MATH 201 Chapter 14 Summary, Fall 2011

Chapter 14: Partial Derivatives Section 1: Definition 1: R is a region in xy-plane. (x0, y0) a point in the xy-plane. • If there is a disc of center (x0, y0) which lies entirely in R, we say that (x0, y0) is an interior point of R. • If any disc of center (x0, y0) intersect both R and the complement of R, we say that (x0, y0) is a boundary point of R. • The set of all interior points of R is called the interior of R. • The set of all boundary points of R is called the boundary of R. • If each point of R is an interior point, we say that R is an open region. (In this case, R is the same as its interior). • If R contains its boundary, we say that R is a closed region. • If R lies inside a disc of fixed radius, we say that R is a bounded region. Definition 2: The function of 2 variables is a function whose domain is a region in the xy-plane and whose range is a subset of the set IR. Definition 3: f(x, y) is a function of 2 variables. c is in the range of f. The set of all points (x, y, z) in space such that z = f(x, y) is called the graph of f. The graph of f is also called the surface z = f(x, y). The set of all points (x, y) in the plane such that f(x, y) = c is called a level curve of f. Definition 4: f(x, y, z) is a function of 3 variables. Suppose c in Range f. The set of all points (x, y, z) in space such that f(x, y, z) = c is called the level surface of f. Section 2: Definition 1: Suppose R is a region. The point (x0, y0) point in the plane. If (x0, y0) is either an interior point of R or a boundary point of R, we say that (x0, y0) is a limit point of R. Definition 2: f(x, y) is a function of 2 variables. (x0, y0) is a limit point of Domain f. We say f(x, y) has a limit L as (x, y) approaches to (x0, y0) and write ( ) ( ) ( ) L x y f y x x y = → , lim 0 , 0 , if to any given ε > 0, there corresponds an δ > 0 such that: ( ) ( ) ( ) ( ) < ε − ⇒    ∈ < δ < L f x y f y x x y y x , Domain , to , , distance from 0 0 0 i.e.: ( ) ( ) ( ) ( ) < ε − ⇒    ∈ < δ − + − < L f x y f y x y y x x , Domain , 0 2 0 0 2 Theorem 1: (i) ( ) ( ) 0 , , lim0 0 x x y x x y = → (ii) ( ) ( ) 0 , , lim0 0 y y y x x y = → (iii) ( ) ( ) k k y x x y = → , lim0 , 0 Theorem 2: ( ) ( f x y) g x y , , , are two functions of two variables. ( x0, y0 ) is a limit point of Domain f and Domain g. Suppose ( ) ( ) ( ) 1 , , , lim 0 0 L x y f y x x y = → and ( ) ( ) ( ) 2 , , , lim 0 0 L g x y y x x y = → Then: (i) ( ) ( ) ( ) ( ) 2 1 , , , , lim 0 0 L L g x y x y f y x x y ± = ± → (ii) ( ) ( ) ( ) ( ) 2 1 , , , , lim 0 0 L L x y g x y f y x x y = → (iii) ( ) ( ) ( ) ( ) 2 1 , , , , lim 0 0 L L g x y x y f y x x y = → provided that both ( ) and 2 , L g x y are different from zero. Definition 3: Suppose ( f x y) , is a function of two variables. Suppose ( x0, y0 ) is a limit point of Domain f. We say that f is continuous at ( x0, y0 ) if : (i) ( ) ( ) ( x y) f y x x y , lim 0 , 0 , → (ii) ( x0, y0 ) is actually in Domain f. (iii) ( ) ( ) ( ) ( 0 ) 0 , , , , lim 0 0 y f x x y f y x x y = → Section 3: Definition 1: Suppose ( f x y) , is a function of 2 variables. ( x0, y0 ) is an interior point of Domain f. For ( x, y) in Domain f we define 0 0 and y y y x x x − ∆ = − ∆ = ( ) ( ) ( y) x y f x y f x f x y f + ∆ + ∆ = − = ∆ 0 0 0 0 , , , The partial derivatives of f at ( x0, y0 ) are defined as: ( ) ( ) ( ) ( ) ( ) ( )        ∆ − + ∆ = ∆ ∆ = ∂ ∂ ∆ − + ∆ = ∆ ∆ = ∂ ∂ ∆ = ∆ → ∆ = → ∆ ∆ = ∆ → ∆ = → ∆ y y f x y f x y y f y y x f x y f x x y f x x f y x x f y x x y y x y x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , lim lim , , , lim lim , Provided that these two limit exist. Definition 2: Suppose ( f x y) , is a function of 2 variables. ( x0, y0 ) is an interior point of Domain f. Then: We say that f is differentiable at ( x0, y0 ) if: (i) y f x f ∂ ∂ ∂ ∂ and both exist at ( x0, y0 ) . (ii) ( ) ( ) y x y x x y f x y x x f f ∆ + ε ∆ + ε ∆ ∂ ∆ + ∂ ∂ = ∂ ∆ 2 1 0 0 0, 0 where ( ) ( 0,0 ) , 0 as 0 and 2 0 → ∆ ∆ ε → ε → y x Section 4: Theorem 1: Chain Rule: ( )z f x y , , is a differentiable function of 3 variables. ( ) ( ) ( )t g t h t , and l , differentiable functions of one variable. If ( ) ( ) ( ) ( )z f x y w t z h t y g t x , , , and , , = = = = l Then: w is a differentiable function of one variable and : dt f dz dt f dy dt f dx dt dw z y x + + = Corollary To Chain Rule: ( f x y z) , , is a differentiable function of 3 variables. ( ) ( ) ( )s r g r s h r s , , and , , , l differentiable functions of two variable. If ( ) ( ) ( ) ( )z f x y w r s z h r s y g r s x , , , and , , , , , = = = = l Then: w is a differentiable function of two variable and : r z f r y f r x f dr dw z y x ∂ ∂ ∂ + ∂ ∂ + ∂ = s z f s y f s x f ds dw z y x ∂ ∂ ∂ + ∂ ∂ + ∂ = Theorem 2: ( F x y) , differentiable function of 2 variables: Suppose the equation ( ) 0 , F x y = defines y implicitly as a function of x. Say ( ) y = h x then: y x F F dx dy = − Provided that Fy ≠ 0 Section 5: Definition 1: ( f x y z) , , differentiable function of 3 variables. Suppose k j i c b a u + + = is a unit vector in space (i.e. 1 2 2 2 = + + c b a ) ( 0 ) 0 0 , , z y x is an interior point of Domain f. L is the line through ( 0 ) 0 0 , , z y x parallel to u. For ( )z x y , , in L, define s ∆ as the directed distance from ( 0 ) 0 0 , , z y x to ( )z x y , , . The directed derivation of f at ( 0 ) 0 0 , , z y x in the direction of u is defined as: ( ) s f z y f x D s u ∆ ∆ = ∆ → 0 0 0 0 lim , , provided that this limit exists. Formula for the Directed Derivation : ( 0 ) 0 0 , , . z y f x Du f = u ∇ where ( ) ( ) ( ) ( )k j i 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , z y x f z y x f z y x f z y x f z y x + + = ∇ Some Important Observations: 1. f increases most rapidly at ( x0, y0 ) in the direction of ( ∇f x0, y0 ) . The value of the derivation in this direction is ( ∇f x0, y0 ) . 2. f decreases most rapidly at ( x0, y0 ) in the direction of ( − ∇f x0, y0 ) . The value of the derivation in this direction is ( − ∇f x0, y0 ) . Section 7: Definition 1: ( f x y) , a differentiable function of 2 variables. ( a,b) interior point of Domain f. If ( ) ( ) 0 , , = = a b f a b f y x then we say that the point ( a,b) is a critical point of f. Theorem 1: ( f x y) , a differentiable function of 2 variables. ( a,b) interior point of Domain f. Then: f has a local max or a local min at ( ) ( a b) a b , , ⇒ is a critical point of f. Theorem 2: (Second Derivation Test): ( f x y) , a differentiable function of 2 variables. ( a,b) is a critical point of f. Then: (i) ( ) ( ) ⇒    > > − 0 0 at , 2 xx xy xx yy f a b f f f f has a local min at ( a,b) . (ii) ( ) ( ) ⇒    < > − 0 0 at a,b 2 xx xy xx yy f f f f f has a local max at ( a,b) . (iii) ( ) ( ) ⇒ < − 0 at a,b 2 xy xx yy f f f f has a saddle point at ( a,b) . (iv) ( ) ( ) ⇒ = − 0 at a,b 2 xy xx yy f f f Test fails. Theorem 3: ( f x y) , a function of 2 variables. R is a region in the plane. Then: f R R f ⇒    is closed and bounded continuous everywhere in has an absolute max and an absolute min in R. Section 8: Theorem 1: Lagrange Multipliers: ( f x y z) , , , ( g x y z) , , are 2 differentiable functions of 3 variables. Let S be the level surface ( ) 0 , , g x y z = and suppose that S is contained in the Domain f. Then: if a point ( ) S z y x ∈ 0 0 0 , , is a point where f has a local min or local max (as a function of S), then: ( ) ( 0 ) 0 0 0 0 0 , , , , z y g x z y f x = λ∇ ∇