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8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
1/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
MATH 209Calculus, III
Volker Runde
University of Alberta
Edmonton, Fall 2011
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
2/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, I
Recall...
Let z=f(x, y) be a function of two variables. Then:fx
: slope of the tangent line to the graph off in the
direction of the x-axis;fy
: slope of the tangent line to the graph off in thedirection of the y-axis.
QuestionWhat about the slope of a tangent line to the graph off in thedirection of an arbitrary line in the xy-plane?
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
3/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, II
DefinitionThedirectional derivativeoff at (x0, y0) in the direction of theunit vector u=a, b is the limit
Duf(x0, y
0) = lim
h0
f(x0+ha, y0+hb)f(x0, y0)h
if it exists.
Example
What is Duf(1, 2), for f(x, y) =x3
3xy+ 4y2
and ugiven bythe angle = 6 , i.e.,
u= cos
6, sin
6 =
3
2 ,
1
2
?
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
4/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, III
An application of the chain rule
Set g(t) :=f(x0+ta, y0+tb), so that
Duf(x0, y0) = limh0
f(x0+ha, y0+hb)f(x0, y0)h
= limh0
g(h)g(0)h
= dgdt
(0).
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
5/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, IV
An application of the chain rule (continued)
With g(t) =f(x, y), x=x0+ta, and y=y0+tb, the chainrule yields
Duf(x0, y0) = dg
dt(0)
=f
x(x0, y0)
dx
dt(0) +
f
y(x0, y0)
dy
dt(0)
=f
x(x0, y0)a+
f
y(x0, y0)b.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
6/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, V
Theorem
If f is differentiable, then Duf exists for any unit vectoru=a, b and is computed as
Duf(x0, y0) =f
x(x0, y0)a+
f
y(x0, y0)b.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
7/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Directional derivatives, V
Example (resumed)
For f(x, y) =x3 3xy+ 4y2 and u=
32 ,
12
, we have
f
x = 3x2 3y and f
y3x+ 8y,
so that
Duf(1, 2) = (36)32
+ (3 + 16) 12
=13332
.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
8/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Definition of the gradient
Definition
Thegradientoff is the vector function defined by
f(x, y) =fx(x, y), fy(x, y)= fx(x, y)i+fy(x, y)j.
Important
Iff is differentiable, and u is a unit vector, then
Duf(x, y) =f(x, y)u.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
9/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, I
ExampleFind the directional derivative off(x, y) = ln x+e(x1)y at(1, 0) in the direction ofv=i+ 2j.As
fx= 1x
+ye(x1)y and fy= (x1)e(x1)y,
we havef(1, 0) =1, 0.Since|v|= 5, theunitvector u in the direction ofv isu= 15v= 15 i+ 25j.All in all:
Duf(1, 0) =1, 0
15,
25
= 1
5.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
10/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Geometric meaning of the gradient, I
A geometric observation
Let fbe a differentiable function of two variables, let u be aunit vector, and let be the angle between u and
f.
We have:
Duf =fu=|f||u| cos =
|f|
cos .
This becomes maximal if cos = 1, i.e., = 0, which meansthat u has the same direction asf.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
11/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Geometric meaning of the gradient, II
Theorem
Let f be a differentiable function of two variables, and letu be
a unit vector. Then Duf(x, y) is maximal when u has the same
direction asf(x, y). This maximal value of Duf(x, y) is|f(x, y)|.
Important
Everything we said about directional derivatives and gradientsof functions of two variables remains valid for functions of three(or more) variables.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
12/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, II
Example
The temparature in a room at (x, y, z) is given by
T = 80
1 +x2 + 2y2 + 3z2,
with T in centigrade and x, y, z in meters.In which direction increases the temparature fastest at(1, 1,2)? What is the maximum rate of change?The gradient is
T = Tx
i+T
y j+
T
zk,
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
13/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, III
Example (continued)
...i.e.,
T =
160x
(1 +x2
+ 2y2
+ 3z2
)2i
320y
(1 +x2
+ 2y2
+ 3z2
)2j
480z(1 +x2 + 2y2 + 3z2)2
k
= 160
(1 +x2 + 2y2 + 3z2)2(xi2yj3zk).
It follows that
T(1, 1,2) =58
(i2j+ 6k).
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
14/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, IV
Example (continued)
This vector has the same direction as1,2, 6.The unit vector in this direction is
u= 141
i 241
j+ 6
41k,
which gives the direction of the maximum increase.The maximum increase is
|T(1, 1,2)|= 58
414 C/m.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
15/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
More geometric meaning of the gradient, I
Another geometric observation
Let Sbe a level surface of a function F(x, y, z), i.e., given bythe equation F(x, y, z) =Cwith a constant C.Let P(x0, y0, z0) be a point on S, and let
r(t) =x(t), y(t), z(t) be a curve in Spassing through P.Then F(x(t), y(t), z(t)) =C, so that
0 = dC
dt
= Fx
dxdt
+ Fy
dydt
+ Fz
dzdt
=Fr(t).
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
16/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
More geometric meaning of the gradient, II
Another geometric observation (continued)
Let t0 be such that r(t0) =x(t0), y(t0), z(t0).Then
0 =F(x0, y0, z0)r(t0).This means thatF(x0, y0, z0) is perpendicular to the tangent
vector r(t0) to any curve r(t) passing through P.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
17/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Definition of tangent plane and normal line
DefinitionThetangent planeto the level surface Sgiven byF(x, y, z) =C at P(x0, y0, z0) is the plane passing through Pwith normal vectorF(x0, y0, z0), i.e., with the equation
Fx
(x0, y0, z0)(xx0)
+F
y(x0, y0, z0)(yy0) + F
z(x0, y0, z0)(zz0) = 0.
Thenormal lineto S through P is the line passing through Pperpendicularly to the tangent plane, i.e., with the symmetricequations
xx0Fx(x0, y0, z0)
= yy0
Fy(x0, y0, z0)
= zz0
Fz(x0, y0, z0)
.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
18/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, IV
ExampleLet Ebe the ellipsoid given by
x2
2 +
(y1)24
+ (z+ 2)2 = 4.
Find the tangent plane and the normal line to E atP(2, 3,1).Set
F(x, y, z) = x2
2
+(y1)2
4
+ (z+ 2)2.
Then E is the level surface F(x, y, z) = 4.We have
F
x
=x, F
y
=1
2
(y
1), and
F
z
= 2(z+ 2).
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
19/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, V
Example (continued)
We obtain:
F
x(2, 3,1) =2,
F
y(2, 3,1) = 1, and F
z(2, 3,1) = 2.
8/13/2019 Lec6[1]DerivadaDireccional y Gradiente
20/20
MATH 209
Calculus,III
Volker Runde
Directionalderivatives
The gradient
Tangentplanes to levelsurfaces
Examples, VI
Example (continued)
Hence:
the tangent plane has the equation
2(x+ 2) + (y3) + 2(z+ 1) = 0,i.e.,
2x+y+ 2z= 5;
the normal line has the symmetric equations
x+ 22
=y3 = z+ 12
.