Thursday,
April 13, 2006
More on Infinity, Rates,
and Slopes
Recall from last class:
Arithmetic
growth: 2, 7, 12, 17, 22, ...
Examples?
Formula: A (x) = 5x + 2; Y(x) = mx + b, m>0
Geometric growth: 2, 10, 50 , 250, 1250, 6250,
Examples?
Formula: P(n) = 2*5n ; P(x) = A rx , r>1.
For each type of growth, when x >
> 0.
Infinite
curves: The snowflake.
Activity on snowflakes.
WinFeed (old version)
Space
filling curves.
Other examples using approximations- measurements of curves and surfaces:
Archimedes: The area of the circle- the area of a triangle.
Kepler: The volume of a torus- the volume of a cylinder.
Growth Rates: m - linear, r -
exponential.
Graphing and rates: slopes of lines, approximating curves with
lines.
slope of straight line:
m = (change in y)/(change in x)
Motion: time: t ; position:s
constant and average velocity:
v = (change in s)/(change in t)
Questions: What if line is curved?
"zooming"
What if motion does not proceed at constant velocity?
Example: What line does the graph
of y = x2 look like near (3,9).
Two related measurement concepts:
- Instantaneous
Rate of growth or motion.
- Slope
(gradient) of a curved line, tangent line to a curve.
