Day 35, Mat128: Calculus A

• You have an assignment due Wednesday, over 2.9 and 3.1.
• The quizzes are returned:
  1. No global max; local min at $x=2$.
  2. If $f$ is continuous, then it will achieve its global max and min on a closed and bounded interval $[a,b]$.

     The "continuity" is not about the domain -- which is a closed and bounded interval, $[a,b]$.

  3. That local extrema correspond to critical points, where $f'(c)=0$ or $f'(c)$ DNE (does not exist).

Just a reminder from the quiz, however: since many of you missed these:

• Extreme Value Theorem: If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains a global maximum and global minimum on $[a,b]$.

  Notice that the interval is finite; notice that the extrema promised are global.

• Fermat's Theorem: If $f$ has a local max or min at $c$ and $f^\prime(c)$ exists, then $f^\prime(c)=0$.

• #8, p. 204
• #22
• #27
• #31 (and let's decide whether they're local maxes or mins)
• #45
• #63 (an important story problem)

• We've already seen that derivatives are related to slopes. If the slope is positive (i.e. the derivative is positive) at a point, then the function's graph is increasing there. Similarly, if the derivative is negative at a point, then the function is decreasing there.

  This is the relationship that we explore in this section.

• Similarly the second derivative has much to tell about the shape of a graph.

• Let's work through an example and see how the derivatives are related to the function itself. We'll need a smooth curve....

  Let's look at the graph of #5, p. 220. This is a graph of the derivative:

  

  Questions:

  1. On what intervals is $f$ increasing or decreasing?
  2. At what values of $x$ does $f$ have a local max or min?
  You might notice that the derivative looks quadratic, and hence think to yourself that the function itself probably looks cubic....

• As we work through the example above, we should see how the first and second derivatives are related to the function itself. So doing, we discover the first and second derivative tests:
  • 1st: When a function is differentiable at a point x=a and has zero derivative (tangent line slope) there, then if the first derivative changes sign from
    • positive to negative, there's a max at x=a;
    • negative to positive, there's a min at x=a;
    • no change in sign, there's neither max nor min at x=a.

  • 2nd: When a function is differentiable at a point x=a and has zero derivative (tangent line slope) there, then if the second derivative is
    • positive at x=a, there's a min at x=a;
    • negative at x=a, there's a max at x=a;
    • zero, we're not sure what's going on....
      • 
      • 

• Inflection points:
  1. A graph is concave up if it's "bowlish"; concave down if it's "umbrellaish" (see Figures 5, 6, p. 216)
     

  2. inflection point: Point P on a curve is an inflection point if the curve changes concavity at P.
  3. ``There is a point of inflection at any point where the second derivative changes sign.'' (p. 218)
     

     You might notice that the function itself looks cubic, and hence think to yourself that the derivative probably looks quadratic....

• Examples:
  • #6, p. 220
    

  • #27
    

  • #23
  • #39
  • #50
  • #51