Quote of the Day:
   "One should always generalize."  -- Carl Jacobi

Objectives:
The student will sketch a slope field for a given 
   differential equation. 

Given a slope field, the student will sketch a solution            
   curve through a given point. 

The student will be able to match a slope field to a 
   differential equation.

The student will be able to match a slope field to a 
   solution of a differential equation.


1. Collect Homework.

2. Today we are going to study a concept called slope 
   fields.
  
   Slope fields provide an excellent way to visualize a 
   family of solutions of differential equations. 
  
   Some differential equations can be solved algebraically, 
   and the slope field for those differential equations can 
   be used to verify that the algebraic solution matches 
   the graphical solution. 
  
   Other differential equations cannot be solved 
   algebraically, and the slope field for those 
   differential equations provides a way to solve the 
   differential equation graphically. 

   Slope fields also give us a great way to visualize a 
   family of antiderivatives.

3. Recall how to graph a line given a point and the slope.
   Graph a line passing through (-2, 3)  with a slope of 2.

4. Hand out rulers and slope field packets.

   Start with a differential equation such as 
   dy/dx = x + 1. 

   Pick a starting point on our grid and substitute it into 
   the differential equation to determine the slope at that 
   point. 
   Then we draw a tiny segment that passes through our 
   point and has the slope that we found. 

   Next ask the students to name other points that have the 
   same slope. They notice that all of the points that have 
   the same x-coordinate will have the same slope because 
   our differential equation has an x-term but no y-term. 

   After completing the slope field for dy/dx = x + 1,  try 
   another differential equation, such as dy/dx = 2y. 
   This time the students notice that all of the points 
   that have the same y-coordinate will have the same slope 
   because our differential equation contains a y-term but 
   no x-term. 

   This knowledge helps the students when they are asked to 
   match a differential equation to a slope field. 
   The student looks at the slope field to see if all of 
   the segments in the vertical direction have the same 
   slope; if they do, then the differential equation 
   contains an x-term but no y-term. 
   
   If all of the segments in the horizontal direction on a 
   slope field have the same slope, then the differential 
   equation contains a y-term but no x-term. 
   After making these observations, move on to differential 
   equations that contain both an x-term and a y-term, such 
   as dy/dx = x +y and look for points that have the same 
   slope as we draw the slope field for this differential 
   equation. 

   The students like to use a ruler at first to help draw 
   their segments so that they have the correct slope, but 
   soon they are able to draw them without using a ruler.


5. Let students work on the slope field packets.

6. Assignment

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