Calculus of Multivariable Functions, Slides of Mathematics

Download Calculus of Multivariable Functions and more Slides Mathematics in PDF only on Docsity!

Calculus of multivariable functions

Lê Xuân Trường

Outline

Partial derivatives and total differential

Definition of partial derivatives Intepretations of partial derivatives Higher-order partial derivatives Total differential

Applications

Relative Minimums And Maximums Absolute Minimums and Maximums Lagrange Multipliers

Partial derivatives

Examples: Find the partial derivatives in following cases

a/ z = f (x, y ) = 3 x^2 y 3 + xexy b/ z = g (x, y ) =

x^2 + 3 y − sin( 2 x + y 2 )

Giải fx (x, y ) = 6 xy 3 + ( 1 + xy )exy

fy (x, y ) = 9 x^2 y 2 + x^2 exy

gx (x, y ) = √ x x^2 + 3 y

− 2 cos( 2 x + y 2 )

gx (x, y ) =

x^2 + 3 y

− 2 y cos( 2 x + y 2 )

Intepretations of partial derivatives

The rates of change of the function z = f (x, y )

The rate of change of z with respect to x when y held fixed

f (^) x′ (x 0 , y 0 ) ≈

f (x 0 + ∆x, y 0 ) − f (x 0 , y 0 ) ∆x The rate of change of z with respect to y when x held fixed

f (^) y′ (x 0 , y 0 ) ≈ f^ (x^0 ,^ y^0 +^ ∆ ∆y^ y)^ −^ f^ (x^0 ,^ y^0 )

Intepretations of partial derivatives

Example: Maginal Productivity A manufacturer of a popular toy has determined that the production function is Q =

LK ,

where L is the number of labour-hours per week K is the capital required for a weekly production of Q gross of the toy (one gross is 144 units)

Determine the marginal productivity functions, and evaluate them when

L = 400 , K = 16.

Interpret the results.

Intepretations of partial derivatives

Partial (Elasticity) of the function z = f (x, y )

Elasticity with respect to x

Ezx (x, y ) = (^) f (xx, y ) f (^) x′ (x, y ) ≈ (^) f (xx, y ). ∆ ∆xz

Elasticity with respect to y

Ezy (x, y ) = y f (x, y )

f (^) y′ (x, y ) ≈ y f (x, y )

. ∆z ∆y

Higher-Order Partial Derivatives

Second-Order Partial Derivatives Given a function z = f (x, y ). The second-order partial derivatives of f are defined as follows

fxx (x, y ) ≡

∂ 2 f ∂ x^2 (x,^ y^ )^ :=^

∂ x (fx^ (x,^ y^ )) fyx (x, y ) ≡ ∂^

(^2) f ∂ y ∂ x (x,^ y^ )^ :=^

∂ y (fx^ (x,^ y^ ))

fxy (x, y ) ≡ ∂^

(^2) f ∂ x∂ y

(x, y ) := ∂ ∂ x

(fy (x, y ))

fyy (x, y ) ≡ ∂^

(^2) f ∂ y 2

(x, y ) := ∂ ∂ y

(fy (x, y ))

Higher-Order Partial Derivatives

Example: Find the four second-order partial derivatives of

z = f (x, y ) = 2 x^2 y + 6 x^1 /^3 y 2 /^3

Solution Since fx (x, y ) = 4 xy + 2 x−^2 /^3 y 2 /^3 we have

fxx (x, y ) = 4 y − 4 3

x−^5 /^3 y 2 /^3 fyx (x, y ) = 4 x + 4 3

x−^2 /^3 y −^1 /^3.

Also, since fy (x, y ) = 2 x^2 + 4 x^1 /^3 y −^1 /^3 we have

fyy (x, y ) = − 4 3

x^1 /^3 y −^4 /^3 fxy (x, y ) = 4 x + 4 3

x−^2 /^3 y −^1 /^3.

Total differential

Definition

Given the function z = f (x, y ). Let dx and dy represent changes in x and y , respectively. The total differential of z at (x, y ) is defined by

df (x, y ) = fx (x, y )dx + fy (x, y )dy

The second-order differential of z given by

d^2 f (x, y ) = fxx (x, y )dx^2 + 2 fxy (x, y )dxdy + fyy (x, y )dy 2

Total differential

Applications Approximation for the change in z

df (x, y ) ≈ ∆z := f (x + dx, y + dy ) − f (x, y ).

Linear approximation of two variable functions

f (x, y ) ≈ f (x 0 , y 0 ) + fx (x 0 , y 0 )(x − x 0 ) + fy (x 0 , y 0 )(y − y 0 ).

Maxima and Minima for functions of Two Variables

Global and Local Extrema

The function z = f (x, y ) has a local maximum at (x 0 , y 0 ) if

f (x 0 , y 0 ) ≥ f (x, y )

for all points (x, y ) within some disk centered at (x 0 , y 0 ). If the preceding inequality holds for every point (x, y ) in the do- main of f , then f has a global maximum at (x 0 , y 0 ). Similarly, we can define the local minimum and the global minimum

Maxima and Minima for functions of Two Variables

Saddle points The point (x 0 , y 0 ) is a saddle point of z = f (x, y ) if it is a critical point and, for every disk D containing (x 0 , y 0 ), there are points (x 1 , y 1 ), (x 2 , y 2 ) in D such that

f (x 1 , y 1 ) < f (x 0 , y 0 ) < f (x 2 , y 2 )

Maxima and Minima for functions of Two Variables

Find extrema of the function z = f (x, y ) Rule 2: (Second Derivative Test for Local Extrema) For every critical point P(x 0 , y 0 ), we define

D(x 0 , y 0 ) = fxx (x 0 , y 0 )fyy (x 0 , y 0 ) − (fxy (x 0 , y 0 ))^2

If D > 0 and fxx (x 0 , y 0 ) > 0 then f has a local minimum at P If D > 0 and fxx (x 0 , y 0 ) < 0 then f has a local maximum at P If D < 0 then P is a saddle point of f If D = 0 then the test is inconclusive.

Maxima and Minima for functions of Two Variables

Find extrema of the function z = f (x, y ) Rule 3: (Second Derivative Test for Global Extrema) Define D(x, y ) = fxx (x, y )fyy (x, y ) − (fxy (x, y ))^2

If D(x, y ) > 0 and fxx (x, y ) > 0, for all points (x, y ) in the domain of f then f has a global minimum at P If D(x, y ) > 0 and fxx (x, y ) < 0, for all points (x, y ) in the domain of f then f has a global maximum at P