The document discusses Lagrange multipliers, a method for finding the maximum or minimum value of a function subject to a constraint. It provides examples of using Lagrange multipliers to find: (1) the maximum volume of a box given a fixed surface area, (2) the extreme values of a function on a circle, and (3) the extreme values of a function on a disk. It also gives an example of using Lagrange multipliers to find the points on a sphere closest to and farthest from a given point.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
This document provides an overview of functions of several variables. It discusses notation for functions with multiple independent variables, domains of such functions, and graphs of functions with two or more variables. Specifically, it gives examples of finding the domain of functions defined by equations, sketching the graph of a function as a surface in 3D space using traces in coordinate planes and parallel planes, and creating a contour map using level curves representing different values of the dependent variable.
This document discusses infinite limits, limits at infinity, and limit rules. It begins by explaining that the limits of 1/x as x approaches 0 from the left and right do not exist as real numbers, but it is useful to describe the behavior as approaching positive and negative infinity. It then discusses properties of infinite limits, including one-sided limits and examples. The document proceeds to define vertical asymptotes and provide examples of determining asymptotes. It concludes by covering limit rules that can be used to evaluate limits more easily, such as sum, difference, product, and quotient rules, as well as the sandwich theorem.
The Lagrange multiplier method provides a strategy for finding the maxima and minima of a function subject to constraints. It involves setting up a system of equations involving the function, its derivatives, and the constraints and their derivatives. Solving this system of equations yields candidate maxima/minima points, which are then checked in the original function to determine if they are actually maxima or minima. The document provides examples of applying the Lagrange multiplier method to problems with single and multiple constraints.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
A partial differential equation contains one dependent variable and more than one independent variable. The partial derivatives of a function f(x,y) with respect to x and y at a point (x,y) are represented as ∂f/∂x and ∂f/∂y. Higher order partial derivatives can be found by taking partial derivatives multiple times with respect to the independent variables. The chain rule can be used to find partial derivatives when the dependent variable is a function of other variables that are themselves functions of the independent variables.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
This document provides an overview of functions of several variables. It discusses notation for functions with multiple independent variables, domains of such functions, and graphs of functions with two or more variables. Specifically, it gives examples of finding the domain of functions defined by equations, sketching the graph of a function as a surface in 3D space using traces in coordinate planes and parallel planes, and creating a contour map using level curves representing different values of the dependent variable.
This document discusses infinite limits, limits at infinity, and limit rules. It begins by explaining that the limits of 1/x as x approaches 0 from the left and right do not exist as real numbers, but it is useful to describe the behavior as approaching positive and negative infinity. It then discusses properties of infinite limits, including one-sided limits and examples. The document proceeds to define vertical asymptotes and provide examples of determining asymptotes. It concludes by covering limit rules that can be used to evaluate limits more easily, such as sum, difference, product, and quotient rules, as well as the sandwich theorem.
The Lagrange multiplier method provides a strategy for finding the maxima and minima of a function subject to constraints. It involves setting up a system of equations involving the function, its derivatives, and the constraints and their derivatives. Solving this system of equations yields candidate maxima/minima points, which are then checked in the original function to determine if they are actually maxima or minima. The document provides examples of applying the Lagrange multiplier method to problems with single and multiple constraints.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
A partial differential equation contains one dependent variable and more than one independent variable. The partial derivatives of a function f(x,y) with respect to x and y at a point (x,y) are represented as ∂f/∂x and ∂f/∂y. Higher order partial derivatives can be found by taking partial derivatives multiple times with respect to the independent variables. The chain rule can be used to find partial derivatives when the dependent variable is a function of other variables that are themselves functions of the independent variables.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
The document provides an overview of key concepts in calculus limits including:
1) Limits describe the behavior of a function as its variable approaches a constant value.
2) Tables of values and graphs can be used to evaluate limits by showing how the function values change as the variable nears the constant.
3) Common limit laws are described such as addition, multiplication, and substitution which allow evaluating limits of combined functions.
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoint positions and total length.
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoints and total length.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
The document discusses limits and continuity of functions. It provides examples of computing one-sided limits, limits at points of discontinuity, and limits involving algebraic, trigonometric, exponential and logarithmic functions. The key rules for limits include the properties of limits, the sandwich theorem, and limits of compositions of functions. Continuity of functions is defined as a function having a limit equal to its value at a point. Polynomials, trigonometric functions and exponentials are shown to be continuous everywhere they are defined.
The document discusses several rules for derivatives:
1) The derivative of a sum of functions is the sum of the derivatives of each term.
2) The derivative of a product of functions is the first function times the derivative of the second plus the second function times the derivative of the first.
3) To find critical points, set the derivative equal to 0 and solve for x. Then evaluate the second derivative at those points to determine if they are maxima or minima.
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others fixed. It provides definitions, examples of computing partial derivatives, and interpretations as rates of change.
2) Techniques covered include implicit differentiation, using the chain rule to find derivatives of implicitly defined functions, and computing second order partial derivatives.
3) Diagrams and tables are referenced to illustrate level curves and contour maps for functions of two variables.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
The document presents three solutions to finding the curve that represents the fastest path between two points under the influence of gravity, known as the brachistochrone curve.
The first solution uses a variational approach with a Lagrangian of 1 + y^2/√y to derive the Euler-Lagrange equation and obtain the parametrization x = a(θ - sinθ), y = a(1 - cosθ) of a cycloid curve.
The second solution also uses a variational approach but with y as the independent variable, obtaining the same solution.
The third solution notes the first Lagrangian is independent of x, so the quantity E = -1/√y(1+y
The document presents three solutions to finding the curve that represents the fastest path between two points under the influence of gravity, known as the brachistochrone curve.
The first solution uses a variational approach with a Lagrangian of 1 + y^2/√y to derive the Euler-Lagrange equation and obtain the parametrization x = a(θ - sinθ), y = a(1 - cosθ) of a cycloid curve.
The second solution also uses a variational approach but with y as the independent variable, obtaining the same solution.
The third solution notes the first Lagrangian is independent of x, so the quantity E = -1/√y(1+y
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
I am Duncan V. I am a Calculus Homework Expert at mathshomeworksolver.com. I hold a Master's in Mathematics from Manchester, United Kingdom. I have been helping students with their homework for the past 8 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if |g'(α)| < 1, where α is the root and g' is the derivative of g. This ensures the error decreases at each iteration.
S3. Examples show the method can converge rapidly, as in Newton's method, or diverge, depending on the properties of g near the root. Aitken extrapolation can provide a better estimate of the root than the current iterate xn.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if g(x) is continuous and λ, the maximum absolute value of the derivative of g(x), is less than 1.
S3. Examples show that fixed point iteration can converge slowly if the derivative of g(x) at the root is close to 1, and Aitken's method can be used to accelerate convergence by extrapolating the iterates.
1) The document discusses various methods for graphing polynomials, including using function shift rules to graph even and odd powers, using the leading term test to predict behavior, and graphing using known zeros and the multiplicity rules.
2) The multiplicity rules state that a zero with even multiplicity will cause the graph to "bounce off" the x-intercept, while an odd multiplicity will cause the graph to pass through the intercept.
3) An example graphs a polynomial by factoring it, finding the zeros, and applying the multiplicity rules to the graph.
This document provides information about Calculus 2, including lessons on indeterminate forms, Rolle's theorem, the mean value theorem, and differentiation of transcendental functions. It defines Rolle's theorem and the mean value theorem, provides examples of applying each, and discusses how Rolle's theorem can be used to find the value of c. It also defines inverse trigonometric functions and their derivatives. The document is for MATH 09 Calculus 2 and includes exercises for students to practice applying the theorems.
The document provides an overview of key concepts in calculus limits including:
1) Limits describe the behavior of a function as its variable approaches a constant value.
2) Tables of values and graphs can be used to evaluate limits by showing how the function values change as the variable nears the constant.
3) Common limit laws are described such as addition, multiplication, and substitution which allow evaluating limits of combined functions.
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoint positions and total length.
The document presents four solutions to finding the shape of a hanging chain under its own weight. The first solution uses force balancing on small segments of the chain. This leads to two differential equations that are solved to get a hyperbolic cosine function for the shape. The other three solutions use variational arguments to minimize the chain's potential energy, subject to its total length constraint. This also results in a hyperbolic cosine shape function. The constant parameter in this function can be determined from the chain's endpoints and total length.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
The document discusses limits and continuity of functions. It provides examples of computing one-sided limits, limits at points of discontinuity, and limits involving algebraic, trigonometric, exponential and logarithmic functions. The key rules for limits include the properties of limits, the sandwich theorem, and limits of compositions of functions. Continuity of functions is defined as a function having a limit equal to its value at a point. Polynomials, trigonometric functions and exponentials are shown to be continuous everywhere they are defined.
The document discusses several rules for derivatives:
1) The derivative of a sum of functions is the sum of the derivatives of each term.
2) The derivative of a product of functions is the first function times the derivative of the second plus the second function times the derivative of the first.
3) To find critical points, set the derivative equal to 0 and solve for x. Then evaluate the second derivative at those points to determine if they are maxima or minima.
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others fixed. It provides definitions, examples of computing partial derivatives, and interpretations as rates of change.
2) Techniques covered include implicit differentiation, using the chain rule to find derivatives of implicitly defined functions, and computing second order partial derivatives.
3) Diagrams and tables are referenced to illustrate level curves and contour maps for functions of two variables.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
The document presents three solutions to finding the curve that represents the fastest path between two points under the influence of gravity, known as the brachistochrone curve.
The first solution uses a variational approach with a Lagrangian of 1 + y^2/√y to derive the Euler-Lagrange equation and obtain the parametrization x = a(θ - sinθ), y = a(1 - cosθ) of a cycloid curve.
The second solution also uses a variational approach but with y as the independent variable, obtaining the same solution.
The third solution notes the first Lagrangian is independent of x, so the quantity E = -1/√y(1+y
The document presents three solutions to finding the curve that represents the fastest path between two points under the influence of gravity, known as the brachistochrone curve.
The first solution uses a variational approach with a Lagrangian of 1 + y^2/√y to derive the Euler-Lagrange equation and obtain the parametrization x = a(θ - sinθ), y = a(1 - cosθ) of a cycloid curve.
The second solution also uses a variational approach but with y as the independent variable, obtaining the same solution.
The third solution notes the first Lagrangian is independent of x, so the quantity E = -1/√y(1+y
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
I am Duncan V. I am a Calculus Homework Expert at mathshomeworksolver.com. I hold a Master's in Mathematics from Manchester, United Kingdom. I have been helping students with their homework for the past 8 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if |g'(α)| < 1, where α is the root and g' is the derivative of g. This ensures the error decreases at each iteration.
S3. Examples show the method can converge rapidly, as in Newton's method, or diverge, depending on the properties of g near the root. Aitken extrapolation can provide a better estimate of the root than the current iterate xn.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if g(x) is continuous and λ, the maximum absolute value of the derivative of g(x), is less than 1.
S3. Examples show that fixed point iteration can converge slowly if the derivative of g(x) at the root is close to 1, and Aitken's method can be used to accelerate convergence by extrapolating the iterates.
1) The document discusses various methods for graphing polynomials, including using function shift rules to graph even and odd powers, using the leading term test to predict behavior, and graphing using known zeros and the multiplicity rules.
2) The multiplicity rules state that a zero with even multiplicity will cause the graph to "bounce off" the x-intercept, while an odd multiplicity will cause the graph to pass through the intercept.
3) An example graphs a polynomial by factoring it, finding the zeros, and applying the multiplicity rules to the graph.
This document provides information about Calculus 2, including lessons on indeterminate forms, Rolle's theorem, the mean value theorem, and differentiation of transcendental functions. It defines Rolle's theorem and the mean value theorem, provides examples of applying each, and discusses how Rolle's theorem can be used to find the value of c. It also defines inverse trigonometric functions and their derivatives. The document is for MATH 09 Calculus 2 and includes exercises for students to practice applying the theorems.
Similar to Chap14_Sec8 - Lagrange Multiplier.ppt (20)
How to Fix the Import Error in the Odoo 17Celine George
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
2. PARTIAL DERIVATIVES
In Example 6 in Section 14.7, we maximized
a volume function V = xyz subject to
the constraint 2xz + 2yz + xy = 12—which
expressed the side condition that the surface
area was 12 m2.
3. In this section, we present Lagrange’s method
for maximizing or minimizing a general
function f(x, y, z) subject to a constraint
(or side condition) of the form g(x, y, z) = k.
PARTIAL DERIVATIVES
4. 14.8
Lagrange Multipliers
In this section, we will learn about:
Lagrange multipliers for two and three variables,
and given one and two constraints.
PARTIAL DERIVATIVES
6. So, we start by trying to find the extreme
values of f(x, y) subject to a constraint of
the form g(x, y) = k.
In other words, we seek the extreme values
of f(x, y) when the point (x, y) is restricted to lie
on the level curve g(x, y) = k.
LAGRANGE MULTIPLIERS—TWO VARIABLES
7. The figure shows this curve together with
several level curves of f.
These have the
equations f(x, y) = c,
where c = 7, 8, 9,
10, 11
LAGRANGE MULTIPLIERS—TWO VARIABLES
8. To maximize f(x, y) subject to g(x, y) = k
is to find:
The largest value of c
such that the level
curve f(x, y) = c
intersects g(x, y) = k.
LAGRANGE MULTIPLIERS—TWO VARIABLES
9. It appears that this happens when these
curves just touch each other—that is, when
they have a common tangent line.
Otherwise, the value
of c could be
increased further.
LAGRANGE MULTIPLIERS—TWO VARIABLES
10. This means that the normal lines at
the point (x0 , y0) where they touch are
identical.
So the gradient vectors are parallel.
That is,
for some scalar λ.
0 0
0 0
( , ) ( , )
f x y g x y
LAGRANGE MULTIPLIERS—TWO VARIABLES
11. This kind of argument also applies to the
problem of finding the extreme values of
f(x, y, z) subject to the constraint g(x, y, z) = k.
Thus, the point (x, y, z) is restricted to lie on
the level surface S with equation g(x, y, z) = k.
LAGRANGE MULTIPLIERS—THREE VARIABLES
12. Instead of the level curves in the previous
figure, we consider the level surfaces
f(x, y, z) = c.
We argue that, if the maximum value of f is
f(x0, y0, z0) = c, then the level surface f(x, y, z) = c
is tangent to the level surface g(x, y, z) = k.
So, the corresponding gradient vectors are parallel.
LAGRANGE MULTIPLIERS—THREE VARIABLES
13. This intuitive argument can be
made precise as follows.
LAGRANGE MULTIPLIERS—THREE VARIABLES
14. Suppose that a function f has an extreme
value at a point P(x0, y0, z0) on the surface S.
Then, let C be a curve with vector equation
r(t) = <x(t), y(t), z(t)> that lies on S and passes
through P.
LAGRANGE MULTIPLIERS—THREE VARIABLES
15. If t0 is the parameter value corresponding to
the point P, then
r(t0) = <x0, y0, z0>
The composite function
h(t) = f(x(t), y(t), z(t))
represents the values that f
takes on the curve C.
LAGRANGE MULTIPLIERS—THREE VARIABLES
16. f has an extreme value at (x0, y0, z0).
So, it follows that h has an extreme value at t0.
Thus, h’(t0) = 0.
LAGRANGE MULTIPLIERS—THREE VARIABLES
17. However, if f is differentiable, we can use
the Chain Rule to write:
0
0 0 0 0 0 0 0 0
0 0 0 0
0 0 0 0
0 '
, , ' , , '
, , '
, , '
x y
z
h t
f x y z x t f x y z y t
f x y z z t
f x y z t
r
LAGRANGE MULTIPLIERS—THREE VARIABLES
18. This shows that the gradient vector
is orthogonal to the tangent
vector r’(t0) to every such curve C.
0 0 0
, ,
f x y z
LAGRANGE MULTIPLIERS—THREE VARIABLES
19. However, we already know from Section 14.6
that the gradient vector of g, ,
is also orthogonal to r’(t0) for every such
curve.
See Equation 18 from Section 6.
This means that the gradient vectors
and must be parallel.
0 0 0
, ,
g x y z
0 0 0
, ,
f x y z
0 0 0
, ,
g x y z
LAGRANGE MULTIPLIERS—THREE VARIABLES
20. Therefore, if , there is
a number λ such that:
The number λ in the equation is called
a Lagrange multiplier.
The procedure based on Equation 1 is as follows.
0 0 0
, , 0
g x y z
0 0 0 0 0 0
, , , ,
f x y z g x y z
LAGRANGE MULTIPLIER Equation 1
21. LAGRANGE MULTIPLIERS—METHOD
To find the maximum and minimum values
of f(x, y, z) subject to the constraint
g(x, y, z) = k [assuming that these extreme
values exist and on the surface
g(x, y, z) = k], we proceed as follows.
g
0
22. a. Find all values of x, y, z, and λ such that
and
b. Evaluate f at all the points (x, y, z) that
result from step a.
The largest of these values is the maximum value of f.
The smallest is the minimum value of f.
LAGRANGE MULTIPLIERS—METHOD
, , , ,
, ,
f x y z g x y z
g x y z k
23. In deriving Lagrange’s method, we
assumed that .
In each of our examples, you can check that
at all points where g(x, y, z) = k.
0
g
0
g
LAGRANGE’S METHOD
24. If we write the vector equation
in terms of its components, then the equations
in step a become:
fx = λgx fy = λgy fz = λgz g(x, y, z) = k
This is a system of four equations in the four unknowns
x, y, z, and λ.
However, it is not necessary to find explicit values for λ.
LAGRANGE’S METHOD
f g
25. For functions of two variables,
the method of Lagrange multipliers is
similar to the method just described.
LAGRANGE’S METHOD
26. To find the extreme values of f(x, y) subject
to the constraint g(x, y) = k, we look for values
of x, y, and λ such that:
This amounts to solving three equations
in three unknowns:
fx = λgx fy = λgy g(x, y) = k
, , and ,
f x y g x y g x y k
LAGRANGE’S METHOD
27. Our first illustration of Lagrange’s method
is to reconsider the problem given in
Example 6 in Section 14.7
LAGRANGE’S METHOD
28. A rectangular box without a lid is to be
made from 12 m2 of cardboard.
Find the maximum volume of such a box.
LAGRANGE’S METHOD Example 1
29. As in Example 6 in Section 14.7, we let
x, y, and z be the length, width, and height,
respectively, of the box in meters.
Then, we wish to maximize V = xyz
subject to the constraint
g(x, y, z) = 2xz + 2yz +xy = 12
LAGRANGE’S METHOD Example 1
30. Using the method of Lagrange multipliers,
we look for values of x, y, z, and λ
such that:
and ( , , ) 12
V g g x y z
LAGRANGE’S METHOD Example 1
31. This gives the equations
Vx = λgx
Vy = λgy
Vz = λgz
2xz + 2yz + xy = 12
LAGRANGE’S METHOD Example 1
32. The equations become:
yz = λ(2z + y)
xz = λ(2z + x)
xy = λ(2x + 2y)
2xz + 2yz + xy = 12
LAGRANGE’S METHOD E. g. 1—Eqns. 2-5
33. There are no general rules for solving
systems of equations.
Sometimes, some ingenuity is required.
LAGRANGE’S METHOD Example 1
34. In this example, you might notice that
if we multiply Equation 2 by x, Equation 3 by y,
and Equation 4 by z, then left sides of
the equations will be identical.
LAGRANGE’S METHOD Example 1
35. Doing so, we have:
xyz = λ(2xz + xy)
xyz = λ(2yz + xy)
xyz = λ(2xz + 2yz)
LAGRANGE’S METHOD E. g. 1—Eqns. 6-8
36. We observe that λ ≠ 0 because λ = 0
would imply yz = xz = xy = 0 from Equations
2, 3, and 4.
This would contradict Equation 5.
LAGRANGE’S METHOD Example 1
37. Therefore, from Equations 6 and 7,
we have
2xz + xy = 2yz + xy
which gives xz = yz.
However, z ≠ 0 (since z = 0 would give V = 0).
Thus, x = y.
LAGRANGE’S METHOD Example 1
38. From Equations 7 and 8,
we have
2yz + xy = 2xz + 2yz
which gives 2xz = xy.
Thus, since x ≠ 0, y = 2z.
LAGRANGE’S METHOD Example 1
39. If we now put x = y = 2z in Equation 5,
we get:
4z2 + 4z2 + 4z2 = 12
Since x, y, and z are all positive, we therefore
have z = 1, and so x = 2 and y = 2.
This agrees with our answer in Section 14.7
LAGRANGE’S METHOD Example 1
40. Another method for solving the system
of equations 2–5 is to solve each of Equations
2, 3, and 4 for λ and then to equate
the resulting expressions.
LAGRANGE’S METHOD
41. Find the extreme values of the function
f(x, y) = x2 + 2y2 on the circle x2 + y2 = 1.
We are asked for the extreme values of f
subject to the constraint
g(x, y) = x2 + y2 = 1
LAGRANGE’S METHOD Example 2
42. Using Lagrange multipliers, we solve
the equations and g(x, y) = 1.
These can be written as:
fx = λgx
fy = λgy
g(x, y) = 1
f g
LAGRANGE’S METHOD Example 2
43. They can also be written as:
2x = 2xλ
4y = 2yλ
x2 + y2 = 1
LAGRANGE’S METHOD E. g. 2—Eqns. 9-11
44. From Equation 9, we have
x = 0 or λ = 1
If x = 0, then Equation 11 gives y = ±1.
If λ = 1, then y = 0 from Equation 10;
so, then Equation 11 gives x = ±1.
LAGRANGE’S METHOD Example 2
45. Therefore, f has possible extreme values
at the points
(0, 1), (0, –1), (1, 0), (–1, 0)
Evaluating f at these four points,
we find that:
f(0, 1) = 2 f(0, –1) = 2 f(1, 0) = 1 f(–1, 0) = 1
LAGRANGE’S METHOD Example 2
46. Therefore, the maximum value of f
on the circle x2 + y2 = 1 is:
f(0, ±1) = 2
The minimum value is:
f(±1, 0) = 1
LAGRANGE’S METHOD Example 2
47. Checking with the figure, we see that
these values look reasonable.
LAGRANGE’S METHOD Example 2
48. LAGRANGE’S METHOD
The geometry behind the use of Lagrange
multipliers in Example 2 is shown here.
The extreme values
of f(x, y) = x2 + 2y2
correspond to the
level curves that
touch the circle
x2 + y2 = 1
49. Find the extreme values of
f(x, y) = x2 + 2y2 on the disk x2 + y2 ≤ 1
According to the procedure in Equation 9
in Section 14.7, we compare the values of f
at the critical points with values at the points
on the boundary.
LAGRANGE’S METHOD Example 3
50. Since fx = 2x and fy = 4y, the only critical
point is (0, 0).
We compare the value of f at that point
with the extreme values on the boundary
from Example 2:
f(0, 0) = 0 f(±1, 0) =1 f(0, ±1) = 2
LAGRANGE’S METHOD Example 3
51. Therefore, the maximum value of f
on the disk x2 + y2 ≤ 1 is:
f(0, ±1) = 2
The minimum value is:
f(0, 0) = 0
LAGRANGE’S METHOD Example 3
52. Find the points on the sphere
x2 + y2 + z2 = 4 that are closest to
and farthest from the point (3, 1, –1).
LAGRANGE’S METHOD Example 4
53. The distance from a point (x, y, z) to the point
(3, 1, –1) is:
However, the algebra is simpler if we
instead maximize and minimize the square
of the distance:
2 2 2
3 1 1
d x y z
LAGRANGE’S METHOD Example 4
2
2 2 2
( , , )
3 1 1
d f x y z
x y z
54. The constraint is that the point (x, y, z)
lies on the sphere, that is,
g(x, y, z) = x2 + y2 + z2
= 4
LAGRANGE’S METHOD Example 4
55. According to the method of Lagrange
multipliers, we solve:
, 4
f g g
LAGRANGE’S METHOD Example 4
57. The simplest way to solve these equations
is to solve for x, y, and z in terms of λ from
Equations 12, 13, and 14, and then
substitute these values into Equation 15.
LAGRANGE’S METHOD Example 4
58. From Equation 12, we have:
x – 3 = xλ or x(1 – λ) = 3 or
Note that 1 – λ ≠ 0 because λ = 1 is impossible
from Equation 12.
3
1
x
LAGRANGE’S METHOD Example 4
61. These values of λ then give the corresponding
points (x, y, z):
It’s easy to see that f has a smaller value
at the first of these points.
6 2 2 6 2 2
, , and , ,
11 11 11 11 11 11
LAGRANGE’S METHOD Example 4
62. Thus, the closest point is:
The farthest is:
6/ 11,2/ 11, 2/ 11
6/ 11, 2/ 11,2/ 11
LAGRANGE’S METHOD Example 4
63. LAGRANGE’S METHOD
The figure shows the sphere and
the nearest point in Example 4.
Can you see how to
find the coordinates
of P without using
calculus?
64. TWO CONSTRAINTS
Suppose now that we want to find
the maximum and minimum values of
a function f(x, y, z) subject to two constraints
(side conditions) of the form g(x, y, z) = k
and h(x, y, z) = c.
65. TWO CONSTRAINTS
Geometrically, this means:
We are looking for the extreme values of f
when (x, y, z) is restricted to lie on the curve
of intersection C of the level surfaces
g(x, y, z) = k and
h(x, y, z) = c.
66. Suppose f has such an extreme value at
a point P(x0, y0, z0).
We know from the beginning of this section
that is orthogonal to C at P.
f
TWO CONSTRAINTS
67. However, we also know that is orthogonal
to g(x, y, z) = k and is orthogonal to
h(x, y, z) = c.
So, and are both orthogonal to C.
g
h
TWO CONSTRAINTS
g
h
68. This means that the gradient vector
is in the plane determined
by and
We assume that these gradient vectors
are not zero and not parallel.
0 0 0
, ,
f x y z
TWO CONSTRAINTS
0 0 0
, ,
g x y z
0 0 0
, ,
h x y z
69. So, there are numbers λ and μ
(called Lagrange multipliers)
such that:
0 0 0
0 0 0 0 0 0
, ,
, , , ,
f x y z
g x y z h x y z
TWO CONSTRAINTS Equation 16
70. In this case, Lagrange’s method is to look
for extreme values by solving five equations
in the five unknowns
x, y, z, λ, μ
TWO CONSTRAINTS
71. These equations are obtained by writing
Equation 16 in terms of its components and
using the constraint equations:
fx = λgx + μhx fy = λgy + μhy fz = λgz + μhz
g(x, y, z) = k h(x, y, z) = c
TWO CONSTRAINTS
72. Find the maximum value of the function
f(x, y, z) = x + 2y + 3z on the curve of
intersection of the plane x – y + z = 1
and the cylinder x2 + y2 = 1
TWO CONSTRAINTS Example 5
73. We maximize the given function subject
to the constraints
g(x, y, z) = x – y + z = 1
h(x, y, z) = x2 + y2 = 1
TWO CONSTRAINTS Example 5
74. The Lagrange condition is
So, we solve the equations
1 = λ + 2xμ
2 = –λ + 2yμ
3 = λ
x – y + z = 1
x2 + y2 = 1
f g h
TWO CONSTRAINTS E. g. 5—Eqns. 17-21
75. Putting λ = 3 (from Equation 19)
in Equation 17, we get 2xμ = –2.
Thus, x = –1/μ.
Similarly, Equation 18 gives y = 5/(2μ).
TWO CONSTRAINTS Example 5
76. Substitution in Equation 21 then
gives:
Thus,
2 2
1 25
1
4
2 29
4 , 29 / 2
TWO CONSTRAINTS Example 5
77. Then,
and, from Equation 20,
2/ 29
x
1
1 7 / 29
z x y
TWO CONSTRAINTS Example 5
5/ 29
y
78. The corresponding values of f are:
Hence, the maximum value of f
on the given curve is:
2 5 7
2 3 1 3 29
29 29 29
3 29
TWO CONSTRAINTS Example 5
79. TWO CONSTRAINTS
The cylinder x2 + y2 = 1
intersects the plane
x – y + z = 1 in an
ellipse.
Example 5 asks for
the maximum value of f
when (x, y, z) is restricted
to lie on the ellipse.