Questions over the Directed reading handout, Section 2.3 (html) (pdf)?
- Continuity:
- The limit at x=c exists and is equal to f(c)
- When a function is continuous at for all values in its domain, we say simply that it is a continuous function.
- Discontinuity: three main types:
- jump
- removable
- infinite
We define one-sided continuity, which helps resolve some of these discontinuities. Where does the function fail to be continuous, but demonstrate continuity from the left or the right?
- Theorems:
- Theorem 1: concerns continuity at x=c, for
sums, differences, products, and quotients:
If two functions are continuous at x=c, then the NOSE of the two functions is continuous there.
(replace the word NOSE in the statement above with any of the other vocab words).
- Theorem 2: All polynomials and rational functions are continuous on their domains.
- Theorem 3: sin, cos, exponential functions, and power functions are continuous on their domains.
- Theorem 4: compositions: if
g is continuous at x
and
f is continuous at g(x)
then
f(g(x)) is continuous at x.
- Theorem 1: concerns continuity at x=c, for
- Notes:
- The usual problem with the quotient, just to make sure that the limit of the denominator function is not zero....
- To evaluate limits of continuous functions is easy: simply substitute the value of c for x in the formula for f(x) (called the substitution method)
- Examples:
- #14, p. 71
- #20, p. 71
- #77, p. 73
- I learned something about our atmosphere:
- Figure 13
- The Thermosphere is where the shuttle flies....
- Temperature over the Antarctic, summer and winter
- #56-70 even, p. 73
- Questions over the Directed reading handout, Section 2.4 (html) (pdf)?
