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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. 
  1. Given the differential equation \(\dfrac{dy}{dt}=ty-1.\)
    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 \((0,0)\)?
    2. Find all the points where the tangents to the solution curves are horizontal.
    3. Describe the nature of the critical points.​

      1. Slopes:
        • \((t,y)=(0,1),\ m_1=-1\)
        • \((t,y)=(1,1),\ m_2=0\)
        • \((t,y)=(3,-1),\ m_3=-4\)
        • \((t,y)=(0,0),\ m_4=-1\)
      2. \(\left\{\left(t,\dfrac{1}{t}\right)\ |\ t\neq 0\right\}\)
      3. 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.

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      MATH 308 WIR22A V1