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Find the differential d z of the function f ( x , y ) = 4 y 2 + x 2 y 2 x y and use it to approximate Δ z at point ( 1 , −1 ) . Use Δ x = 0.03 and Δ y = −0.02 . What is the exact value of Δ z ?

d z = 0.18 Δ z = f ( 1.03 , −1.02 ) f ( 1 , −1 ) = 0.180682

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Differentiability of a function of three variables

All of the preceding results for differentiability of functions of two variables can be generalized to functions of three variables. First, the definition:

Definition

A function f ( x , y , z ) is differentiable at a point P ( x 0 , y 0 , z 0 ) if for all points ( x , y , z ) in a δ disk around P we can write

f ( x , y ) = f ( x 0 , y 0 , z 0 ) + f x ( x 0 , y 0 , z 0 ) ( x x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y y 0 ) + f z ( x 0 , y 0 , z 0 ) ( z z 0 ) + E ( x , y , z ) ,

where the error term E satisfies

lim ( x , y , z ) ( x 0 , y 0 , z 0 ) E ( x , y , z ) ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 = 0 .

If a function of three variables is differentiable at a point ( x 0 , y 0 , z 0 ) , then it is continuous there. Furthermore, continuity of first partial derivatives at that point guarantees differentiability.

Key concepts

  • The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
  • Tangent planes can be used to approximate values of functions near known values.
  • A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
  • The total differential can be used to approximate the change in a function z = f ( x 0 , y 0 ) at the point ( x 0 , y 0 ) for given values of Δ x and Δ y .

Key equations

  • Tangent plane
    z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 )
  • Linear approximation
    L ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 )
  • Total differential
    d z = f x ( x 0 , y 0 ) d x + f y ( x 0 , y 0 ) d y .
  • Differentiability (two variables)
    f ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) + E ( x , y ) ,
    where the error term E satisfies
    lim ( x , y ) ( x 0 , y 0 ) E ( x , y ) ( x x 0 ) 2 + ( y y 0 ) 2 = 0 .
  • Differentiability (three variables)
    f ( x , y ) = f ( x 0 , y 0 , z 0 ) + f x ( x 0 , y 0 , z 0 ) ( x x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y y 0 ) + f z ( x 0 , y 0 , z 0 ) ( z z 0 ) + E ( x , y , z ) ,
    where the error term E satisfies
    lim ( x , y , z ) ( x 0 , y 0 , z 0 ) E ( x , y , z ) ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 = 0 .

For the following exercises, find a unit normal vector to the surface at the indicated point.

f ( x , y ) = x 3 , ( 2 , −1 , 8 )

( 145 145 ) ( 12 i k )

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ln ( x y z ) = 0 when x = y = 1

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For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point P .

x 2 + x y + y 2 = 3 , P ( −1 , −1 )

Normal vector: i + j , tangent vector: i j

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( x 2 + y 2 ) 2 = 9 ( x 2 y 2 ) , P ( 2 , 1 )

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x y 2 2 x 2 + y + 5 x = 6 , P ( 4 , 2 )

Normal vector: 7 i 17 j , tangent vector: 17 i + 7 j

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2 x 3 x 2 y 2 = 3 x y 7 , P ( 1 , −2 )

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z e x 2 y 2 3 = 0 , P ( 2 , 2 , 3 )

−1.094 i 0.18238 j

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For the following exercises, find the equation for the tangent plane to the surface at the indicated point. ( Hint: Solve for z in terms of x and y . )

−8 x 3 y 7 z = −19 , P ( 1 , −1 , 2 )

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z = −9 x 2 3 y 2 , P ( 2 , 1 , −39 )

−36 x 6 y z = −39

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x 2 + 10 x y z + y 2 + 8 z 2 = 0 , P ( −1 , −1 , −1 )

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z = ln ( 10 x 2 + 2 y 2 + 1 ) , P ( 0 , 0 , 0 )

z = 0

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z = e 7 x 2 + 4 y 2 , P ( 0 , 0 , 1 )

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x y + y z + z x = 11 , P ( 1 , 2 , 3 )

5 x + 4 y + 3 z 22 = 0

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x 2 + 4 y 2 = z 2 , P ( 3 , 2 , 5 )

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x 3 + y 3 = 3 x y z , P ( 1 , 2 , 3 2 )

4 x 5 y + 4 z = 0

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z = a x y , P ( 1 , 1 a , 1 )

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z = sin x + sin y + sin ( x + y ) , P ( 0 , 0 , 0 )

2 x + 2 y z = 0

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h ( x , y ) = ln x 2 + y 2 , P ( 3 , 4 )

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z = x 2 2 x y + y 2 , P ( 1 , 2 , 1 )

−2 ( x 1 ) + 2 ( y 2 ) ( z 1 ) = 0

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For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, P 0 ( x 0 , y 0 , z 0 ) , and a vector n = a , b , c that is parallel to the line. Then the equation of the line is x x 0 = a t , y y 0 = b t , z z 0 = c t . )

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Smarajit Reply
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can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 4

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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