Day 21, Mat129: Calculus I

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Today:

Announcements
- You have a homework due today.
- You have another one to hand in on Friday (section 3.1). We'll go over that one in class, as you need it for the exam.
- Your homework is returned: pp. 154--, graded #17, 24, and 78
- Class this Friday will held in here, rather than in the computer lab. Actually it will be review, because we have an exam on the Monday upon return from spring break.
  Also a reminder that the math/calc lab will be open from 9:00am - 3:00pm that day, for anyone who needs extra help just before the exam.
- The exam will cover up through max/mins, section 3.1.
Last time we began talking about section 3.1: Maximum and Minimum Values.
- Example application of extrema:
  - The Chickens of Manitoba
  - How was the flock egg production over time?
  - How long should we let the flock live to maximize the average daily return?
- Maxima and minima are both types of extrema. Our author begins by distinguishing between global (or absolute) extrema and local extrema:
  - Local is the biggest thing in the neighborhood;
  - Global is the biggest thing anywhere!
- Take a moment and sketch a continuous function on the interval [-2,2]. Make it look kind of interesting -- maybe even ugly.
  1. Does your function have a global maximum and a global minimum?
  2. Does your function have local extrema which are not global extrema?
  3. If your function does have extrema, how would you characterize the derivative of the function at those extrema?
- By the way, our author requires that the function be defined in an open interval about a local extremum. So in this figure from p. 199, the right endpoint is not considered a local maximum:
- Now I want you to sketch a function defined on the interval [-2,2] that does not have any global extrema. What is true about your function?
- Now I want you to sketch a smooth function on [-2,2] that has a value of x=a where $f'(a)=0$ but no extremum there. What is true about your function?
By now we should have a good idea of what these two theorems say:
- Extreme Value Theorem
- Fermat's Theorem
We have hopefully deduced these. We might also have deduced the closed interval method, to find the extrema of a function on a closed interval [a,b]:
1. Find the critical numbers for f (that is, those values of x at which the derivative is zero or doesn't exist).
2. Find the values of f at the critical numbers of f in (a,b).
3. Find the values of f at the endpoints of the interval (at a and b).
4. The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.
Now let's look at some standard problems:
- #31, p. 205 (and let's decide whether they're local maxes or mins)
- #45 (closed interval method)
- #63
Today's new topic: section 3.3: The derivative and the shape of a function
- 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 (at left):
  
  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 on the left looks quadratic, and hence think to yourself that the function $f$ 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....
      - $x^3$
      - $x^4$
  - If you'll just remember the two functions
    - $x^2$
    - $-x^2$
    You'll be able to keep these two rules straight.
    
    $f'(x)$:
    - to 0 to + $f'(x)$:
    + to 0 to -
    
    $f''(0)$: + $f''(0)$: -
- 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

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$f'(x)$: - to 0 to +	$f'(x)$: + to 0 to -
$f''(0)$: +	$f''(0)$: -