1. Warmup
describe
the interval(s)
on which the
function is
continuous
2) which of the following points of discontinuity of
are not removable?
0 c) -2 e) -5
6 d) 4

2. 2.4 Rates of Change and Tangent Lines

3. The slope of a line is given by:
The slope at (1,1) can be approximated by the slope of the secant through (4,16).
We could get a better approximation if we move the point closer to (1,1). ie: (3,9)
Even better would be the point (2,4).

4. The slope of a line is given by:
If we got really close to (1,1), say (1.1,1.21), the approximation would get better still
How far can we go?

5. slope
slope at
The slope of the curve at the point is:

6. The slope of the curve at the point is:
is called the difference quotient of f at a.
If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.
Sometimes, you will see the problem already in the difference
quotient, and have to figure out the limit.

7. The slope of a curve at a point is the same as the slope of the tangent line at that point.
In the previous example, the tangent line could be found using
If you want the normal line (perpendicular line), use
the negative reciprocal of the slope. (in this case, )

8. Example 4:
Let b
Where is the slope ?

9. Review: velocity = slope
These are often mixed up by Calculus students!
average slope:
slope at a point:
average velocity:
(slope)
So are these!
instantaneous velocity:
(slope at 1 point)
If is the position function:
velocity = slope

10. (slope of the tangent line to graph
at the point

14. Suggested Review Problems for Ch.2 from the book:
The End
Day 1:
p. 87 (1-5 odd, 7, 9-23 odd)
Day 2:
p. 87 (2-6 even, even, 29, 30)
Suggested Review Problems for Ch.2 from the book:
p. 91 (6, 7, 9, 21-32, 39-43, 52)