Multivariable calculus
VIETNAM NATIONAL UNIVERSITY-HO CHI MINH
CITY
INTERNATIONAL UNIVERSITY
Chapter 2. Multivariable calculus
Calculus 2B for Business Administration
Lecturer: Nguyen Minh Quan, PhD
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
1 / 80
Contents
1
Functions of several variables
2
Partial derivatives
3
Maxima and minima. Optimization
4
Constrained Optimization
5
Total differentials and approximations
6
Double integrals
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
2 / 80
Introduction
Example
If the company produces two products, with x of one product at a cost of
$10 each, and y of another product at a cost of $15 each, then the total cost to the firm is a function of two independent variables, x and y. By generalizing notation, the total cost can be written.
C (x, y ) = 10x + 15y
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
3 / 80
Introduction
Example (Production cost)
A company is developing a new soft drink. The cost in dollars to produce a batch of the drink is approximated by
C (x, y ) = 2200 + 27x 3 − 72xy + 8y 2 , where x is the number of kilograms of sugar per batch and y is the number of grams of flavoring per batch. Find the amounts of sugar and flavoring that result in minimum cost per batch.
What is the minimum cost?
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
4 / 80
Functions of two variables
Definition
A function of two variables f 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 variables x and y are independent variables, and z is the dependent variable. The set D is the domain of f and its range is the set of values that f takes on, that is R = {f (x, y )|(x, y ) ∈ D}.
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2.
CITY
INTERNATIONAL UNIVERSITY
Chapter 2. Multivariable calculus
Calculus 2B for Business Administration
Lecturer: Nguyen Minh Quan, PhD
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
1 / 80
Contents
1
Functions of several variables
2
Partial derivatives
3
Maxima and minima. Optimization
4
Constrained Optimization
5
Total differentials and approximations
6
Double integrals
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
2 / 80
Introduction
Example
If the company produces two products, with x of one product at a cost of
$10 each, and y of another product at a cost of $15 each, then the total cost to the firm is a function of two independent variables, x and y. By generalizing notation, the total cost can be written.
C (x, y ) = 10x + 15y
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
3 / 80
Introduction
Example (Production cost)
A company is developing a new soft drink. The cost in dollars to produce a batch of the drink is approximated by
C (x, y ) = 2200 + 27x 3 − 72xy + 8y 2 , where x is the number of kilograms of sugar per batch and y is the number of grams of flavoring per batch. Find the amounts of sugar and flavoring that result in minimum cost per batch.
What is the minimum cost?
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2. Multivariable calculus
Summer 2013
4 / 80
Functions of two variables
Definition
A function of two variables f 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 variables x and y are independent variables, and z is the dependent variable. The set D is the domain of f and its range is the set of values that f takes on, that is R = {f (x, y )|(x, y ) ∈ D}.
Dr. Nguyen Minh Quan (HCMIU-VNU)
Chapter 2.