- So: what are
- Limits,
- Rates of Change, and
- Tangent lines,
- We start with velocity:
- How is it defined in terms of position and time?
- What are its units?
- Consider first uniform motion: each time interval T gives
the same change in position:
- What happens as motion is no longer uniform?
- Instantaneous ROC -- how do we get at it?
- By successive approximations: A sequence of secant lines converges on a tangent line (applet)
-
Here's an applet where you can choose the function, window,
etc.
Let's use it for exercise #11, p. 47.
- Questions regarding Homework/directed reading handout? Section 2.1 (html) (pdf)
- How do the values of a function f(x) behave when x approaches a
number c, whether or not f(c) is defined?
- Definition of a Limit
- Important idea: as we approach c along the x-axis, what are the y values doing?
- Theorem 1, p. 52 (demonstration)
- Important Cases:
- Limit exists and equals function value
Example: All polynomials, at every value of c!
- Limit exists and does not equal function value
- where function value exists
- where function value does not exist
- Limit does not exist
- one-sided limits exist and are unequal
- Infinite limits
- Limit exists and equals function value
- Questions regarding Homework/directed reading handout? Section 2.2 (html) (pdf)
