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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
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Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
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Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Section 8: Lagrange Multipliers
Section 8: Lagrange Multipliers
Instructions
First, you should watch the concepts videos below explaining the topics in the section.
Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
This is the last section so you should return to the
main several variable calculus page
or the
main workshop page
when you have finished the material below.
Concepts
Explanation of Lagrange's Theorem and Lagrange multipliers
Finding the absolute maximum or absolute minimum values of \(z=f(x,y)\) subject to the constraint \(g(x,y)=k\)
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Conceptual V1
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find the extreme values of \(f(x,y)=3x+y\) subject to the constraint \(x^2+y^2=10.\)
Reveal Answer
Absolute Maximum: \(z=10\)
Absolute Minimum: \(z=-10\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Exercise V1
Find the extreme values of \(f(,y)=x^2+2y^2\) subject to the constraint \(x^2+16y^2=16.\)
Reveal Answer
Absolute Maximum: \(z=16\)
Absolute Minimum: \(z=2\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Exercise V2
Find the minimum values of \(f(x,y,z)=x^2+y^2+z^2\) subject to the constraint \(x+3y-2z=12.\)
Reveal Answer
Absolute Minimum: \(f\left(\frac{6}{7},\frac{18}{7},-\frac{12}{7}\right)=\dfrac{72}{7}\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Exercise V3
Find the volume of the largest rectangular box with faces parallel to the coordinate planes that can be inscribed in the ellipsoid \(16x^2+4y^2+9z^2=144.\)
Reveal Answer
The maximum volume is \(V=64\sqrt{3}\) units\(^3.\)
Watch Video Solution
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Lagrange Multipliers Exercise V4
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
How do we know when it is proper to use the method of Lagrange to find the absolute extrema of \(z=f(x,y)\)?
What system of equations must be solved in order to find the absolute extrema of \(z=f(x,y)\) subject to \(g(x,y)=k\)?
Choose an exercise from
Section 7
that uses The Extreme Value Theorem, and instead use the method of Lagrange. Determine whether the method of Lagrange is superior over The Extreme Value Theorem method for the problem you chose.
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