Calculus 3 5

							       3.5 Concavity and Inflection
                Points
              Tues Nov 29
                         Do Now
      Find the 2nd derivative of each function
     1) f (x)  3x  2x  4
                  2




     2) f (x)  x 4  3x 3  5x  2




HW Review: worksheet p.287
   #19-24, 27-28, 31-38
     Applications of the 2nd
           derivative
• So far, we’ve only talked about one
  application of the 2nd derivative, which
  is the acceleration function
• The second derivative can also be used
  to describe the behavior of functions as
  well.
    Concavity and Inflections
• The 1st derivative is used to describe
  slope. But since it is also a function, it
  also has its own “slope” or derivative.
• The 2nd derivative can be used to
  model the behavior of the slope, as it is
  ALSO changing with the function
  – Some slopes can be steep, while others
    rather flat
              Concavity
• The 2nd derivative can be used to
  describe concavity
• Concavity is the rate at which the slope
  increases or decreases
• There are two types of concavity
  – Concave up (looks like a smile)
  – Concave down (looks like a frown)
        Concavity and f’’(x)
• Thm- Suppose f(x) is differentiable on
  an interval I and f ’’(x) exists,
  – If f ’’(x) > 0, then the graph is concave up
  – If f ’’(x) < 0, then the graph is concave
    down
  Note: Second derivative only
          Inflection Points
• An inflection point is a point on the
  graph where a graph alternates
  between concave up and concave down
• Think of the inflection point as the 2nd
  derivative’s critical point
• We will find all of our inflection points
  when f ’’(x) = 0
                Example 5.1
     • Determine where the graph is concave
       up and concave down
            f (x)  2x 3  9x 2  24x 10




           Example 5.2
• Determine where the graph is concave
  up and down, and find any inflection
  points     f (x)  x 4  6x 2 1



    
                Ex 5.3
• Determine the concavity and inflection
  points of     f (x)  x 4



      
                Closure
• Journal Entry: What can the 2nd
  derivative tell us about the behavior of
  the function?

• HW: p.284 #9-14 (Don’t sketch the
  graphs)
          3-5 2nd Derivative Test
               Wed Nov 30
     • Do Now
     • Find the intervals of concavity and
       inflection points of the following function

               f (x)  x  4 x 1
                          4       3





   HW Review p.284 #9-14
• book
        2nd Derivative Test
• The 2nd Derivative can also be used to
  determine if a critical point is a local
  max or min.
• Thm- Suppose that f(x) is continuous on
  an interval (a,b) and f’( c) = 0, then
  – If f’’( c) < 0, then c is a local max
     • Concave down means a local max!
  – If f’’( c) > 0, then c is a local min
     • Concave up means a local min!
              Practice
• Bookwork p.285 #15, 16, 19-28
                    Closure:
     • Hand in: Find intervals of increase,
       decrease, and concavity, local extrema,
       and inflection points of
              f (x)  x  4 x 1
                        4       2


     • HW: p.285 #15, 16, 19-28



       Sketching Curves
         Thurs Dec 01
• Do Now