- I hope that you had a particularly large infinity of fun over Labor Day weekend!
- Reminder: turn off your rectangles.
- You have a homework due.
- Your quizzes are returned.
- Disaster struck... you don't know the most important
definition in calculus!
Some of you suspected that I was asking the definition of a function again, which is one reason why I started it for you: $f'(x)=$....
I also gave you the big hint that it involved limits. It's why we spend so much time on limits.
\[ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \]
A Haiku, by A. Long.
Derivative Function limit -- as $h$ goes
to zero -- change in function
over step-size $h$.
- Quotient! Division!
- Use words, too. Just because I say "diagram" doesn't mean that you can't provide a caption. Also label your graphs.
- Squeezing can be useful as we saw last time. Label graphs, and don't let your "functions" fail the vertical line test (because then they're only "functions", and not functions).
- Disaster struck... you don't know the most important
definition in calculus!
- Section 1.8: Continuity
- What is the difference between continuous and discontinuous things?
- A new administration?
- 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:
- The limit of $f$ exists at $a$,
- The function value $f(a)$ exists at $a$, and
- 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"
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:
- 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
for
.
- 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
.
- Roughly: A composition of continuous functions is continuous.
- Examples: pp. 90-93, #3, 10, 12, 19, 23, 36, 46 (with Mathematica)
- 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
- 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 $f(c)=N$.
- What is the difference between continuous and discontinuous things?