Section 1.1 – Some Basic Mathematical Models; Direction Fields
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. 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.
- 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.