Quote of the Day:
"I had been to school ... and could say the multiplication table 
      up to 6 x 7 = 35, and I don't reckon I could ever get any further 
      than that if I was to live forever.  I don't take no stock in 
      mathematics anyway." -- Huckleberry Finn (by Mark Twain)

Objectives:
The student will test for relative extrema using the first derivative.
The student will test for relative extrema using the second derivative.
The student will sketch curves from given information and without using a calculator.

1. Collect homework.

2. Definitions:
   Critical points occur when f'(x) = 0 or when f(x) is not 
      differentiable. (Slopes are either zero or infinite – 
      e.g., parabolas or cusps). 
   Stationary points are those where f'(x) = 0.
 
   The relative extrema of a function (if any) occur at 
critical points. But this does not mean that relative 
extrema occur at every critical point (they could be 
points of inflection, for example).

   Relative maximums and minimums may occur when the first 
derivative is not zero (like a cusp). 

3. To Find relative max/min:
     (A) Take the first derivative and set equal to zero.
         Solve for x.  These are critical points.

     (B) 1st derivative test:
           Test points on each side of the critical points
            found in (A) by substituting in the first 
            derivative.  

           If the value of the derivative of the point to          
            the left of the critical point is positive and 
            the value of the derivative for the point to 
            the right is negative, then the critical point 
            is a relative maximum.

           If the value of the derivative of the point to          
            the left of the critical point is negative and 
            the value of the derivative for the point to
            the right is positive, then the critical point 
            is a relative minimum.

           If the values of the derivative of the points to 
            the left and the right of the critical point 
            are the same (i.e., both positive or both 
            negative), then the critical point is a point 
            of inflection.

     (C) 2nd derivative test:
           Take the first derivative and set it equal to 
            zero to solve for critical points.
           
           Take the second derivative of the function.

           Substitute the critical point in the second 
            derivative.
              If this value is negative, the critical point 
               is a relative maximum.
              If this value is positive, the critical point 
               is a relative minimum.
              If this value is zero, the critical point 
               is a possible point of inflection.  Test 
               points on either side of the critical point 
               by substituting them into the second 
               derivative to verify that the concavity 
               changed.

4. Example:

5. Look at the Top Ten List of Pick Up Lines for Math 
    Chicks (in math humor) – 

   #4. My love for you is like the slope of a concave up 
       function because it is always increasing.

6. Distribute a copy of the Curve Sketching Techniques and
     the Curve Sketching Worksheet.

7. Assignment:
    p. 304 (7, 24, 29)

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