- Written problems returned
- Watch units
- Explain! Don't just write down an answer....
- What do things mean? Seriously! If the problem's about flu, talk about flu....
- Exams have been tentatively scheduled as follows:
- Monday, 2/15
- Monday, 3/22
- Monday, 4/19
- New assignment:
Mon 1/25 Section 2.3/2.4 - Read 2.5.
- Begin problem set 2.4 on IMathAS (due Monday, 2/1, at midnight).
- By hand, for Tuesday, 2/2: pp. 73, #77, 78, 79
- Sum law
- We'll actually prove this one.
- Constant multiple law
- Product law
- Now that we've got the sum and product and constant multiple rules, what can we deduce?
- Affine rule, for one.
- Then we can deduce that all polynomials have limits, and furthermore that
- Quotient law
- Examples:
- p. 62, #8
- p. 62, #10
- p. 63, #23
- p. 63, #31
- Definition: continuity at a point
Otherwise
is discontinuous at
There are various kinds of discontinuity:
- Removable discontinuities
- Jump discontinuities
- Infinite discontinuities
- Maybe a better definition: a function is continuous on a domain if its graph can be traced there without lifting the pencil from the paper.
- Most of the functions we've looked at so far satisfy this, although not all: recall the function of 2.2, #2:
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 limits
- we have sum laws, product, and quotient laws.
- Classes of functions that are continuous on their domains:
- Polynomials
- Rational functions: (ratios of polynomials)
- Trigonometric functions: sine, cosine, tangent
- Root functions:
- Exponential functions:
- For all these functions one can evaluate limits using the
so-called "substitution method": simply plug in
for
.
- An important theorem: Theorem 4, p. 68, RE composite functions
- Roughly: A composition of continuous functions is continuous.
- Let
be a composite function. If
is continous at
and
, then
is continuous at