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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
Quantitative Finance
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 2: Limits and Continuity
Section 2: Limits and Continuity
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 on
Section 3
or return to the
main several variable calculus page
.
Concepts
Calculating the limit of a surface
The definition of the limit of a two-variable function
Limits at infinity and infinite limits of two-variable functions
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video 1
Watch Concepts Video 2
Watch Concepts Video 3
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find the limit. If the limit does not exist, support your answer.
\(\displaystyle \lim_{(x,y)\rightarrow (1,2)} \ln \left(1+x^2y^2\right)\)
\(\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{x^3+xy^2}{x^2+y^2}\)
Reveal Answer
\(\displaystyle \lim_{(x,y)\rightarrow (1,2)} \ln \left(1+x^2y^2\right)=\ln 5\)
\(\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{x^3+xy^2}{x^2+y^2}=0\)
Watch Video
Find the limit. If the limit does not exist, support your answer.\[ \lim_{(x,y)\rightarrow (0,0)} \frac{\sqrt{x^2+y^2+1}-1}{x^2+y^2}\]
Reveal Answer
\(\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{\sqrt{x^2+y^2+1}-1}{x^2+y^2}=\frac{1}{2}\)
Watch Video
Find the limit. If the limit does not exist, support your answer. \[\lim_{(x,y)\rightarrow (0,0)}\frac{3xy}{x^2+y^2}\]
Reveal Answer
The limit does not exist since the limit along \(y=0\) does not equal the limit along the path \(y=x\).
Watch Video
Find the limit. If the limit does not exist, support your answer. \[\lim_{(x,y)\rightarrow (0,0)} \frac{x^2-y^2}{x^2+y^2}\]
Reveal Answer
The limit does not exist since the limit along the \(x\)-axis does not equal the limit along the \(y\)-axis.
Watch Video
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.
When we want to study the \(z\) values of a surface \(z=f(x,y)\) near a point \((a,b)\) in the \(xy\)-plane using a table of values, how can we determine whether \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) exists? If the limit does exist, how can we estimate the value of the limit?
Does \(f(a,b)\) have to exist in order for \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) to exist?
How can we prove \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) does not exist?
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