==Method of Lagrange Multipliers==
If <math>f(x,y)</math> has a maximum or minimum subject to the constraint <math>g(x,y)=0</math>, then it will occur at one of the critical numbers of the function defined by
<math>F(x,y,\lambda)=f(x,y)-\lambda g(x,y)</math>.

In this section, we need to set up the system of equations:

<math>F_x(x,y,\lambda)=0</math>
<math>F_y(x,y,\lambda)=0</math>
<math>F_{\lambda}(x,y,\lambda)=0</math>

'''Example:''' Set up the Lagrange Multipliers:

'''1)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+y-14=0</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>\frac{\partial z}{\partial x}=4x^2-4y</math>
|-
|<math>\frac{\partial z}{\partial y}=-4x</math>
|}

'''2)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+3y-6=0</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>\frac{\partial z}{\partial x}=2xy^3</math>
|-
|<math>\frac{\partial z}{\partial y}=3x^2y^2</math>
|}

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22 Lagrange Multipliers

2020-08-18T16:45:45Z

Tphan046:

2020-08-18T16:22:30Z

Tphan046:

Math 22 Extrema of Functions of Two Variables

2020-08-18T16:04:24Z

Tphan046:

Math 22

2020-08-18T16:01:12Z

Tphan046:

== Concepts ==

'''Some useful pages'''

[[Graphs_of_equations_and_Symmetry| '''This page has a description of even and odd functions''']]

[[Graphing_Rational_Functions| '''Graphing Rational Functions''']]

'''Lectures:'''

The pages below are made for reviewing purposes only. If you find any errors, please send me an email at tphan046@ucr.edu

Special thanks to [https://profiles.ucr.edu/app/home/profile/mcurtis Mr. Curtis] for his consultation on the content of Math 22 wiki.

[[Math 22 Graph of Equation|'''1.2 Graphs of Equations''']]

[[Lines in the Plane and Slope|'''1.3 Lines in the Plane and Slope''']]

[[Math 22 Functions|'''1.4 Function''']]

[[Math 22 Limits|'''1.5 Limits''']]

[[Math 22 Continuity|'''1.6 Continuity''']]

[[Math 22 The Derivative and the Slope of a Graph|'''2.1 The Derivative and the Slope of a Graph''']]

[[Math 22 Differentiation|'''2.2 Differentiation''']]

[[Math 22 Rates of Change|'''2.3 Rates of Change''']]

[[Math 22 Product Rule and Quotient Rule|'''2.4 Product Rule and Quotient Rule''']]

[[Math 22 Chain Rule|'''2.5 Chain Rule''']]

[[Math 22 Higher-Order Derivative|'''2.6 Higher-Order Derivative''']]

[[Math 22 Implicit Differentiation|'''2.7 Implicit Differentiation''']]

[[Math 22 Related Rates|'''2.8 Related Rates''']]

[[Math 22 Increasing and Decreasing Functions|'''3.1 Increasing and Decreasing Functions''']]

[[Math 22 Extrema and First Derivative Test|'''3.2 Extrema and First Derivative Test''']]

[[Math 22 Concavity and the Second-Derivative Test|'''3.3 Concavity and the Second-Derivative Test''']]

[[Math 22 Optimization Problems|'''3.4 Optimization Problems''']]

[[Math 22 Business and Economics Applications|'''3.5 Business and Economics Applications''']]

[[Math 22 Asymptotes|'''3.6 Asymptotes''']]

[[Math 22 Differentials and Marginal Analysis|'''3.8 Differentials and Marginal Analysis''']]

[[Math 22 Exponential Functions|'''4.1 Exponential Functions''']]

[[Math 22 Natural Exponential Functions|'''4.2 Natural Exponential Functions''']]

[[Math 22 Derivatives of Exponential Functions|'''4.3 Derivatives of Exponential Functions''']]

[[Math 22 Logarithmic Functions|'''4.4 Logarithmic Functions''']]

[[Math 22 Derivatives of Logarithmic Functions|'''4.5 Derivatives of Logarithmic Functions''']]

[[Math 22 Exponential Growth and Decay|'''4.6 Exponential Growth and Decay''']]

[[Math 22 Antiderivatives and Indefinite Integrals|'''5.1 Antiderivatives and Indefinite Integral''']]

[[Math 22 Integration by Substitution and the General Power Rule|'''5.2 Integration by Substitution and the General Power Rule''']]

[[Math 22 Exponential and Logarithmic Integrals|'''5.3 Exponential and Logarithmic Integrals]]

[[Math 22 Area and the Fundamental Theorem of Calculus|''' 5.4 Area and the Fundamental Theorem of Calculus]]

[[Math 22 The Area of a Region Bounded by Two Graphs|''' 5.5 The Area of a Region Bounded by Two Graphs]]

[[Math 22 Integration by Parts and Present Value|''' 6.1 Integration by Parts and Present Value]]

[[Math 22 The Three-Dimensional Coordinate System|''' 7.1 The Three-Dimensional Coordinate System]]

[[Math 22 Functions of Several Variables|''' 7.3 Function of Several Variables]]

[[Math 22 Partial Derivatives|'''7.4 Partial Derivatives]]

[[Math 22 Extrema of Functions of Two Variables|'''7.5 Extrema of Functions of Two Variables]]
== Sample Exams ==
[[022_Exam_1_Sample_A|'''Sample Exam 1, A''']]

[[022_Exam_2_Sample_A|'''Sample Exam 2, A''']]

[[022_Exam_2_Sample_B|'''Sample Exam 2, B''']]

[[022 Sample Final A|'''Sample Final A''']]

Contributors

2020-08-18T15:49:59Z

Tphan046:

Math 22 Extrema of Functions of Two Variables

2020-08-18T15:49:47Z

Tphan046: Created page with " '''Return to Topics Page''' '''This page were made by Tri Phan'''"

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22

2020-08-18T15:49:32Z

Tphan046:

== Concepts ==

'''Some useful pages'''

[[Graphs_of_equations_and_Symmetry| '''This page has a description of even and odd functions''']]

[[Graphing_Rational_Functions| '''Graphing Rational Functions''']]

'''Lectures:'''

The pages below are made for reviewing purposes only. If you find any errors, please send me an email at tphan046@ucr.edu

Special thanks to [https://profiles.ucr.edu/app/home/profile/mcurtis Mr. Curtis] for his consultation on the content of Math 22 wiki.

[[Math 22 Graph of Equation|'''1.2 Graphs of Equations''']]

[[Lines in the Plane and Slope|'''1.3 Lines in the Plane and Slope''']]

[[Math 22 Functions|'''1.4 Function''']]

[[Math 22 Limits|'''1.5 Limits''']]

[[Math 22 Continuity|'''1.6 Continuity''']]

[[Math 22 The Derivative and the Slope of a Graph|'''2.1 The Derivative and the Slope of a Graph''']]

[[Math 22 Differentiation|'''2.2 Differentiation''']]

[[Math 22 Rates of Change|'''2.3 Rates of Change''']]

[[Math 22 Product Rule and Quotient Rule|'''2.4 Product Rule and Quotient Rule''']]

[[Math 22 Chain Rule|'''2.5 Chain Rule''']]

[[Math 22 Higher-Order Derivative|'''2.6 Higher-Order Derivative''']]

[[Math 22 Implicit Differentiation|'''2.7 Implicit Differentiation''']]

[[Math 22 Related Rates|'''2.8 Related Rates''']]

[[Math 22 Increasing and Decreasing Functions|'''3.1 Increasing and Decreasing Functions''']]

[[Math 22 Extrema and First Derivative Test|'''3.2 Extrema and First Derivative Test''']]

[[Math 22 Concavity and the Second-Derivative Test|'''3.3 Concavity and the Second-Derivative Test''']]

[[Math 22 Optimization Problems|'''3.4 Optimization Problems''']]

[[Math 22 Business and Economics Applications|'''3.5 Business and Economics Applications''']]

[[Math 22 Asymptotes|'''3.6 Asymptotes''']]

[[Math 22 Differentials and Marginal Analysis|'''3.8 Differentials and Marginal Analysis''']]

[[Math 22 Exponential Functions|'''4.1 Exponential Functions''']]

[[Math 22 Natural Exponential Functions|'''4.2 Natural Exponential Functions''']]

[[Math 22 Derivatives of Exponential Functions|'''4.3 Derivatives of Exponential Functions''']]

[[Math 22 Logarithmic Functions|'''4.4 Logarithmic Functions''']]

[[Math 22 Derivatives of Logarithmic Functions|'''4.5 Derivatives of Logarithmic Functions''']]

[[Math 22 Exponential Growth and Decay|'''4.6 Exponential Growth and Decay''']]

[[Math 22 Antiderivatives and Indefinite Integrals|'''5.1 Antiderivatives and Indefinite Integral''']]

[[Math 22 Integration by Substitution and the General Power Rule|'''5.2 Integration by Substitution and the General Power Rule''']]

[[Math 22 Exponential and Logarithmic Integrals|'''5.3 Exponential and Logarithmic Integrals]]

[[Math 22 Area and the Fundamental Theorem of Calculus|''' 5.4 Area and the Fundamental Theorem of Calculus]]

[[Math 22 The Area of a Region Bounded by Two Graphs|''' 5.5 The Area of a Region Bounded by Two Graphs]]

[[Math 22 Integration by Parts and Present Value|''' 6.1 Integration by Parts and Present Value]]

[[Math 22 The Three-Dimensional Coordinate System|''' 7.1 The Three-Dimensional Coordinate System]]

[[Math 22 Functions of Several Variables|''' 7.3 Function of Several Variables]]

[[Math 22 Partial Derivatives|'''7.4 Partial Derivatives]]

[[Math 22 Extrema of Functions of Two Variables|'''7.4 Extrema of Functions of Two Variables]]
== Sample Exams ==
[[022_Exam_1_Sample_A|'''Sample Exam 1, A''']]

[[022_Exam_2_Sample_A|'''Sample Exam 2, A''']]

[[022_Exam_2_Sample_B|'''Sample Exam 2, B''']]

[[022 Sample Final A|'''Sample Final A''']]

Math 22

2020-08-18T15:49:15Z

Tphan046:

== Concepts ==

'''Some useful pages'''

[[Graphs_of_equations_and_Symmetry| '''This page has a description of even and odd functions''']]

[[Graphing_Rational_Functions| '''Graphing Rational Functions''']]

'''Lectures:'''

The pages below are made for reviewing purposes only. If you find any errors, please send me an email at tphan046@ucr.edu

Special thanks to [https://profiles.ucr.edu/app/home/profile/mcurtis Mr. Curtis] for his consultation on the content of Math 22 wiki.

[[Math 22 Graph of Equation|'''1.2 Graphs of Equations''']]

[[Lines in the Plane and Slope|'''1.3 Lines in the Plane and Slope''']]

[[Math 22 Functions|'''1.4 Function''']]

[[Math 22 Limits|'''1.5 Limits''']]

[[Math 22 Continuity|'''1.6 Continuity''']]

[[Math 22 The Derivative and the Slope of a Graph|'''2.1 The Derivative and the Slope of a Graph''']]

[[Math 22 Differentiation|'''2.2 Differentiation''']]

[[Math 22 Rates of Change|'''2.3 Rates of Change''']]

[[Math 22 Product Rule and Quotient Rule|'''2.4 Product Rule and Quotient Rule''']]

[[Math 22 Chain Rule|'''2.5 Chain Rule''']]

[[Math 22 Higher-Order Derivative|'''2.6 Higher-Order Derivative''']]

[[Math 22 Implicit Differentiation|'''2.7 Implicit Differentiation''']]

[[Math 22 Related Rates|'''2.8 Related Rates''']]

[[Math 22 Increasing and Decreasing Functions|'''3.1 Increasing and Decreasing Functions''']]

[[Math 22 Extrema and First Derivative Test|'''3.2 Extrema and First Derivative Test''']]

[[Math 22 Concavity and the Second-Derivative Test|'''3.3 Concavity and the Second-Derivative Test''']]

[[Math 22 Optimization Problems|'''3.4 Optimization Problems''']]

[[Math 22 Business and Economics Applications|'''3.5 Business and Economics Applications''']]

[[Math 22 Asymptotes|'''3.6 Asymptotes''']]

[[Math 22 Differentials and Marginal Analysis|'''3.8 Differentials and Marginal Analysis''']]

[[Math 22 Exponential Functions|'''4.1 Exponential Functions''']]

[[Math 22 Natural Exponential Functions|'''4.2 Natural Exponential Functions''']]

[[Math 22 Derivatives of Exponential Functions|'''4.3 Derivatives of Exponential Functions''']]

[[Math 22 Logarithmic Functions|'''4.4 Logarithmic Functions''']]

[[Math 22 Derivatives of Logarithmic Functions|'''4.5 Derivatives of Logarithmic Functions''']]

[[Math 22 Exponential Growth and Decay|'''4.6 Exponential Growth and Decay''']]

[[Math 22 Antiderivatives and Indefinite Integrals|'''5.1 Antiderivatives and Indefinite Integral''']]

[[Math 22 Integration by Substitution and the General Power Rule|'''5.2 Integration by Substitution and the General Power Rule''']]

[[Math 22 Exponential and Logarithmic Integrals|'''5.3 Exponential and Logarithmic Integrals]]

[[Math 22 Area and the Fundamental Theorem of Calculus|''' 5.4 Area and the Fundamental Theorem of Calculus]]

[[Math 22 The Area of a Region Bounded by Two Graphs|''' 5.5 The Area of a Region Bounded by Two Graphs]]

[[Math 22 Integration by Parts and Present Value|''' 6.1 Integration by Parts and Present Value]]

[[Math 22 The Three-Dimensional Coordinate System|''' 7.1 The Three-Dimensional Coordinate System]]

[[Math 22 Functions of Several Variables|''' 7.3 Function of Several Variables]]

[[Math 22 Partial Derivatives|'''7.4 Partial Derivatives]]

[[Math 22 Extrema of Functions of Two Variables|'''7.4 Extrema of Functions of Two Variables]]]
== Sample Exams ==
[[022_Exam_1_Sample_A|'''Sample Exam 1, A''']]

[[022_Exam_2_Sample_A|'''Sample Exam 2, A''']]

[[022_Exam_2_Sample_B|'''Sample Exam 2, B''']]

[[022 Sample Final A|'''Sample Final A''']]

Math 22 Partial Derivatives

2020-08-18T15:48:00Z

Tphan046: /* Higher-Order Partial Derivatives */

Math 22 Partial Derivatives

2020-08-18T15:47:20Z

Tphan046:

==Partial Derivatives of a Function of Two Variables==
If <math>z=f(x,y)</math>, then the first partial derivatives of with respect to <math>x</math> and <math>y</math> are the functions <math>\frac{\partial z}{\partial x}</math> and <math>\frac{\partial z}{\partial x}</math>, defined as shown.

<math>\frac{\partial z}{\partial x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}</math>

<math>\frac{\partial z}{\partial y}=\lim_{\Delta y\to 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}</math>

We can denote <math>\frac{\partial z}{\partial x}</math> as <math>f_x(x,y)</math> and <math>\frac{\partial z}{\partial y}</math> as <math>f_y(x,y)</math>
'''Example:''' Find <math>\frac{\partial z}{\partial x}</math> and <math>\frac{\partial z}{\partial y}</math> of:

'''1)''' <math>z=f(x,y)=2x^2-4xy</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>\frac{\partial z}{\partial x}=4x^2-4y</math>
|-
|<math>\frac{\partial z}{\partial y}=-4x</math>
|}

'''2)''' <math>z=f(x,y)=x^2y^3</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>\frac{\partial z}{\partial x}=2xy^3</math>
|-
|<math>\frac{\partial z}{\partial y}=3x^2y^2</math>
|}

'''3)''' <math>z=f(x,y)=x^2e^{x^2y}</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>\frac{\partial z}{\partial x}=2xe^{x^2y}+x^2e^{x^2y}2xy</math> (product rule +chain rule)
|-
|<math>\frac{\partial z}{\partial y}=x^2e^{x^2y}(x^2)=x^4e^{x^2y}</math>
|}

==Higher-Order Partial Derivatives==
1. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial x^2}=f_{xx}</math>

2. <math>\frac{\partial}{\partial y}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial y^2}=f_{yy}</math>

3. <math>\frac{\partial}{\partial y}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial y\partial x}=f_{xy}</math>

4. <math>\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial x\partial y}=f_{yx}</math>

'''1)''' <math>f(x,y)=2x^2-4xy</math>, find <math>f_{xy}</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f_x=4x-4y</math>
|-
|Then, <math>f_{xy}=-4</math>
|}

'''2)''' <math>z=f(x,y)=3xy^2-2y+5x^2y^2</math>, find <math>f_{yx}</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f_y=6xy-2+10x^2y</math>
|-
|Then, <math>f_{yx}=6y+20xy</math>
|}

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22 Partial Derivatives

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22

2020-08-18T15:07:41Z

Tphan046:

== Concepts ==

'''Some useful pages'''

[[Graphs_of_equations_and_Symmetry| '''This page has a description of even and odd functions''']]

[[Graphing_Rational_Functions| '''Graphing Rational Functions''']]

'''Lectures:'''

The pages below are made for reviewing purposes only. If you find any errors, please send me an email at tphan046@ucr.edu

Special thanks to [https://profiles.ucr.edu/app/home/profile/mcurtis Mr. Curtis] for his consultation on the content of Math 22 wiki.

[[Math 22 Graph of Equation|'''1.2 Graphs of Equations''']]

[[Lines in the Plane and Slope|'''1.3 Lines in the Plane and Slope''']]

[[Math 22 Functions|'''1.4 Function''']]

[[Math 22 Limits|'''1.5 Limits''']]

[[Math 22 Continuity|'''1.6 Continuity''']]

[[Math 22 The Derivative and the Slope of a Graph|'''2.1 The Derivative and the Slope of a Graph''']]

[[Math 22 Differentiation|'''2.2 Differentiation''']]

[[Math 22 Rates of Change|'''2.3 Rates of Change''']]

[[Math 22 Product Rule and Quotient Rule|'''2.4 Product Rule and Quotient Rule''']]

[[Math 22 Chain Rule|'''2.5 Chain Rule''']]

[[Math 22 Higher-Order Derivative|'''2.6 Higher-Order Derivative''']]

[[Math 22 Implicit Differentiation|'''2.7 Implicit Differentiation''']]

[[Math 22 Related Rates|'''2.8 Related Rates''']]

[[Math 22 Increasing and Decreasing Functions|'''3.1 Increasing and Decreasing Functions''']]

[[Math 22 Extrema and First Derivative Test|'''3.2 Extrema and First Derivative Test''']]

[[Math 22 Concavity and the Second-Derivative Test|'''3.3 Concavity and the Second-Derivative Test''']]

[[Math 22 Optimization Problems|'''3.4 Optimization Problems''']]

[[Math 22 Business and Economics Applications|'''3.5 Business and Economics Applications''']]

[[Math 22 Asymptotes|'''3.6 Asymptotes''']]

[[Math 22 Differentials and Marginal Analysis|'''3.8 Differentials and Marginal Analysis''']]

[[Math 22 Exponential Functions|'''4.1 Exponential Functions''']]

[[Math 22 Natural Exponential Functions|'''4.2 Natural Exponential Functions''']]

[[Math 22 Derivatives of Exponential Functions|'''4.3 Derivatives of Exponential Functions''']]

[[Math 22 Logarithmic Functions|'''4.4 Logarithmic Functions''']]

[[Math 22 Derivatives of Logarithmic Functions|'''4.5 Derivatives of Logarithmic Functions''']]

[[Math 22 Exponential Growth and Decay|'''4.6 Exponential Growth and Decay''']]

[[Math 22 Antiderivatives and Indefinite Integrals|'''5.1 Antiderivatives and Indefinite Integral''']]

[[Math 22 Integration by Substitution and the General Power Rule|'''5.2 Integration by Substitution and the General Power Rule''']]

[[Math 22 Exponential and Logarithmic Integrals|'''5.3 Exponential and Logarithmic Integrals]]

[[Math 22 Area and the Fundamental Theorem of Calculus|''' 5.4 Area and the Fundamental Theorem of Calculus]]

[[Math 22 The Area of a Region Bounded by Two Graphs|''' 5.5 The Area of a Region Bounded by Two Graphs]]

[[Math 22 Integration by Parts and Present Value|''' 6.1 Integration by Parts and Present Value]]

[[Math 22 The Three-Dimensional Coordinate System|''' 7.1 The Three-Dimensional Coordinate System]]

[[Math 22 Functions of Several Variables|''' 7.3 Function of Several Variables]]

[[Math 22 Partial Derivatives|'''7.4 Partial Derivatives]]
== Sample Exams ==
[[022_Exam_1_Sample_A|'''Sample Exam 1, A''']]

[[022_Exam_2_Sample_A|'''Sample Exam 2, A''']]

[[022_Exam_2_Sample_B|'''Sample Exam 2, B''']]

[[022 Sample Final A|'''Sample Final A''']]

Math 22 Functions of Several Variables

2020-08-18T15:06:47Z

Tphan046:

==Definition of a Function of Two Variables==
Let <math>D</math> be a set of ordered pairs of real numbers.
If to each ordered pair <math>(x,y)</math> in <math>D</math> there corresponds a unique real number <math>f(x,y)</math>, then <math>f</math> is a function of <math>x</math> and <math>y</math>.
The set <math>D</math> is the domain of <math>f</math>, and the corresponding set of values for <math>f(x,y)</math> is the range of <math>f</math>. Functions of three, four, or more variables are defined similarly.

'''Exercises 1''' Given <math>f(x,y)=2x+y-3</math>. Evaluate:

'''1)''' <math>f(0,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(0,2)=2(0)+2-3=-1</math>
|}

'''2)''' <math>f(5,20)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(5,20)=2(5)+20-3=27</math>
|}

'''3)''' <math>f(-1,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(-1,2)=2(-2)+2-3=-5</math>
|}

'''4)''' <math>f(4,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(4,2)=2(3)+2-3=5</math>
|}
==The Domain and Range of a Function of Two Variables==
'''Example:''' Find the domain of <math>f(x,y)=\sqrt{9-x^2-y^2}</math>

Notice that : The radicand should be non-negative. So, <math>9-x^2-y^2\ge 0</math>, hence the domain is <math>x^2+y^2\le 9</math> (or the set of all point that lie inside the circle).

Notice: <math>x^2+y^2= 9</math> is the circle center at <math>(0,0)</math>, radius 3.

Since the point <math>(0,0)</math> satisfies the inequality <math>x^2+y^2\le 9</math>. Hence the range is <math>0\le x\le 3</math>

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22 Functions of Several Variables

2020-08-18T15:06:00Z

Tphan046:

==Definition of a Function of Two Variables==
Let <math>D</math> be a set of ordered pairs of real numbers.
If to each ordered pair <math>(x,y)</math> in <math>D</math> there corresponds a unique real number <math>f(x,y)</math>, then <math>f</math> is a function of <math>x</math> and <math>y</math>.
The set <math>D</math> is the domain of <math>f</math>, and the corresponding set of values for <math>f(x,y)</math> is the range of <math>f</math>. Functions of three, four, or more variables are defined similarly.

'''Exercises 1''' Given <math>f(x,y)=2x+y-3</math>. Evaluate:

'''1)''' <math>f(0,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(0,2)=2(0)+2-3=-1</math>
|}

'''2)''' <math>f(5,20)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(5,20)=2(5)+20-3=27</math>
|}

'''3)''' <math>f(-1,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(-1,2)=2(-2)+2-3=-5</math>
|}

'''4)''' <math>f(4,2)</math>
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
|-
|<math>f(x,y)=2x+y-3</math>
|-
|So, <math>f(4,2)=2(3)+2-3=5</math>
|}
==The Domain and Range of a Function of Two Variables==
'''Example:''' Find the domain of <math>f(x,y)=\sqrt{9-x^2-y^2}</math>

Notice that : The radicand should be non-negative. So, <math>9-x^2-y^2\ge 0</math>, hence the domain is <math>x^2+y^2\le 9</math>.

Notice: <math>x^2+y^2= 9</math> is the circle center at <math>(0,0)</math>, radius 3.

Since the point <math>(0,0)</math> satisfies the inequality <math>x^2+y^2\le 9</math>. Hence the range is <math>0\le x\le 3</math>

[[Math_22| '''Return to Topics Page''']]

'''This page were made by [[Contributors|Tri Phan]]'''

Math 22 Functions of Several Variables

2020-08-18T14:50:06Z

Tphan046: