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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
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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 1.1 – Some Basic Mathematical Models; Direction Fields
Section 1.1 – Some Basic Mathematical Models; Direction Fields
Directions.
The following are review problems for the section. 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.
Given the differential equation \(\dfrac{dy}{dt}=ty-1.\)
What is the slope of the graph of the solutions at \((0,1)\), at the point \((1,1)\), at the point \((3,-1)\), at the point \((0,0)\)?
Find all the points where the tangents to the solution curves are horizontal.
Describe the nature of the critical points.
Answer
Slopes:
\((t,y)=(0,1),\ m_1=-1\)
\((t,y)=(1,1),\ m_2=0\)
\((t,y)=(3,-1),\ m_3=-4\)
\((t,y)=(0,0),\ m_4=-1\)
\(\left\{\left(t,\dfrac{1}{t}\right)\ |\ t\neq 0\right\}\)
At the critical points, \(y''=y\). If \(y>0\) then \(y''=y>0\) so we have a local minimum at critical points. If \(y<0\) then \(y''=y<0\) so we have a local maximum at critical points.
Video
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MATH 308 WIR22A V1