77(x,y,z)=0,x=-2.5..1,y=-3..3, z=0..1.5,color=red):" }}}{EXCHG 
{PARA 256 "> " 0 "" {MPLTEXT 1 0 67 "display3d([Z7,Z3],axes=framed,lab
els=[x,y,z],orientation=[142,65]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
274 75 "Parametric representation in terms of \"x\" for the curve at t
he intersection" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
256 "> " 0 "" {MPLTEXT 1 0 18 "aux4:=f(x,-2*x-2);" }}}{EXCHG {PARA 
256 "> " 0 "" {MPLTEXT 1 0 37 "r4:=unapply(x*i+(-2*x-2)*j+aux4*k,x);" 
}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
276 49 "Direction Vector of the curve at the intersection" }}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
6 "D(r4);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 16 "aux5:=D(r4)(
-1);" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 17 "hv3:=evalm(aux5);
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
275 95 " In the graph below, a direction vector of the tangent line at
 the given point has been drawn. " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 66 "tv4:=arrow([-1,0,exp(-1)
+1],hv3,width=.04,color=black,length=2.5):" }}}{EXCHG {PARA 256 "> " 
0 "" {MPLTEXT 1 0 71 "display3d([Z3,Z7,tv4],axes=framed,labels=[x,y,z]
,orientation=[142,65]);" }}}{PARA 256 "" 0 "" {TEXT 277 73 "The surfac
e with the four tangent vector at the point (-1,0) is shown now" }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 
1 0 80 "display3d([Z3,tv,tv2,tv3,tv4],axes=framed,labels=[x,y,z],orien
tation=[-153,75]);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 
278 368 "These four vectors lie in plane. That plane is defined as the
 tangent plane to the surface of the function f(x,y) at the point (x,y
)=(-1,0). However, a condition for the existence of this plane is that
 the surface have a tangent line in every direction, and that the tang
ent vectors lie in that plane. A function satisfying this condition is
 said to be differentiable." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 286 0 "" }{TEXT 287 0 "" }
{TEXT 288 14 "Tangent Plane:" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 103 "Now, we will obtain an equation for th
e tangent plane to the surface S, at the point (-1,0,exp(-1)+1). " }}
{PARA 256 "" 0 "" {TEXT -1 110 "From theoretical results, we know that
 a normal vector to this plane is given by the gradient of the functio
n " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 21 "F:=(x,y,z)->z-f(x,y);" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 9 "F(x,y,z);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 
31 "grad(F(x,y,z),vector([x,y,z]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 57 "Nv:=(x,y,z)->ve
ctor([2*x*exp(-x^2), 8*y*exp(-4*y^2), 1]);" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT -1 30 "Evaluated at (-1,0,exp(-1)+1)." }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 26 "aux55:=Nv
(-1,0,exp(-1)+1);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 71 "Therefore the tangent plane at the point  (-1
,0,exp(-1)+1) is given by " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 55 "aux6:=dotprod(aux55,vector
([(x+1),(y),(z-exp(-1)-1)]));" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 
1 0 23 "P:=unapply(aux6,x,y,z);" }}}{EXCHG {PARA 256 "> " 0 "" 
{MPLTEXT 1 0 66 "NVV:=arrow([-1,0,exp(-1)+1],aux55,width=.06,color=red
,length=1.8):" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 83 "PL:=impl
icitplot3d(P(x,y,z)=0,x=-2..2,y=-2..2,z=0..2,color=yellow,grid=[15,15,
15]):" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 71 "display3d([Z3,PL
,NVV],axes=framed,labels=[x,y,z],orientation=[-99,65]);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 68 "Now, we will draw the norma
l line to S passing through the point P. " }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 46 "r1:=t->vector([-2
*exp(-1)*t-1,0,t+exp(-1)+1]);" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 
1 0 81 " NL:=spacecurve(evalm(r1(t)), t=0..1.8, color=blue,axes=NORMAL
, labels=[x,y,z]):\n" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 74 "d
isplay3d([Z3,PL,NVV,NL],axes=framed,labels=[x,y,z],orientation=[-99,65
]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "84 0 1" 
253 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }