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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
Mathematical Probability
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 7: Maximum and Minimum Values
Section 7: Maximum and Minimum Values
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.
When you have finished the material below, you can start onthe
next section
or return to the
main several variable calculus page
.
Concepts
Local and absolute extrema of a function \(z=f(x,y)\)
The Second Derivative Test for Local Extrema
Extreme Value Theorem for Functions of Two Variables
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video 1
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.
Maximum and Minimum Value (multivariable) Conceptual V1
Watch Concepts Video 2
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.
Maximum and Minimum Value (multivariable) Conceptual V2
Watch Concepts Video 3
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.
Maximum and Minimum Value (multivariable) Conceptual V3
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find all local extrema or saddle points for \(f(x,y)=y^3-6y^2-2x^3-6x^2+48x+20.\)
Reveal Answer
Saddle Points at \((-4,0,-140)\), \((2,4,44)\)
Local Minimum at \((-4,4,-172)\)
Local Maximum at \((2,0,76)\)
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.
Maximum and Minimum Value (multivariable) Exercise V1
Find all local extrema or saddle points for \(f(x,y)=x^3+6xy-2y^2.\)
Reveal Answer
Saddle Point at \((0,0,0)\)
Local Maximum at \(\left(-3,-\frac{9}{2},\frac{27}{2}\right)\)
No Local Minimum
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.
Maximum and Minimum Value (multivariable) Exercise V2
A box with no lid is to hold 10 cubic meters. Find the dimensions of the box with a minimum surface area.
Reveal Answer
Length\(\; =\sqrt[3]{20}\)
Width\(\; =\sqrt[3]{20}\)
Height\(\; =\dfrac{\sqrt[3]{20}}{2}\)
Watch Video Solution
Find the absolute extrema of \(f(x,y)=x^2+y^2-2x\) on the closed triangular region with vertices \((2,0)\), \((0,2)\), and \((0,-2).\)
Reveal Answer
Absolute Maximum: \(z=4\)
Absolute Minimum: \(z=-1\)
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.
Maximum and Minimum Value (multivariable) Exercise V4
Find the absolute extrema of \(f(x,y)=x^2+y^2-2x\) on the closed circulur region \(x^2+y^2\leq 4.\)
Reveal Answer
Absolute Maximum: \(z=8\)
Absolute Minimim: \(z=-1\)
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.
Maximum and Minimum Value (multivariable) Exercise V5
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.
What is the difference between local extrema and absolute extremam of a surface \(z=f(x,y)\)?
Can a surface \(z=f(x,y)\) have more than one local maximum?
Does a critical point always yield either a local maximum, minimum, or saddle point? How do we classify a critical point as a local maximum, minimum or saddle point?
Under what conditions are we guaranteed a surface will attain both an absolute maximum and an absolute minimum?
Under what conditions does \(z=f(x,y)\) attain an absolute maximum or minimum on the boundary of the closed set \(D\)?
Outline the steps in locating the absolute extrema of \(z=f(x,y)\) on a closed, bounded set \(D\).
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