HUT – DEP DEPART ARTMENT MENT OF MATH MATH.. APPLIE APPLIED D --------------------------------------------------------------------------------------------------------
GIAÛI TÍCH 2 (QUOÁC TEÁ) LECTURE 01: FUNCTIONS OF SEVERAL VARIABLES: PARTIAL DERIVATIVES STEWART: PAGES 855 PhD. NGUYEÃN QUOÁC LAÂN (August, 2015) 2015)
878
LECTURE 1 CONTENTS ---------------------------------------------------------------------------------------------------------------------------------
1- FUNCTION OF 2 VARIABLES: DEFINITION – GRAPH. 2- PARTIAL DERIVATIVES 3- IMPLICIT DIFFERENTIATION 4- INCREMENTAL FORMULA - DIFFERENTIAL
FUNCTIONS OF TWO VARIABLES --------------------------------------------------------------------------------------------------------------------------------------
A function f of two variables is a formula z = f ( x, y ), ( x, y ) ∈ D. The set D = {( x, y ) | f ( x, y ) is defined } ⊂ R 2 is the domain of f Example : For f ( x, y ) = y − x , find f (1,2), f (2,1) & domain D.
The graph of y = f(x) is a curve
But the graph of z = f(x, y) is a surface
EXAMPLE --------------------------------------------------------------------------------------------------------------------------------------
EXAMPLE --------------------------------------------------------------------------------------------------------------------------------------
PARTIAL DERIVATIVES OF z = f(x, y) --------------------------------------------------------------------------------------------------------------------------------------
⎡ f / = f = ∂ f = D f = D f ⎫ x x 1 ⎪⎪ ⎢ x ∂ x Notation : z = f ( x, y ) : ⎢ ⎬ f ∂ ⎢ f y/ = f y = = D2 f = D y f ⎪ ⎪⎭ ⎢⎣ ∂ y
Partial Derivatives with respect to x (or y )
Rule for finding partial derivatives of z = f(x, y) 1/ To find f x, regard y as a constant and differentiate z = f(x, y) with respect to x 2/ To find f y, regard x as a constant and differentiate z = f(x, y) with respect to y ⎡ f x = ? Example 1 : f = x + x y − 2 y ⇒ ⎢ ⎣ f y = ? 3
2 3
2
⎡ f x = ? Exam. 2 : f = x ⇒ ⎢ ⎣ f y = ? y
HIGHER – ORDER PARTIAL DERIVATIVES --------------------------------------------------------------------------------------------------------------------------------------
⎡( z x ) x = z xx , ( z x ) y = z xy ⎡ z x ⎫ z = f ( x, y ) ⇒ ⎢ ⎬Functions of x, y ⇒ ⎢ : ⎣ z y ⎭ ⎢⎣( z y ) x = z yx , ( z y ) y = z yy Second - order partial derivatives of f . Similarly, we define 3rd ,4 th K
∂ 2 f ∂ 2 f ∂ 3 f Symbol : f xx = 2 , f xy = : mixed second partial, f xyy = K 2 ∂ x ∂ x∂ y ∂ x∂ y Example : Find the second partial derivatives of f = x 3 + x 2 y 3 − 2 y 2 Answer : We have f x = 3 x 2 + 2 xy 3 , f y = 3 x 2 y 2 − 4 y therefore
∂ 2 z ∂ 2 z 3 2 6 2 , 6 , x y xy = + = 2 ∂ x∂ y ∂ x
∂ 2 z = 6 xy 2 , ∂ y∂ x
∂ 2 z 2 6 xy = −4 2 ∂ y
∂ 3 f ∂ 3 f ∂ 3 f Equality with mixed partials : f xy = f yx ⇒ ... = 2 = 2 ∂ x∂ y∂ x ∂ x ∂ y ∂ y∂ x
IMPLICIT PARTIAL DIFFERENTIATION --------------------------------------------------------------------------------------------------------------------------------------
Function z = z ( x, y ) is defined implicitly by F ( x, y, z ) = 0 ⇒ z x , z y = ? Example : Find z x , z y if z = z ( x, y ) is defined by x 3 + y 3 + z 3 + 6 xyz = 1 Answer : To find z x , we differentiate the equation implicitly with respect to x, considering y as a constant : 3 x 2 + 3 z 2 z x + 6 yz + 6 xyz x = 0 (1) Solving (1) for z x : z x = −
x + 2 yz 2
z + 2 xy 2
Similarly for z y : Differentiate respect to y and solve for z y ⇒ z y = −
2
y + 2 xz 2
z + 2 xy
APPROXIMATION: INCREMENTAL FORMULA --------------------------------------------------------------------------------------------------------------------------------------
z = f ( x, y ) : Δ z = f (a + Δ x, b + Δ y ) − f (a, b ) ≈ f x (a, b ) ⋅ Δ x + f y (a, b ) ⋅ Δy 144 4 4 244 4 4 3
The change in z
At some factory, the daily output is
Q =
1 1 60 K 2 L3
(Cobb Douglas!)
units, where K – capital measured in units of $1000 and L – labor force measured in worked – hours. The current capital is $900.000 and 1000 worker – hours of labor are used each day. Estimate the change in output that will result if capital is increased by $1000 and labor is increased by 2 worker – hour Ans. : ΔQ ≈
∂Q ∂Q ΔK + Δ L = 30 K −1 2 L1 3ΔK + 20 K 1 2 L− 2 3Δ L = 22 units ∂K ∂ L
DIFFERENTIAL ---------------------------------------------------------------------------------------------------------------------------
Given z = f(x, y), x & y: independent variables ⇒ dz = f x dx + f y dy =
Differential
∂ f ∂ f dx + dy (Consider dx, dy as real number!) ∂ x ∂ y
2 2 Example : Given z = x + 2 y let find a/dz b/dz (1, 2 ) 2 y x , z = Answer : Partial derivatives z x = ⇒ 2 2 y 2 2 x + 2 y x + 2 y
a/ dz = z x dx + z y dy =
⎧ x = 1 1 4 b/ ⎨ ⇒ dz (1, 2 ) = dx + dy 3 3 ⎩ y = 2
xdx + 2 ydy x 2 + 2 y 2
Example : Given f = x 3 + x 2 y 3 − 2 y 2
Second order differential d2z 2
2
d z = z xx dx + 2 z xy dxdy + z yy dy
2
find d 2 z (2,1) Answer : d 2 z = 14dx 2 + 24dxdy + 8dy 2
