Day 38, Mat128: Calculus A

• Your homework that was originally due for today was moved to Wednesday.

• Quiz is returned:
  1. EVT
  2. that $c$ is critical.
  3. Fermat!

• Let's talk about the future for a bit:
  • We will have our third exam next Monday.
  • There will be no quiz on Friday, since there's an exam on Monday before Thanksgiving.
  • There are two more "topics" we need to cover, then we'll finish off with story problems (optimization):
    • Section 3.4 -- Limits at infinity (asymptotic behavior)
    • Section 3.9 -- Anti-derivatives
    • Section 3.7 -- Optimization problems
    You'll just have two more problem sets, which will be due Monday, 11/30 (3.3) and Wednesday, 12/2 (3.4 and 3.9).

    We'll do story problems the last week of classes, with a short review on Friday.

  • Our final is Wednesday, 12/9, at 1:00. I often do a review the evening before the final, if students are interested. That would be Tuesday, say from 7:00 to 9:00. Are folks interested in that?

    I'm also available all finals week for questions. Make an appointment if interested.

• #67, p. 206 (see p. 1083, for some context)

• 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.

     Concavity test:

     • If $f^{\prime\prime}>0$ for all $x$ in interval $I$, then $f$ is concave up on $I$;
     • if $f^{\prime\prime}<0$ for all $x$ in interval $I$, then $f$ is concave down on $I$.

  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:
  • #39: whoops! My apologies: I said that $1+\cos{x}=0$ at $x=\frac{3\pi}{2}$ -- that's crazy talk! Let's fix that, and finish properly....
  • #50
  • #51

We've been working on closed, bounded intervals; now let's talk about what happens when we let $x$ become unbounded. How will a function behave as $x$ races off to $\infty$ or $-\infty$?

• Examples of functions without finite limits at infinity:
  • non-constant polynomials (go to infinity, in the positive or negative sense)
  • the usual trig functions (while sine and cosine are bounded, but don't settle down to some nice limiting value)

• But rational functions may have horizontal asymptotes, as do exponential functions.

  Another example with a horizontal asymptote is knowledge as a function of time -- #51, p. 222. We might guess that accumulated knowledge in studying for an exam looks something like this:

  

  We might imagine that this physical process becomes less productive from hour to hour as the evening wears on (the law of diminishing returns).

  Other Examples:

  • The hyperbola: $f(x)=\frac{1}{x}$

    More generally, If $r>0$ is a rational number, then

    $\lim_{x \to \infty} \frac{1}{x^r}=0$ If, furthermore, $x^r$ is defined for negative numbers, then $\lim_{x \to -\infty} \frac{1}{x^r}=0$

  • The most general form of an hyperbola is this: $f(x)=\frac{ax+b}{cx+d}$
    • What is its horizontal asymptote?
    • Where is its vertical asymptote?

  • A rational function will have a horizontal asymptote any time that the degree of the denominator equals or exceeds that of the numerator: if
    $r(x)=\frac{p(x)}{q(x)}$

    then if the degree of q exceeds that of p, there is a horizontal asymptote, and the value of the asymptote is given by examining the approximating function given by the ratio of leading terms alone.

  • The usual rules of limits apply, with the caveat that one must be careful when dealing with $\infty$. It is not a number, and cannot be added to anything else (e.g. it is not valid to write that $\infty=0$).

• Examples:
  • #7, 8, p. 234
  • #44, p. 235
  • #49