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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
Math for 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
Python Instructional Video Series
Double Integrals
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Practice Problems for Module 2
Practice Problems for Module 2
Section 6.2
Directions.
The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
The base of a solid is the region enclosed by \(x = y^2 − 9\) and \(x = 7\). Its cross-sections are perpendicular to the \(x\)-axis and are equilateral triangles. Set up an integral to find the volume of the solid.
Answer
An integral giving the volume is \( \displaystyle \int_{-9}^7 \! \sqrt{3}(x+9) \, dx\).
Video
To see the full video page and find related videos, click the following link.
WIR 20B M152 V11
The base of a solid is the region enclosed by \(y = \sqrt{x + 5}\), \(x = 4\) and the \(x\)-axis. Its cross-sections are perpendicular to the \(y\)-axis and are semicircles. Set up an integral to find the volume of the solid.
Answer
An integral giving the volume is \( \displaystyle \int_{0}^3 \! \frac{1}{8} \pi \left( 9 - y^ 2\right)^2 \, dy\).
Video
To see the full video page and find related videos, click the following link.
WIR 20B M152 V12
Set up an integral representing the volume of the solid obtained by rotating the region bounded by the following curves about the \(x\)-axis: \(y = 2/x\), the \(x\)-axis, \(x = 1\) and \(x = 4\).
Answer
An integral giving the volume is \( \displaystyle \int_{1}^4 \! \dfrac{4\pi}{x^2} \, dx\).
Video
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WIR 20B M152 V13
Set up an integral representing the volume of the solid obtained by rotating the region bounded by the following curves about the line \(x=7\): \(x=y^2 +3\) and \(x=7\).
Answer
An integral giving the volume is \( \displaystyle \int_{-2}^2 \! \pi \left(4-y^2\right)^2\, dy\).
Video
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WIR 20B M152 V14
Set up an integral representing the volume of the solid obtained by rotating the region bounded by the following curves about the \(y\)-axis: \(y = \ln(x)\), the \(x\)-axis and \(x = e\).
Answer
An integral giving the volume is \( \displaystyle \int_{0}^1 \! \pi \left(e^2-e^{2y}\right)\, dy\).
Video
Video Errata:
The presenter repeatedly says disk, but should have said washer.
To see the full video page and find related videos, click the following link.
WIR 20B M152 V15
Set up an integral representing the volume of the solid obtained by rotating the region bounded by the following curves about the line \(y=5\): \(y = \sqrt{x+1}\), the \(x\)-axis and \(x = 8\).
Answer
An integral giving the volume is \( \displaystyle \int_{-1}^8 \! \pi \left[25-\left(5-\sqrt{x+1}\right)^2\right]\, dx\).
Video
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WIR 20B M152 V16