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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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Additional Exam 1 Problems
Additional Exam 1 Problems
These are a few additional problems for Exam 1, but please note this review does not cover all sections on your Exam 1.
Directions.
It is recommended 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.
For the initial value problem \((t^2-4)y'+2ty=3t^2, \hskip.2in y(1)=-3\)
Determine an interval in which the solution to the initial value problem is certain to exist.
Solve the initial value problem.
Answer
\(I=(-2,2)\)
\(y=\dfrac{t^2-2t+4}{t-2}, \quad I.V.=(-\infty,2)\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V38
A large tank initially contains 10 L of fresh water. A brine containing 20 g/L of salt flows into the tank at a rate of 3 L/min. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 2 L/min. Determine the concentration of salt in the tank as a function of time.
Answer
\(C(t)=20-20,000(t+10)^{-3}\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V39
Given the differential equation \[\dfrac{dy}{dt}=7y-y^2-10\]
Find the equilibrium solutions.
Sketch the phase line and determine whether the equilibrium solutions are stable, unstable, or semistable.
Sketch the graph of some solutions.
Determine the behavior of \(y(t)\) as \(t\) increases for all possible values of \(y(0) = y_0\).
Do any solutions admit a vertical asymptote?
Solve the equation.
Answer
Equilibrium solutions: \(y=2\), \(y=5\)
\(y=2\) is unstable and \(y=5\) is stable. See the video for the sketch of the phase line.
See the video.
\(\begin{cases} y_0<2 & \Rightarrow & y\to -\infty \quad \text{as }t \text{ increases}\\ y_0=2 & \Rightarrow & y(t)=2 \\ 2< y_0<5 & \Rightarrow & y(t) \to 5\quad \text{as }t \text{ increases} \\ y_0=5 & \Rightarrow & y(t)= 5 \\ y_0>5 & \Rightarrow & y(t)\to 5\quad \text{as }t \text{ increases} \end{cases}\)
Yes. See video for explanation.
\(y=\dfrac{5-2Ce^{-3t}}{1-Ce^{-3t}}\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V40
Find an integrating factor for the equation \[(y^2+xy)+(x^2+3xy)y'=0\] and then solve the equation.
Answer
\(\dfrac{1}{2}x^2y^2+xy^3=C\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V41