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The Implicit Function Theorem - Examples with Resolution | MTH 254, Study notes of Calculus

Material Type: Notes; Class: Vector Calculus I; Subject: Math; University: Portland Community College; Term: Unknown 2007;

Typology: Study notes

Pre 2010

Uploaded on 08/17/2009

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Miscellaneous stuff from sections 11.4 and 11.6
Page 1 of 5
The Implicit Function Theorem
The slope at any point
along the level curve
(
)
,
f
xy k
=
is given by
()
()
,
,
x
y
f
xy
dy
dx f x y
=−
.
Example 1
Let's illustrate the Implicit Function Theorem using the curve
22
24 4xy x y
=−
at the point
(
)
1, 4
.
Note: The Implicit Function Theorem has
nothing
to do with the slope along a surface
(
)
,zfxy=
.
The Implicit Function Theorem gives you the slope
of a curve in the xy-plane!
Figure 1
pf3
pf4
pf5

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The Implicit Function Theorem

The slope at any point along the level curve f ( x y, )= kis given by

x

y

dy f^ x y

dx f x y

Example 1

Let's illustrate the Implicit Function Theorem using the curve

2 2 2 x y − 4 x = y − 4 at the point

( −1,4^ ).

Note: The Implicit Function Theorem has nothing to do with the slope along a surface z = f ( x y, ).

The Implicit Function Theorem gives you the slope of a curve in the xy-plane!

Figure 1

Fact Jack

Assuming everything relevant is differentiable, ∇f ( x 0 ,y 0 )

G

is perpendicular to the tangent line to

the level curve f ( x y, ) = f ( x 0 ,y 0 )at the point ( x 0 ,y 0 ).

Example 2

Let’s illustrate Jack’s Fact for the function

2

z = 3 − 2 x − 2 y − 4 y at the point ( 0, − 1 ). Figure 3

shows the level curves z = − 1 , z = 1 , and z = 3.

Please note that this action is taking place in the xy-plane. Despite that, this fact does have

repercussions vis-à-vis the surface z^ =^ f^ ( x y, ); the maximal slope along the surface at the point

( x 0 ,^ y 0 ,z 0 )always occurs perpendicularly to the level curve at that point (relative to thexy-plane).

Figure 2 Figure 3

Definition

The gradient of the four-dimensional function w = f ( x y z, , ) at the point ( xO , yO ,zO ) is the

three-dimensional vector ∇f ( xO , yO , zO ) = f x ( xO , yO , zO ) , f y ( xO , yO , zO ) , f z ( xO , yO ,zO). At

any point the gradient exists the gradient has both of the following properties.

• ∇f ( xO , yO ,zO) points in the direction the function w = f ( x y z, , ) increases most rapidly

through the point ( xO , yO , zO , f ( xO , yO ,zO )).

• The instantaneous rate of change in w = f ( x y z, , )at the point ( xO , yO , zO , f ( xO , yO ,zO ))in

the direction of ∇f ( xO , yO ,zO) is ∇f ( xO , yO ,zO). Consequently ∇ f ( xO , yO ,zO) is the

maximum rate of change along w = f ( x y z, , )at the point ( xO , yO ,z O).

Example 4

Missylaneous is one salt seeking microbe. A sample of the microbe is stuck in a tank whose salt

concentration (ppm) at the point ( x, y z, )is given by the function ( ) ( )

, , sin 2

x f x y z x y x z

where an axis system has been established so that all three independent variables always have

values between 0 and 3. Assuming that the microbe always moves in the direction that the salt

concentration increases most rapidly, let’s determine the direction at which the ‘crobe moves when

it is at the point

.

Word Jill

Assuming everything relevant is differentiable, ∇f ( xO , yO ,zO) is perpendicular to the tangent

plane to the level surface f ( x y z, , ) = f ( xO , yO ,zO)at the point ( xO , yO ,zO ).

Example 6

Let's use Jill’s Word to help us find an equation for the tangent plane to the surface

2 z = x + yat

the point ( −1, − 1,0).

Figure 5