Day 12, Mat128: Calculus A

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Today:

Announcements
- Your exams are returned:
  - Curve: Add 6 to your score (which is a percentage).
  - There's a key on-line, made up of your peers' work.
  - The Exam
  - See me if you need to discuss your exam.
  - In particular, algebra proved a problem, as expected. Some of you have some errors that must be expunged! If you have made a major algebra blunder, see me to get squared away.
    I have found that some of us have internalized truly horrible algebra rules -- and we have to drive them out like demons! They are difficult to let go of, and that's what makes them so horrible and terrible.
  - My biggest disappointment was that many couldn't (or didn't) draw the secant and tangent lines in #5.
    And of course there were those who didn't write the equation of the secant line in point-slope form.
  - My biggest surprise was the number who didn't get the domain right on the very first problem.
- You have a new assignment, due next Monday.
Today: we begin (and maybe even finish) Section 1.8
- Section 1.8: Continuity
  - What is the difference between continuous and discontinuous things?
    - A new administration in government?
    - A sidewalk?
    - Coffee poured into a cup?
  - Definition: continuity at the point $x=a$: \[ \lim_{x\rightarrow{a}}f(x)=f(a) \]
    Note: three things have to happen:
    1. The limit of $f$ exists at $a$,
    2. The function value $f(a)$ exists at $a$, and
    3. The two values are equal.
    Otherwise $f$ is discontinuous at $a$.
    There are various kinds of discontinuity (which we've already seen):
    - Removable discontinuities (e.g. holes)
    - Jump discontinuities
    - Infinite discontinuities
    - Crazy discontinuities like the function we discussed last time, which was $x^2$ for rational values of $x$, and 0 for irrational ones.
  - Maybe a better definition:
    
    A function is continuous on an interval if its graph can be traced there without lifting the pencil from the paper. (Try one! Make one smooth, and one really "jaggy".)
  - Many of the functions we've considered so far satisfy this, although not all do: consider the "holy function" ${f(x)=\frac{\cos(x)-1}{x^2}}$
    
    This function has a limit at zero (-.5), but is not defined there. If $f$ is not defined at $x=0$, then it cannot be continuous there. We can fix this, by the way.... Just define $f(0)$ to be the limit (the closed dot in the graph above).
  - Since continuity is wrapped up with limits, we have the same kinds of results that we did in relation to limits:
    - We define one-sided continuity
    - we have sum laws, product, and quotient laws. We've already used these to demonstrate that polynomials are continuous on all real numbers:
      ${\lim_{x\to{a}}p(x)=p(a)}$
  - Classes of functions that are continuous on their domains:
    - Polynomials
    - Rational functions: (ratios of polynomials)
    - Trigonometric functions: sine, cosine, tangent
    - Root functions: $f(x)=x^{1/n}$
    - Exponential functions: $f(x)=b^x$
  - For all these functions one can evaluate limits using the so-called "substitution method": simply plug in $a$ for $x$.
  - An important theorem (p. 88), RE composite functions (remember that I told you how important compositions are? here's one place they show up....):
    - Roughly: A composition of continuous functions is continuous.
    - Let $F(x)=f(g(x))$ be a composite function.
      If $g$ is continous at $x=c$ and $f$ is continous at $g(c)$ then
      $F(x)$ is continuous at $x=c$.
  - Examples:
    - pp. 90-93, #3, 10, 12, 19, 23, 36, 46 (with Mathematica)
    - That crazy function which was $x^2$ for rational values of $x$, and 0 for irrational ones, has an interesting property: it is continuous at only a single point! It satisfies the definition only at $x=0$: it's discontinuous everywhere else! That's pretty weird.
  - The Intermediate Value Theorem is an important result of continuity:
    - Theorem: Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where $f(a)\ne{f(b)}$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.
    - Example: pp. 93, #69

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