- Sorry that we missed out on the computer lab on Friday. Better
safe than sorry?
I was going to have you do two things, both of which we'll hit briefly today:
- listen to some sound files with Mathematica, and
- check some limits and limit law calculations with Mathematica.
- Has anyone used the calculus lab (math lab) yet?
- I'm returning your homework problems -- let's take a look with Mathematica -- pp. 61--, #38, 44, 46.
- I'm also returning our first quiz today. You may recall that the
most important definition in calculus was actually on the front wall
during the quiz, but in a slightly different form:
$f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}$ Biggest problem on that problem:
Some of you forgot the limit! Remember, this definition is the reason that we start out where we do -- with limits.
Once we're done with continuity, we'll move along to chapter 2 and the definition of the derivative -- only you've already seen it!:)
- We start with two especially trivial special limits:
- Constant rule $(f(x)=c)$:
Here the function $f(x)=1$ is plotted. As $x \to 0$, the $y$-values head to 1. Well, more to the point, they never vary from 1! Quite a boring function....
- Identity rule $(f(x)=x)$:
Here the function $f(x)=x$ is plotted. As $x \to 0$, the $y$-values head to 0. And the closer $x$ gets to 0, the closer $y$ gets to 0. That's the way it's supposed to work! Also a rather boring function: so predictable!
What can we do with those? We need more horsepower....
- Constant rule $(f(x)=c)$:
- Sum law: "the limit of a sum is the sum of the limits"
So what is this limit:
This sum (and many other operations) satisfy the same pattern:
"The limit of a STUFF is the STUFF of the limits." So, in particular,
"The limit of a sum is the sum of the limits." - Difference law
- Product law
- Constant multiple law
- Quotient law (slight caveat here -- no zero denominators)
- Power law
- Root law -- again, a caveat: why can't we compute
\[ \lim_{x\to{0}}\sqrt{x} \]
Because the function isn't even defined as x approaches 0 from the left (square roots of negative numbers are imaginary). So we have only a one-sided limit, that \[ \lim_{x\to{0^+}}\sqrt{x}=0 \] Of course we can say that \[ \lim_{x\to{1}}\sqrt{x}=1 \] Limits exists everywhere else on the domain of the sqrt function. - Putting these laws together
- Now that we've got the sum and product and constant multiple rules, what can we deduce?
- "Affine" rule, for one:
\[ \lim_{x\to{a}}(mx+b)=ma+b \] - Then we can proceed to deduce that all polynomials have limits, and
furthermore that
This is the conclusion of Exercise #55, p. 71, which says that polynomials are continuous, which we'll consider more in section 1.8.
Exercise #56 says it's also true for rational functions, for every element of their domain.
- Take a look at a couple of the examples in the text, examples 5
and 6 on page 66. These both illustrate the limiting
process applied to the most
important definition in calculus:
$f'(x)=\lim_{h\to 0}{\frac{f(x+h)-f(x)}{h}}$ In this case, both show examples of a slightly different version of this most important definition, the derivative at a point $x=a$:
$f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}$ - Other Laws
- The squeeze theorem
What's happening to the $y$-values?
- The squeeze theorem
- Now let's take a look at some of the examples from the section
exercises.
- Example 3, p. 65, illustrates the important point in the box, same page:
If $f(x)=g(x)$ when $x\ne a$, the $\lim_{x\to a}f(x)=\lim_{x\to a}g(x)$ provided either limit exists. So a common strategy is to use algebra to rewrite an expression so that we can actually use our limit laws to evaluate an essentially equivalent expression.
- #7, p. 69
- #10
- #25
- #33
- #47
- Example 3, p. 65, illustrates the important point in the box, same page:
- Additional examples:
- #63, p. 71
- #59, p. 71 (squeeze it!)
- What is the difference between continuous and discontinuous things?
- Missing days in calculus class?
- A sidewalk?
- Coffee poured into a cup?
- Global Temperature?
- How does a function sound?
- Definition: continuity at a point
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.
is discontinuous at
There are various kinds of discontinuity (which we've already seen):
- Removable discontinuities (e.g. holes)
- Jump discontinuities
- Infinite discontinuities
- Maybe a better intuitive definition: a function is continuous
on an interval if its graph can be traced on
that interval without lifting the pencil from the paper.
- 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.
- 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 interesting new function (the greatest integer function)
has an infinite number of discontinuities on its domain (Example 10,
p. 68). For you computer science types, this function is often called
the floor function -- can you see why? How would the
ceiling function differ?
(This picture is a little deceiving: let's have a look with Mathematica.)
- 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 continuous at $x=c$ and $f$ is continuous 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 $f(c)=N$.