ch11 Partial derivatives-new

ch11 Partial derivatives-new - MULTIVARIATE MULTIVARIATE...

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MULTIVARIATE  CALCULUS  Partial derivatives  Partial derivatives
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In this Chapter: In this Chapter:  11.1 Functions of Several Variables  11.2 Limits and Continuity  11.3 Partial Derivatives  11.4 Tangent Planes and Linear Approximations  11.5 The Chain Rule  11.6 Directional Derivatives and the Gradient Vector  11.7 Maximum and Minimum Values  11.8 Lagrange Multipliers     Review 
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Chapter 11, 11.1, P593 DEFINITION  function f of two variables  is a rule  that assigns to each ordered pair of real numbers (x,  y) in a set D a unique real number denoted by f (x, y).  The set D is the  domain  of f and its  range  is the set of  values that f takes on, that is,                              . { } D y x y x f ) , ( ) , (
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Chapter 11, 11.1, P593 We often write z=f (x, y) to make explicit the value  taken on by f at the general point (x, y) . The variables  x and y are  independent variables  and z is the  dependent variable .
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Chapter 11, 11.1, P593
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Chapter 11, 11.1, P594 Domain of  1 1 ) , ( - + + = x y x y x f
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Chapter 11, 11.1, P594 Domain of  ) ln( ) , ( 2 x y x y x f - =
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Chapter 11, 11.1, P594 Domain of  2 2 9 ) , ( y x y x g - - =
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Chapter 11, 11.1, P594 DEFINITION  If f is a function of two variables with  domain  D , then the  graph  of is the set of all points (x, y,  z) in R 3  such that z=f (x, y) and (x, y) is in  D .
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Chapter 11, 11.1, P595
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Chapter 11, 11.1, P595 Graph of 2 2 9 ) , ( y x y x g - - =
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Chapter 11, 11.1, P595 Graph of 2 2 4 ) , ( y x y x h + =
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Chapter 11, 11.1, P596 2 2 ) 3 ( ) , ( ) ( 2 2 y x e y x y x f a - - + = 2 2 ) 3 ( ) , ( ) ( 2 2 y x e y x y x f b - - + =
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Chapter 11, 11.1, P596 DEFINITION  The  level curves  of a function f of two  variables are the curves with equations f (x, y)=k,  where k is a constant (in the range of f).
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Chapter 11, 11.1, P597
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Chapter 11, 11.1, P597
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Chapter 11, 11.1, P598
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Chapter 11, 11.1, P598 Contour map of y x y x f 2 3 6 ) , ( - - =
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Chapter 11, 11.1, P598 Contour map of 2 2 9 ) , ( y x y x g - - =
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Chapter 11, 11.1, P599 The graph of h (x, y)=4x 2 +y 2 is formed by lifting the level curves.
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Chapter 11, 11.1, P599
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Chapter 11, 11.1, P599
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Chapter 11, 11.2, P604 1. DEFINITION  Let f be a function of two variables whose  domain  includes points arbitrarily close to (a, b).  Then we say that the  limit of f (x, y) as (x, y)  approaches (a ,b)  is L and we write if for every number  ε> 0 there is a corresponding number  δ > 0 such that If               and                           then L y x f b a y x = ) , ( lim ) , ( ) , ( D y x ) , ( δ < - + - < 2 2 ) ( ) ( 0 b y a x ε < - L y x f ) , (
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P605 If f( x, y) L 1  as (x, y)  (a ,b) along a path C 1  and f (x, y)  L 2  as (x, y)  (a, b) along a path C 2 , where L 1 ≠L 2 , then  lim  (x, y)  (a, b)  f (x, y) does not exist.
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Chapter 11, 11.2, P607 4. DEFINITION 
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This note was uploaded on 02/17/2012 for the course ANY 191 taught by Professor Any during the Spring '12 term at Mapúa Institute of Technology.

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ch11 Partial derivatives-new - MULTIVARIATE MULTIVARIATE...

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